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Characterisation of Complex Networks from their Topologies
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{
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A thesis submitted to The University of Manchester for the degree of
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Doctor of Philosophy
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in the Faculty of Science and Engineering
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liningnums{2019}
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LARGE Oliver A Blanthorn
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vspace{4em}
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Large School of Engineering
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Department of Computer Science
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}
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\series bold
Word count: 36,000
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update this before submission!
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\begin_layout Chapter*
Abstract
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Tex note: 
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textin{*} for A, D, F, G, L, N, O, Q, T and (much worse) X.
 produces really beautiful illuminated letters.
 Worth rewording every chapter just for that?
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https://tex.stackexchange.com/questions/228993/remove-space-in-reference-in-et-al#
answer-229135 if 
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et al.
 ,
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 reappears but remember you use authordate4
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Complex networks appear everywhere in the world.
 They describe everything from the Internet and transport networks to neuronal
 networks and social networks.
 Characterising these networks—that is, describing them quantitatively—is
 vitally important to understanding them, comparing them and tracking their
 temporal evolution.
 The connection patterns, or topologies, of networks are often very different
 across different networks and sometimes even between different parts of
 a single network, or over time in temporal networks.
 This work focuses particularly on these connection patterns and how they
 can characterise networks at multiple scales: we use the topology to characteri
se nodes, communities and entire networks.
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On the subject of node characterisation, we demonstrate the merits of using
 colour in graphlet profiles when characterising ego-networks, using this
 technique to discover that, in scientific co-authorship networks, the proportio
n of graphlets of a specific kind centred on an author early in their career
 is correlated with that author's final academic career length.
 These graphlets which are correlated with career length are 4-stars; induced
 subgraphs containing four nodes where one node is the author and the other
 three nodes only have a link to the author.
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For whole-network characterisation, we use matrix decomposition of data
 matrices formed from topological properties such as degree-mixing matrices
 and graphlet profiles to clearly distinguish between real networks and
 network models and track changes in networks over time.
 We introduce the term 
\begin_inset Quotes xld
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piecewise static networks
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 to describe the application of standard static network analysis to temporal
 networks.
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Finally, on community characterisation, we introduce a new way of assessing
 the goodness-of-fit for tensor decompositions of temporal networks by applying
 adjusted mutual information and employ this technique to throw more light
 on software dependency networks—where we use it to highlight the uniqueness
 of the R language's software package ecosystem due to R's 
\begin_inset Quotes xld
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rolling-release cycle
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 policy—and semantic networks—where we start to study the evolution of the
 English language over the past thousand years.
 
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\begin_layout Chapter*
Declaration
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No portion of the work referred to in the thesis has been submitted in support
 of an application for another degree or qualification of this or any other
 university or other institute of learning.
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Copyright
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\begin_layout Enumerate
The author of this thesis (including any appendices and/or schedules to
 this thesis) owns certain copyright or related rights in it (the “Copyright”)
 and s/he has given The University of Manchester certain rights to use such
 Copyright, including for administrative purposes.
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\begin_layout Enumerate
Copies of this thesis, either in full or in extracts and whether in hard
 or electronic copy, may be made only in accordance with the Copyright,
 Designs and Patents Act 1988 (as amended) and regulations issued under
 it or, where appropriate, in accordance with licensing agreements which
 the University has from time to time.
 This page must form part of any such copies made.
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\begin_layout Enumerate
The ownership of certain Copyright, patents, designs, trademarks and other
 intellectual property (the “Intellectual Property”) and any reproductions
 of copyright works in the thesis, for example graphs and tables (“Reproductions
”), which may be described in this thesis, may not be owned by the author
 and may be owned by third parties.
 Such Intellectual Property and Reproductions cannot and must not be made
 available for use without the prior written permission of the owner(s)
 of the relevant Intellectual Property and/or Reproductions.
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Further information on the conditions under which disclosure, publication
 and commercialisation of this thesis, the Copyright and any Intellectual
 Property and/or Reproductions described in it may take place is available
 in the University IP Policy (see 
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), in any relevant Thesis restriction declarations deposited in the University
 Library, The University Library’s regulations (see 
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) and in The University’s policy on Presentation of Theses.
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Acknowledgements
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\backslash
epigraph{``Muad'Dib learned rapidly because his first training was in how
 to learn.
 And the first lesson of all was the basic trust that he could learn.
 It's shocking to find how many people do not believe they can learn, and
 how many more believe learning to be difficult.
 Muad'Dib knew that every experience carries its lesson.
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}{--- 
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textup{Frank Herbert,} Dune}
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This work bears only my name but it is the product of everyone who helped
 me along the way to reach this point.
 I shall do a great disservice to most of these people as I only have room
 to thank a few.
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First and foremost, I must acknowledge that I owe a great debt to my supervisor,
 Dr Eva Navarro-López.
 I'm certain this would be a far lesser work without her unwavering support
 and strength.
 I don't know how she does it.
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I would also like to thank my family.
 Mum, thanks for always pushing me to work hard and ensuring that I always
 had the opportunity to have a great education.
 Thanks for looking after the koi pond and the watercress.
 Tara, keep chasing whatever makes you happy, whether that's ecology, field
 sites, or your daughter Eilah! I'm equal parts sorry and glad that the
 experimental work for thesis was a little less involved than your chasing
 primates across the Congo.
 Eilah, I can't wait to see what you do.
 Dad: this thesis is really for you.
 Now that you can't read this I think it is safe for me to admit that, yes,
 being forced to play 
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Mavis Beacon Teaches Typing
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 on MS-DOS before I was allowed a quota of fun video-games was perhaps a
 little useful and that your abject refusal to give me a straight answer
 to any question probably instilled the curiosity (and, if we're honest,
 intransigency) that meant I was able to complete this thesis.
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Finally, Harri: I am perpetually baffled why you put up with this curmudgeon,
 but I am glad that you have.
 Thanks for all of your support, especially in these last few months while
 I produced this thesis and gradually morphed into a troglodyte.
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I'm grateful that this work was supported as part of an Engineering and
 Physical Sciences Research Council (EPSRC
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) grant, project reference 
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liningnums{EP/I028099/1}
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.
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Availability of data and material
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The source for this thesis is available at 
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.
 I believe this is a fairly comprehensive list of all the code used in this
 work.
 If an interested reader feels we have omitted any crucial routines, please
 file an issue on the repository for the thesis and we will endeavour to
 produce it.
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make all links here public / real.
 Consider Zenodo DOIs.
 Sort out wrapping on this page and throughout thesis.
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The programmes described in this chapter are available from:
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,
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,
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.
 
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.
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.
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The data used in this research are available from:
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 for the ecological networks and
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The data used in this research are available from:
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, and our code is
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, and
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.
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: the proprietary data used in this research are available with agreement
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.
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.
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\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
clearpage 
\backslash
pagestyle{fancy}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Newpage newpage
\end_inset


\end_layout

\begin_layout Part
Introduction
\begin_inset CommandInset label
LatexCommand label
name "part:Background"

\end_inset


\end_layout

\begin_layout Chapter
Introduction
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
epigraph{``I've suffered for my music,
\backslash

\backslash
and now it's your turn"}{--- 
\backslash
textup{Neil Innes,} A protest song}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
MyLettrine{N}{etworks are}
\end_layout

\end_inset

 all around us.
 They characterise the Internet, social networks, patterns of brain activity,
 transport systems and financial networks that define our lives.
 Humanity often meddles with these networks, for example in transportation
 networks where we add traffic lights to control traffic, financial networks
 where politicians suggest transaction taxes and social networks where parents
 try to send their children to a good school in the hope that they will
 be influenced in a good way by friends.
 But we only have a shallow understanding of how these networks work.
 Surprising (and controversial!) results are still published in this very
 young field 
\begin_inset CommandInset citation
LatexCommand citep
key "broido2019scale"
literal "false"

\end_inset

.
 It is our hope that the new tools this thesis provides will be used to
 further this understanding.
\end_layout

\begin_layout Standard
This thesis is particularly concerned with ways of characterising networks
 and parts of networks, i.e.
\begin_inset space \space{}
\end_inset

developing ways of comparing and distinguishing networks or parts of networks,
 from looking at their topologies, or wiring patterns.
 This topological approach is attractive as the wiring patterns of real-world
 networks often change at a much slower rate than the dynamics on those
 networks; for example, the amount of road-traffic changes with the time
 of day but new roads take years to build 
\begin_inset CommandInset citation
LatexCommand citep
key "gross_adaptive_2009"
literal "false"

\end_inset

.
 The connections between parts of a system are often fairly easy to observe
 compared to the equations governing the dynamics—it is far easier to observe
 where a road leads than it is to understand exactly what caused a driver
 to pick that road, for example.
 Data on wiring patterns are therefore often more readily available than
 they are for node dynamics.
 We take a multi-scale approach where we consider characterisation of complex
 networks at three scales: at the scale of nodes, communities and whole
 networks.
\end_layout

\begin_layout Standard
\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: mention csete 2004 bow-ties somewhere?
\end_layout

\end_inset

This first part serves as an introduction to the thesis.
 The first chapter presents a brief introduction to network science and
 the characterisation of networks; the next chapter is a more rigorous and
 mathematically-involved primer on the background of this thesis and the
 state-of-the-art of characterisation in network science and related techniques.
\end_layout

\begin_layout Section
A short history of graph theory and network science 
\end_layout

\begin_layout Standard
Network science has a long and winding history.
 We shall summarise some of it here in the hope of explaining more about
 what network science is and why we think they are subjects of such importance.
\end_layout

\begin_layout Subsection
The seven bridges of Königsberg
\end_layout

\begin_layout Standard
Königsberg, now known as Kaliningrad, is a beautiful Baltic city.
 There are two islands in the centre of the city (see figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Kaliningrad"
plural "false"
caps "false"
noprefix "false"

\end_inset

) which historically had seven bridges connecting them.
 The question whether it was possible to walk across each bridge exactly
 once required new techniques.
 
\begin_inset CommandInset citation
LatexCommand citet
key "euler1741"
literal "false"

\end_inset

 proved the walk was impossible, laying the foundations for graph theory
 along the way.
 He noticed that the topology, what we would now call a multigraph, was
 key to solving the problem: once you represent each bank or island as a
 node, with the bridges between them as links, it follows that, for a node
 which one is not currently on, in order to cross each link exactly once
 and not become stuck on the node, there must be an even number of links
 going to it, which we call an even degree (nodes at which one starts or
 ends the path must have odd degrees for the opposite reason).
 However, all of the nodes in the network have odd degree, as can be seen
 in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Kaliningrad"
plural "false"
caps "false"
noprefix "false"

\end_inset

, and so the walk was not possible.
\end_layout

\begin_layout Standard
Two bridges were lost in the bombing of Königsberg in World War II, along
 with most of the historic city centre, a tragedy compounded about a year
 later by the Soviet expulsion of the native population.
 However, the destruction of the two bridges does at least mean that one
 can enjoy walks crossing each bridge once, as the two banks of the river
 now have even degrees and the islands have odd degrees.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/borrowed/kalinigrad_small.jpg
	display false
	height 15pheight%

\end_inset


\begin_inset Graphics
	filename figures/borrowed/Königsberg_graph.svg
	display false
	height 15pheight%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Modern day Königsberg
\end_layout

\end_inset

Left: A satellite image of the centre of modern day Königsberg, now Kaliningrad.
 Five of the seven bridges from Euler's time remain.
 The last bastions of the two demolished bridges are just visible about
 one-third of the way across the image from the left; one has been turned
 into a berth for a boat.
 Map data from Google and Maxar Technologies
\begin_inset CommandInset label
LatexCommand label
name "fig:Kaliningrad"

\end_inset

.
 Right: Königsberg's seven bridges represented as a network (Chris Martin,
 2006)
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Random graphs
\end_layout

\begin_layout Standard
In the interests of brevity, we now skip over 200 years of fascinating results
 from graph theory to 1959, when 
\begin_inset CommandInset citation
LatexCommand citeauthor
key "Erdos1959"
literal "false"

\end_inset

 first proposed a model of random graphs.
 Their model, explored in more depth in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Network-models"
plural "false"
caps "false"
noprefix "false"

\end_inset

, was the first to be used to rigorously explore what it meant for a graph
 to have randomness.
 They showed that random graphs are very likely to be totally connected
 (i.e.
\begin_inset space \space{}
\end_inset

a route is possible between every pair of nodes) with a relatively low number
 of edges and that the average distance between randomly chosen nodes grows
 very slowly with the total number of nodes.
 This decoupling of the distance between nodes and the total number of nodes
 is a key feature of real world networks and allows information to cross
 networks very quickly.
\end_layout

\begin_layout Subsection
Small-world networks
\end_layout

\begin_layout Standard
These random graphs captured an important property of real-world networks:
 the average distance, or path length, between nodes is often short.
 This has been known intuitively for a very long time: 
\begin_inset Quotes xld
\end_inset

it's a small world
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!small-world
\end_layout

\end_inset


\begin_inset Quotes xrd
\end_inset

 has been a common refrain when one finds that a stranger has a mutual friend
 for at least a hundred years.
 This well-known phenomenon was first proved by 
\begin_inset CommandInset citation
LatexCommand citet
key "milgram1967small"
literal "false"

\end_inset

 in a famous experiment where he gave addressed letters out at random to
 people who were instructed to either send the letter to the addressee directly
 if they knew them personally, or forward the letter to the person they
 thought was most likely to know the addressee.
 Milgram's results are shown in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:SW"
plural "false"
caps "false"
noprefix "false"

\end_inset

: the modal number of intermediaries used was six, so the modal path length
 was seven.
\end_layout

\begin_layout Standard
There is a problem with the random graphs introduced by 
\begin_inset CommandInset citation
LatexCommand citeauthor
key "Erdos1959"
literal "false"

\end_inset

, however.
 It is unlikely that linked nodes share neighbours, a property often found
 in real networks; in the jargon of the field, they have a very low 
\begin_inset Quotes xld
\end_inset

clustering coefficient
\begin_inset Quotes xrd
\end_inset

, whereas real networks often have a high clustering coefficient.
 
\begin_inset CommandInset citation
LatexCommand citet
key "granovetter1977strength"
literal "false"

\end_inset

 explored this issue and argued that social networks should be considered
 as cliques interacting across weak (e.g.
\begin_inset space \space{}
\end_inset

long distance) links.
 
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!Watts-Strogatz
\end_layout

\end_inset


\begin_inset CommandInset citation
LatexCommand citet
key "watts_collective_1998"
literal "false"

\end_inset

 developed a model based on these ideas: a nearest neighbour ring network
 with weak-ties thrown in at random.
 They showed that a network needs very few of these weak ties before the
 path length loses direct proportionality with the number of nodes; this
 slow growth in path length is the small-world effect.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/borrowed/Milgram.png
	display false
	width 30text%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Milgram's small-world effect
\end_layout

\end_inset

The path-length distribution from an experiment on the social network of
 the United States 
\begin_inset CommandInset citation
LatexCommand citep
key "milgram1967small"
literal "false"

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "fig:SW"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Scale-free
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!scale-free
\end_layout

\end_inset

 networks 
\end_layout

\begin_layout Standard
It was soon found that many real-world networks had degree distributions
 that looked nothing like the small-world or random graphs; 
\begin_inset CommandInset citation
LatexCommand citet
key "barabasi1999emergence"
literal "false"

\end_inset

 found that real-world degree distributions often had 
\begin_inset Quotes xld
\end_inset

fat tails
\begin_inset Quotes xrd
\end_inset

 with a few nodes having the lion's share of the links.
 They proposed a model of preferential attachment—that new nodes added to
 a network tend to link to nodes with high degree—which resulted in networks
 with degree distributions which followed a power-law, as they had observed
 in real networks.
 There has been some recent controversy over suggestions that these are
 perhaps less common than was previously thought if one uses a very strict
 definition of what it means to follow a power-law 
\begin_inset CommandInset citation
LatexCommand citep
key "broido2019scale"
literal "false"

\end_inset

.
 This is not a debate we seek to enter.
\end_layout

\begin_layout Subsection
Motifs and graphlets
\end_layout

\begin_layout Standard
\begin_inset CommandInset citation
LatexCommand citet
key "milo_network_2002"
literal "false"

\end_inset

 formalised the intuitive idea that the local structures of networks were
 essential to their function.
 They devised a method of calculating how unusual a local structure was—i.e.
 how unlikely it was to appear by chance—and used this to extract small
 graphs characteristic of graphs from several categories, including the
 Internet, food webs and gene regulatory networks.
\end_layout

\begin_layout Section
Characterisation
\end_layout

\begin_layout Standard
Characterisation is key to many areas of life.
 When we purchase a car, we want to know what colour it is, how much it
 costs, how fast it goes, how much fuel it consumes and whether we can fit
 the shopping in the boot; when purchasing a house we want to know how often
 it floods, how much the asking price is and how close it is to a school.
 We use these characteristics to categorise things: if it goes fast and
 I can't fit my shopping in the boot, it's a sports car; if it was alive
 in the cretaceous period and looks like a giant lizard, I'm calling it
 a dinosaur.
 Characterisation's innate attraction is evident by the success of the 
\begin_inset Quotes xld
\end_inset

Top Trumps
\begin_inset Quotes xrd
\end_inset

 series of card games where one holds fact-sheets for examples of things
 within a certain field, be it sports cars, dinosaurs or Star Wars.
 This desire to categorise and compare is part of being human.
\end_layout

\begin_layout Standard
Perhaps the most famous and successful example of using characterisation
 to more deeply understand part of our universe are Hertzsprung-Russell
 diagrams 
\begin_inset CommandInset citation
LatexCommand citep
key "Russell1914"
literal "false"

\end_inset

, an example of which is shown in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:A-Hertzsprung-Russell-diagram"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 By simply plotting the luminosity (total power radiated) against the surface
 temperature or colour (which are directly related) for observed stars,
 a striking pattern emerges and, with a little interpretation and background
 knowledge, the various paths a star may take throughout its life become
 clear.
 All stars start following the 
\begin_inset Quotes xld
\end_inset

main sequence
\begin_inset Quotes xrd
\end_inset

 in the top left, becoming redder as they age.
 Massive stars become 
\begin_inset Quotes xld
\end_inset

giants
\begin_inset Quotes xrd
\end_inset

 while smaller stars stay on the main sequence.
 Our ultimate goal is to make advances like this possible in the field of
 network science.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/borrowed/HRDiagram_inverted.png
	display false

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
A Hertzsprung-Russell diagram
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Hertzsprung-Russell diagram
\end_layout

\end_inset

 (adapted from Richard Powell, 2007) showing stars with their energy output
 plotted against their surface temperature.
 Approximate trajectories grouping stars with the same life cycles have
 been drawn on in purple
\begin_inset CommandInset label
LatexCommand label
name "fig:A-Hertzsprung-Russell-diagram"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Contributions
\end_layout

\begin_layout Standard
Throughout the course of this thesis, we present numerous novel contributions
 to the field.
 We shall enumerate them here:
\end_layout

\begin_layout Itemize
Software adapting an existing graphlet enumeration routine for use on coloured
 graphs is described in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Graphlets:-colour,-profiles"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Itemize
On academic co-authorship networks, collaboration with authors who do not
 directly collaborate with each other was found to increase the number of
 years in which an author published papers.
 This finding had a robust degree of statistical significance.
 This is detailed in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Career-length-in"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Itemize
The use of coloured graphlet profiles compared to colour-blind graphlet
 profiles when characterising nodes from their ego-networks was found to
 provide more detailed results with a higher degree of statistical significance.
 This is shown in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Career-length-in"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Itemize
Networks can be distinguished using principal component analysis on data
 matrices formed from simple network representations and metrics.
 This is given in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Application-of-PCA"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Itemize
Adjusted mutual information can be used to aid in choosing the number of
 communities in which to cluster a network with tensor decomposition.
 We present this in chapters 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Locally-optimal-numbers"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Software"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Itemize
Software ecosystem evolution: we can see how distinct different software
 dependency ecosystems look, shaped, we theorise, by the decisions made
 by the package-management designers.
 This is given in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Software"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Itemize
English language evolution: we present preliminary results on how we might
 apply tensor decompositions to a temporal linguistic network.
 In our preliminary results presented in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Linguistic-evolution"
plural "false"
caps "false"
noprefix "false"

\end_inset

, we see changes in the language over time.
 We explore how such an analysis could be extended.
\end_layout

\begin_layout Section
Publications and talks
\end_layout

\begin_layout Itemize
Coloured graphlet profiles as a predictor of career length in scientific
 co-authorship networks (talk and paper).
 In: 4th World Conference on Complex Systems: Emergence, self-organization,
 nonlinear dynamics and complexity.
 Institute of Electrical and Electronics Engineers 
\begin_inset CommandInset citation
LatexCommand citep
key "BlanthornMorocco2019"
literal "false"

\end_inset

.
\end_layout

\begin_deeper
\begin_layout Itemize
This paper has been invited for further publication after expansion in a
 special issue of 
\shape italic
Journal of Systems Science and Complexity
\shape default
 in October 2019.
\end_layout

\end_deeper
\begin_layout Itemize
Evolution of communities of software: using tensor decompositions to compare
 software ecosystems.
 Accepted 23 August 2019, awaiting publication.
 Applied Network Science 
\begin_inset CommandInset citation
LatexCommand citep*
key "BlanthornAPNS2019"
literal "false"

\end_inset

.
\begin_inset Note Note
status open

\begin_layout Plain Layout
\begin_inset Quotes xld
\end_inset

in press
\begin_inset Quotes xrd
\end_inset

 — we also have a doi for this somewhere
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Tracking the evolution of the English language as a network with temporal
 clustering (unpublished): presented at Conference on Complex Systems, Nanyang
 Technological University, Singapore 
\begin_inset CommandInset citation
LatexCommand citep
key "BlanthornSingapore2019"
literal "false"

\end_inset

.
\end_layout

\begin_layout Section
Outline of the thesis
\end_layout

\begin_layout Standard
In the rest of this part, the necessary background and the state of the
 art is given.
 The remainder of the thesis is split into three parts following our three
 main areas of contribution.
 In part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Graphlets"
plural "false"
caps "false"
noprefix "false"

\end_inset

, we characterise nodes by their local network topology through the use
 of graphlet profiles.
 In part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Matrix-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

 we characterise whole networks by creating high-dimensional representations
 of them and then collapsing these into low-dimensional representations
 useful for comparison.
 In part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Tensor-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

 we characterise whole temporal networks by the evolution of their communities.
\end_layout

\begin_layout Standard
\begin_inset Newpage newpage
\end_inset


\end_layout

\begin_layout Section*
Bibliography
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "monorefs"
options "authordate4"

\end_inset


\end_layout

\begin_layout Chapter
Background
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
epigraph{``The thing that hath been, it is that which shall be;
\backslash

\backslash
 and that which is done is that which shall be done:
\backslash

\backslash
 and there is no new thing under the sun."}{--- 
\backslash
textup{Ecclesiastes 1:9,} King James Bible}
\end_layout

\end_inset


\end_layout

\begin_layout Standard

\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
MyLettrine{G}{raph theory}
\end_layout

\end_inset

 had a long history before it spawned modern network science at the turn
 of the century.
 We do not have the luxury of enumerating all of its achievements or anything
 close to all of its or network science's achievements in this thesis.
 Instead, we aim only to furnish the reader with the mathematical background
 and advances required to understand this work and the reasons it takes
 the directions it does.
\end_layout

\begin_layout Standard
The reader is advised to skim through this section on first reading.
 It may be useful to revisit this chapter with the aid of the index at the
 rear of this book when unfamiliar terms appear in the later chapters.
 This chapter comprises quite a few sections: section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Basic-mathematical-tools"
plural "false"
caps "false"
noprefix "false"

\end_inset

 introduces some of the basic mathematical foundations needed to understand
 this thesis; section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Networks"
plural "false"
caps "false"
noprefix "false"

\end_inset

 introduces mathematical descriptions of networks and graphs; sections 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Matrix-decomposition-theory"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:bg-Tensor-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

 explore matrix and tensor decomposition which we employ in parts 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Matrix-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Tensor-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

; sections 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Node-characterisation"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Network-characterisation"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Community-characterisation"
plural "false"
caps "false"
noprefix "false"

\end_inset

 describe some current methods for characterising networks at multiple scales
 and are probably the most important sections in this chapter as they form
 the basis of the work we perform in the rest of the thesis; section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Complex-networks"
plural "false"
caps "false"
noprefix "false"

\end_inset

 uses tools introduced in the preceding sections to explain what makes complex
 networks complex and gives some examples of complex networks; section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Statistical-methods"
plural "false"
caps "false"
noprefix "false"

\end_inset

 provides a brief explanation of the statistical methods we employ in chapter
 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Career-length-in"
plural "false"
caps "false"
noprefix "false"

\end_inset

; and finally section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Software-used"
plural "false"
caps "false"
noprefix "false"

\end_inset

 mentions some of the third-party software used to produce this thesis and
 the results therein.
\end_layout

\begin_layout Section
Basic mathematical tools
\begin_inset CommandInset label
LatexCommand label
name "sec:Basic-mathematical-tools"

\end_inset


\end_layout

\begin_layout Standard
Here we specify the most basic mathematical concepts and notations necessary
 for understanding this work.
\end_layout

\begin_layout Subsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
Cullis notation
\end_layout

\end_inset

A note on notation
\end_layout

\begin_layout Standard
\begin_inset Formula $A=[a_{ij}]$
\end_inset

 means that the matrix 
\begin_inset Formula $A$
\end_inset

 comprises the elements 
\begin_inset Formula $\{a_{ij}\}$
\end_inset

, such that the 
\begin_inset Formula $(i,j)^{\text{th}}$
\end_inset

 element of 
\begin_inset Formula $A$
\end_inset

 is 
\begin_inset Formula $a_{ij}$
\end_inset

 
\begin_inset CommandInset citation
LatexCommand citep
key "cullis1913notation"
literal "false"

\end_inset

.
 In this work, we generally use Roman capitals to refer to whole matrices
 or tensors and lowercase letters to refer to scalars or elements of a matrix;
 lowercase bold letters refer to vectors.
 Additionally, we adopt the convention that
\begin_inset Formula 
\[
A=[a_{ij}]\implies\left(\boldsymbol{a}_{j}=\sum_{i}a_{ij}\hat{\boldsymbol{i}}\right)\land\left(\boldsymbol{a}_{i}=\sum_{j}a_{ij}\hat{\boldsymbol{j}}\right)\text{,}
\]

\end_inset

where 
\begin_inset Formula $\hat{\boldsymbol{i}}$
\end_inset

 corresponds to the 
\begin_inset Formula $i^{\text{th}}$
\end_inset

 unit column vector and 
\begin_inset Formula $\hat{\boldsymbol{j}}$
\end_inset

 to the 
\begin_inset Formula $j^{\text{th}}$
\end_inset

 row vector, i.e.
\begin_inset space ~
\end_inset

we form row and column vectors from matrix columns and rows by omitting
 the relevant indices.
\end_layout

\begin_layout Subsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
Kronecker product
\end_layout

\end_inset

Kronecker product
\end_layout

\begin_layout Standard
The Kronecker product is a binary operator, 
\begin_inset Formula $\otimes$
\end_inset

, and is defined thusly for two matrices:
\begin_inset Formula 
\[
\boldsymbol{A}\otimes\boldsymbol{B}=\begin{bmatrix}a_{11}\boldsymbol{B} & \dots & a_{1n}\boldsymbol{B}\\
\vdots & \ddots & \vdots\\
a_{m1}\boldsymbol{B} & \dots & a_{mn}\boldsymbol{B}
\end{bmatrix}\text{,}
\]

\end_inset

i.e.
\begin_inset space ~
\end_inset

each term in the left-hand matrix becomes the coefficient to a block of
 the right-hand matrix.
 In component form, this can be represented as
\begin_inset Formula 
\[
(\boldsymbol{A}\otimes\boldsymbol{B})_{p(r-1)+v,q(s-1)+w}=a_{rs}b_{vw}\text{,}
\]

\end_inset

where 
\begin_inset Formula $A$
\end_inset

 is an 
\begin_inset Formula $m\times n$
\end_inset

 matrix and 
\begin_inset Formula $B$
\end_inset

 is an 
\begin_inset Formula $p\times q$
\end_inset

 matrix 
\begin_inset CommandInset citation
LatexCommand citep
key "van2000ubiquitous"
literal "false"

\end_inset

.
\end_layout

\begin_layout Subsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
tensor product
\end_layout

\end_inset

Tensor product
\end_layout

\begin_layout Standard
The tensor product is an extension of the outer product to two tensors,
 
\begin_inset Formula $A$
\end_inset

 and 
\begin_inset Formula $B$
\end_inset

 is defined component-wise such that
\begin_inset Formula 
\[
(A\otimes B)_{a,\ldots,n,\alpha,\ldots,\nu}=a_{a,\ldots,n}b_{\alpha,\ldots,\nu}\text{.}
\]

\end_inset


\end_layout

\begin_layout Standard
Predominantly, we will use it with vectors (rank 1 tensors) where it becomes
\begin_inset Formula 
\[
(\boldsymbol{a}\otimes\boldsymbol{b}\otimes\boldsymbol{c})_{ijk}=a_{i}b_{j}c_{k}\text{.}
\]

\end_inset


\end_layout

\begin_layout Subsection
Reduction function
\begin_inset Index idx
status open

\begin_layout Plain Layout
reduction function
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
The reduction function, 
\begin_inset Formula $\text{reduce}_{n=1}^{N}(\odot,b_{n})$
\end_inset

, returns the reduction of 
\begin_inset Formula $\{b_{n}|n\in1..N\}$
\end_inset

 by applying an arbitrary associative binary operator 
\begin_inset Formula $\odot$
\end_inset

 repeatedly, i.e.
 
\begin_inset Formula 
\[
\text{reduce}_{n=1}^{N}(\odot,b_{n})=b_{1}\odot b_{2}\odot b_{3}\odot\ldots\odot b_{N}\text{.}
\]

\end_inset


\end_layout

\begin_layout Subsection
\noindent
\begin_inset Index idx
status open

\begin_layout Plain Layout
Frobenius norm
\end_layout

\end_inset

Frobenius norm
\end_layout

\begin_layout Standard
The Frobenius norm of a matrix 
\begin_inset Formula $A$
\end_inset

 is
\begin_inset Formula 
\[
\lVert A\rVert{}_{\text{F}}=\sqrt{\sum_{i,j}\lvert a_{ij}\rvert^{2}}=\sqrt{\text{Tr}(A^{\top}A)}\text{.}
\]

\end_inset


\end_layout

\begin_layout Section
Networks
\begin_inset CommandInset label
LatexCommand label
name "sec:Networks"

\end_inset


\end_layout

\begin_layout Standard
Networks are the key focus of this thesis.
 In this section we introduce them with more rigour than the introduction
 and provide some of their standard mathematical descriptions through the
 lens of graph theory.
\end_layout

\begin_layout Standard
Networks are models which allow us to explore the relationships between
 things and the large-scale impact of local relationships 
\begin_inset CommandInset citation
LatexCommand citep
key "Strogatz2001"
literal "false"

\end_inset

.
 A complex network is a network with a large number of things and non-trivial
 wiring rules, the study of which started with 
\begin_inset CommandInset citation
LatexCommand citet
key "Erdos1959"
literal "true"

\end_inset

 and was reinvigorated in the early 2000s with the discovery of scale-free
 and small world networks 
\begin_inset CommandInset citation
LatexCommand citep
key "watts_collective_1998,barabasi1999emergence"
literal "true"

\end_inset

.
 Complex networks are introduced in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Complex-networks"
plural "false"
caps "false"
noprefix "false"

\end_inset

 after we have the mathematical tools with which to explore them more thoroughly.
\end_layout

\begin_layout Standard
More rigorously, a network, or graph, is an ordered pair 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "G"
symbol "$\\mathcal{G}$"
description "a graph"
literal "true"

\end_inset


\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "V"
symbol "$\\mathcal{V}$"
description "a set of vertices"
literal "true"

\end_inset


\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "E"
symbol "$\\mathcal{E}$"
description "a set of edges "
literal "true"

\end_inset


\begin_inset Formula $\mathcal{G}=(\mathcal{V},\mathcal{E})$
\end_inset

, where 
\begin_inset Formula $\mathcal{V}$
\end_inset

 is a set of 
\begin_inset Formula $N$
\end_inset

 nodes or vertices, with labels such that 
\begin_inset Formula $\mathcal{V}=\{v_{i}|i\in\mathbb{N}\land i\in[1,N]\}$
\end_inset

, and 
\begin_inset Formula $\mathcal{E}$
\end_inset

 is a set of 
\begin_inset Formula $2\text{-element}$
\end_inset

 subsets of 
\begin_inset Formula $\mathcal{V}$
\end_inset

.
 When referring to the node 
\begin_inset Formula $v_{i}$
\end_inset

, one often refers simply to the label 
\begin_inset Formula $i$
\end_inset

 as these are unique.
 The elements of 
\begin_inset Formula $\mathcal{E}$
\end_inset

 may be ordered or unordered depending on whether the network is directed
 or not and represent the links or edges between the nodes.
 This wiring pattern is often represented as the adjacency matrix 
\begin_inset Formula $A=[a_{i\leftarrow j}]$
\end_inset


\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "A"
symbol "$A$"
description "adjacency matrix"
literal "true"

\end_inset

 where the value of the entry 
\begin_inset Formula $a_{i\leftarrow j}$
\end_inset


\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "A"
symbol "$a_{i\\leftarrow j}$"
description "entry of the adjacency matrix corresponding to the link from j to i"
literal "true"

\end_inset

 indicates the weighting of the link from node 
\begin_inset Formula $j$
\end_inset

 to node 
\begin_inset Formula $i$
\end_inset

.
 An entry of zero corresponds to there being no link.
 A small network is shown in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Motifexample"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/ER10N20EUoM.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
A small network
\end_layout

\end_inset

A small network with 10 nodes (purple circles) and and fewer than 20 edges
 (grey lines)
\begin_inset CommandInset label
LatexCommand label
name "fig:Motifexample"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Important concepts
\end_layout

\begin_layout Standard
In this subsection, we introduce the simplest tools we require for the discussio
n of networks.
\end_layout

\begin_layout Subsubsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
centralities!degree
\end_layout

\end_inset

Degree
\begin_inset CommandInset label
LatexCommand label
name "subsec:Degree"

\end_inset


\end_layout

\begin_layout Standard
The degree of a node in a network is the number of edges connected to the
 node.
 It is calculated as
\begin_inset Formula 
\[
k_{i}=\sum_{j=1}^{N}a_{i\leftarrow j}\text{.}
\]

\end_inset


\end_layout

\begin_layout Standard
If the network is directed (i.e.
\begin_inset space ~
\end_inset


\begin_inset Formula $A\neq A^{\intercal}$
\end_inset

), each node has an in-degree, 
\begin_inset Formula $k^{\text{in}}$
\end_inset

, and out-degree, 
\begin_inset Formula $k^{\text{out}}$
\end_inset

, showing the number of inbound and outbound links respectively.
 Intuitively, the first moment of these is identical, 
\begin_inset Formula $\overline{k^{\text{in}}}=\overline{k^{\text{out}}}=\overline{k}$
\end_inset

:
\begin_inset Formula 
\begin{align*}
\overline{k^{\text{in}}} & =\frac{\text{sum of all in-degrees}}{\text{number of nodes}}=\frac{\sum_{j=1}^{N}k_{j}^{\text{in}}}{N}=\frac{\sum_{j=1}^{N}(\sum_{i=1}^{N}a_{i\leftarrow j})}{N}\\
\overline{k^{\text{out}}} & =\frac{\sum_{i=1}^{N}k_{i}^{\text{out}}}{N}=\frac{\sum_{i=1}^{N}(\sum_{j=1}^{N}a_{i\leftarrow j})}{N}=\frac{\sum_{j=1}^{N}\sum_{i=1}^{N}a_{i\leftarrow j}}{N}=\overline{k^{\text{in}}}=\overline{k}\text{,}
\end{align*}

\end_inset

so one often refers just to the 
\begin_inset Quotes xld
\end_inset

mean degree
\begin_inset Quotes xrd
\end_inset

 on a directed network rather than 
\begin_inset Quotes xld
\end_inset

mean in-degree
\begin_inset Quotes xrd
\end_inset

 or 
\begin_inset Quotes xld
\end_inset

mean out-degree
\begin_inset Quotes xrd
\end_inset

.
\end_layout

\begin_layout Standard
The degree distribution of a network, 
\begin_inset Formula $P(k)$
\end_inset

, gives the probability of a randomly-chosen node having a certain degree
 
\begin_inset CommandInset citation
LatexCommand citep
key "newman_networks:_2010"
literal "false"

\end_inset

.
 It is one of the main methods used to characterise networks.
\end_layout

\begin_layout Subsubsection
Adjacency matrix
\begin_inset Index idx
status open

\begin_layout Plain Layout
network representations!adjacency matrix
\end_layout

\end_inset


\end_layout

\begin_layout Standard
As introduced in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Networks"

\end_inset

, the elements of the adjacency matrix, 
\begin_inset Formula $\boldsymbol{A}$
\end_inset

, are defined as:
\begin_inset Formula 
\[
a_{i\leftarrow j}=a_{ij}=\begin{cases}
1 & \text{if there is an edge from node }j\text{ to node }i\\
0 & \text{otherwise.}
\end{cases}
\]

\end_inset

In a weighted network, the values correspond to the weights of the links
 which may be any real numbers (rather than just unity or zero); if 
\shape italic
multi-edges
\shape default
—multiple edges between two nodes—are permitted, the entries in the adjacency
 matrix correspond to the number of edges present; 
\shape italic
self-edges
\shape default
 or 
\shape italic
self-loops
\shape default
 are represented by entries on the diagonal.
 Adjacency matrices and adjacency tensors, as will be introduced later in
 this chapter, are sometimes referred to as 
\begin_inset Quotes xld
\end_inset

network representations
\begin_inset Index idx
status open

\begin_layout Plain Layout
network representations
\end_layout

\end_inset


\begin_inset Quotes xrd
\end_inset

 as they encode all the information one needs to reconstruct the topology
 of a network 
\begin_inset CommandInset citation
LatexCommand citep
key "spinrad2003efficient"
literal "false"

\end_inset

.
 
\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: revise newman, page 136 ish for Viva, eigenvalues for paths
\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
Subgraphs
\end_layout

\begin_layout Standard
A subgraph 
\begin_inset Formula $G=(V,E)$
\end_inset

 is a subset of a graph 
\begin_inset Formula $\mathcal{G}=(\mathcal{V},\mathcal{E})$
\end_inset

 if and only if
\begin_inset Formula 
\[
(V\subseteq\mathcal{V})\land(E\subseteq(\mathcal{E}\cap V^{2}))\text{.}
\]

\end_inset

An induced subgraph is a subgraph wholly specified by the supergraph and
 a subset of nodes; all edges existing between nodes in that subset of the
 supergraph are included in the induced subgraph, so
\begin_inset Formula 
\begin{alignat*}{1}
\mathcal{G}_{\text{induced}}:\mathcal{P}(\mathcal{V}) & \mapsto\{G\vert G\subseteq\mathcal{G}\}\\
S & \to(S,\{(v_{i},v_{j})\vert v_{i},v_{j}\in S\land(v_{i},v_{j})\in\mathcal{E}\})\text{,}
\end{alignat*}

\end_inset

where 
\begin_inset Formula $\mathcal{P}(\mathcal{V})$
\end_inset

 is the powerset of all vertices, i.e.
\begin_inset space ~
\end_inset

the space containing all possible subsets of vertices.
\end_layout

\begin_layout Subsubsection
Paths
\end_layout

\begin_layout Standard
A path 
\begin_inset Formula $p$
\end_inset

 is a sequence of edges such that each consecutive pair of edges is such
 that the first edge ends and the second edge starts at the same node.
 A closed path is a path that starts and ends at a single node.
\end_layout

\begin_layout Subsubsection
Graph colour
\begin_inset Index idx
status open

\begin_layout Plain Layout
graph colour
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Coloured graphs are graphs whose nodes fall into discrete categories.
 A colouring, 
\begin_inset Formula $\pi$
\end_inset

, of 
\begin_inset Formula $\mathcal{G}$
\end_inset

 is a function mapping from 
\begin_inset Formula $\mathcal{V}$
\end_inset

 to some set 
\begin_inset Formula $\Pi$
\end_inset

, such that 
\begin_inset Formula $\pi(i)$
\end_inset

 returns the colour of the node 
\begin_inset Formula $i$
\end_inset

.
\end_layout

\begin_layout Subsubsection
Ego-networks
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego-network
\end_layout

\end_inset


\end_layout

\begin_layout Standard
An 
\begin_inset Formula $n$
\end_inset

-ego-network is a subgraph induced by the node of interest, the ego
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego
\end_layout

\end_inset

, and all of its neighbours a certain distance 
\begin_inset Formula $n$
\end_inset

 or less from the ego; these neighbours are called the alters
\begin_inset Index idx
status open

\begin_layout Plain Layout
alters
\end_layout

\end_inset

 
\begin_inset CommandInset citation
LatexCommand citep
key "Burt1980a"
literal "false"

\end_inset

.
 
\end_layout

\begin_layout Standard
Colour can be applied trivially to ego-networks by setting the ego to be
 one colour and the alters to another.
\end_layout

\begin_layout Subsubsection
Multilayer networks
\begin_inset Index idx
status open

\begin_layout Plain Layout
multilayer network
\end_layout

\end_inset


\end_layout

\begin_layout Standard
A multilayer network is a network with multiple edge types.
 An intuitive example of a multilayer network is a transport network where
 each layer represents a mode of transport (air, rail, road, etc.) and the
 nodes are physical locations and thus shared between all layers.
 We touch on the concept of multilayer networks in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Linguistic-evolution"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Subsection
Temporal
\begin_inset Index idx
status open

\begin_layout Plain Layout
temporal networks
\end_layout

\end_inset

 or 
\begin_inset Index idx
status open

\begin_layout Plain Layout
evolving networks|see
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\end_layout

\end_inset

temporal networks
\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset

evolving networks
\end_layout

\begin_layout Standard
Almost no real-world systems are static: they change over time.
 The topologies of their network representations are no exception—they also
 change with time.
 We call these temporal networks; other authors sometimes call them dynamic
\begin_inset Index idx
status open

\begin_layout Plain Layout
dynamic networks|see
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\end_layout

\end_inset

temporal networks
\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset

, adaptive
\begin_inset Index idx
status open

\begin_layout Plain Layout
adaptive networks|see
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\end_layout

\end_inset

temporal networks
\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset

, or evolving networks.
 There are three main approaches to defining network wiring patterns which
 change with time.
\end_layout

\begin_layout Subsubsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
piecewise static networks
\end_layout

\end_inset

Piecewise static networks
\end_layout

\begin_layout Standard
The simplest way to consider an evolving network is as a totally ordered
 set of adjacency matrices,
\begin_inset Formula 
\[
(A_{t}\vert t\in T)\text{,}
\]

\end_inset


\end_layout

\begin_layout Standard
i.e.
\begin_inset space \space{}
\end_inset

as a sequence of static networks.
 This technique can be beneficial as it allows all of the standard network
 metrics to be employed, which will be used in part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Matrix-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 To the best of our knowledge, the term 
\begin_inset Quotes xld
\end_inset

piecewise static
\begin_inset Quotes xrd
\end_inset

 is our invention.
 We are unaware of any other terms for it although many authors apply it
 without giving it a name (e.g.
\begin_inset space ~
\end_inset


\begin_inset CommandInset citation
LatexCommand citealp
key "bagrow2019information"
literal "false"

\end_inset

).
\end_layout

\begin_layout Subsubsection
Multilayer network
\begin_inset Index idx
status open

\begin_layout Plain Layout
multilayer network
\end_layout

\end_inset

s and adjacency tensors
\begin_inset Index idx
status open

\begin_layout Plain Layout
network representations!adjacency tensor
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "subsec:Adjacency-tensor"

\end_inset


\end_layout

\begin_layout Standard
An adjacency tensor is an extension of the adjacency matrix
\begin_inset Index idx
status open

\begin_layout Plain Layout
network representations!adjacency matrix
\end_layout

\end_inset

 to multilayer temporal networks.
 They can take various forms, but a common one is made of elements
\begin_inset Formula 
\[
a_{i\alpha\leftarrow j\beta}^{t}=\begin{cases}
1 & \text{if there is an edge from node }j\text{ in layer }\beta\text{ to node }i\text{ in layer }\alpha\text{ at time }t\\
0 & \text{otherwise,}
\end{cases}
\]

\end_inset

where the upper and lower indices are used solely for legibility and do
 not imply any special transformation rules.
\end_layout

\begin_layout Subsubsection
Continuous time networks
\end_layout

\begin_layout Standard
The final way to consider temporal networks is in continuous time.
 This is an active area of research that we do not consider in this work
\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: https://www.slideshare.net/masonporter/centrality-in-time-dependent-networks
 porter on continuous time
\end_layout

\end_inset

.
 These networks are often described by a function per pair of nodes such
 that 
\begin_inset Formula 
\begin{align*}
A_{ij}:\mathbb{R} & \to\{1,0\}\\
t & \mapsto A_{ij}(t)\text{.}
\end{align*}

\end_inset

Although most real-world networks operate in continuous time, any data about
 them are usually quantised.
 Continuous time representations of real-world networks are therefore often
 interpolations of adjacency tensors.
\end_layout

\begin_layout Section
Matrix decomposition
\begin_inset Index idx
status open

\begin_layout Plain Layout
matrix decomposition
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:Matrix-decomposition-theory"

\end_inset


\end_layout

\begin_layout Standard
A matrix decomposition factorises a matrix into a product of matrices, e.g.
 a matrix 
\begin_inset Formula $X$
\end_inset

 might be factorised such that 
\begin_inset Formula $X=TW^{-1}$
\end_inset

.
 These decompositions are useful for a variety of purposes; here we focus
 on ones that help us to reduce the number of dimensions required to approximate
 some data contained in a matrix 
\begin_inset Formula $X$
\end_inset

.
 We will exploit this technique in part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Matrix-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
Principal component analysis
\begin_inset Index idx
status open

\begin_layout Plain Layout
principal component analysis
\end_layout

\end_inset

 (
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "PCA"
symbol "PCA"
description "principal component analysis"
literal "false"

\end_inset

PCA) is a type of matrix decomposition 
\begin_inset CommandInset citation
LatexCommand citep
key "wold_principal_1987"
literal "false"

\end_inset

.
 The purpose of PCA is to find a new co-ordinate system such that the greatest
 variance is found on the first co-ordinate, the second greatest variance
 on the second co-ordinate and so on and that each co-ordinate is orthogonal,
 as shown graphically in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:PCA-applied-to"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 The transformation is defined by a set of unit vectors, 
\begin_inset Formula $\{\boldsymbol{w}_{k}\}$
\end_inset

, that map each row vector corresponding to an individual observation, 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "x"
symbol "$\\boldsymbol{x}$"
description "a feature vector corresponding to an example"
literal "true"

\end_inset


\begin_inset Formula $\boldsymbol{x}_{i}$
\end_inset

, where 
\begin_inset Formula $\overline{\boldsymbol{x}}=\boldsymbol{0}$
\end_inset

, to a new vector, 
\begin_inset Formula $\boldsymbol{t}_{i}$
\end_inset


\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "t"
symbol "$\\boldsymbol{t}$"
description "a transformed feature vector, e.g. projected onto its principal components"
literal "true"

\end_inset

, containing the transformed co-ordinates, where
\begin_inset Formula 
\[
\boldsymbol{t}_{i}=\sum_{k}\boldsymbol{x}_{i}\cdot\boldsymbol{w}_{k}\hat{\boldsymbol{k}}\text{,}
\]

\end_inset

where 
\begin_inset Formula $\hat{\boldsymbol{k}}$
\end_inset

 is the 
\begin_inset Formula $k^{\text{th}}$
\end_inset

 unit vector of the transformed space.
 It can therefore be seen that the first transformation vector, 
\begin_inset Formula $\boldsymbol{w}_{1}$
\end_inset

, must satisfy
\begin_inset Formula 
\[
\boldsymbol{w}_{1}=\arg\max\left\{ \sigma^{2}(\boldsymbol{t}'_{1})\right\} =\arg\max\left\{ \sum_{i}(\boldsymbol{t}'_{1})_{i}^{2}\right\} \text{,}
\]

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
this t_1 is wrong - it needs to be the t corresponding to the 1^th unit
 vector of the transformed space
\end_layout

\end_inset

where 
\begin_inset Formula $\sigma^{2}(\boldsymbol{t}'_{1})$
\end_inset

 is the variance of the vector 
\begin_inset Formula $\boldsymbol{t}'_{1}=[\boldsymbol{x}_{i}\cdot\boldsymbol{w}_{1}]$
\end_inset

.
 If we stack the row vectors into a matrix 
\begin_inset Formula $X=\{\boldsymbol{x}_{i}\}$
\end_inset

, we can rewrite this equation such that 
\begin_inset Formula 
\[
\boldsymbol{w}_{1}=\arg\max\left\{ \frac{\sum_{i}(\boldsymbol{x}_{i}\cdot\boldsymbol{w}_{1})^{2}}{\lVert\boldsymbol{w}\rVert^{2}}\right\} =\arg\max\left\{ \frac{\boldsymbol{w}_{1}^{\intercal}X^{\intercal}X\boldsymbol{w}_{1}}{\boldsymbol{w}_{1}^{\intercal}\boldsymbol{w}_{1}}\right\} \text{,}
\]

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: denominator ensures normalisation
\end_layout

\end_inset

which is a Rayleigh quotient and therefore its maximum value is the largest
 eigenvalue of 
\begin_inset Formula $X^{\intercal}X$
\end_inset

 
\begin_inset CommandInset citation
LatexCommand citep
key "strang1993introduction"
literal "false"

\end_inset

, i.e.
\begin_inset space \thinspace{}
\end_inset

the eigenvalue of the sample covariance matrix, which is obtained when 
\begin_inset Formula $\boldsymbol{w}_{1}$
\end_inset

 is the corresponding eigenvector.
 All subsequent eigenvectors and values correspond to the next best orthogonal
 directions of variance.
 These eigenvalues and eigenvectors can be obtained computationally cheaply
 through singular value decomposition (
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "svd"
symbol "SVD"
description "singular value decomposition"
literal "false"

\end_inset

SVD, 
\begin_inset CommandInset citation
LatexCommand citealp
key "halko2011finding"
literal "false"

\end_inset

).
\end_layout

\begin_layout Standard
Therefore, the complete set of transformed co-ordinates is given by
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "X"
symbol "$X$"
description "a data matrix whose rows are feature vectors"
literal "true"

\end_inset


\begin_inset Formula 
\[
T=XW\text{,}
\]

\end_inset

where 
\begin_inset Formula $T=\{\boldsymbol{t}_{i}\}$
\end_inset

, so each row corresponds to a single observation and 
\begin_inset Formula $W=\{\boldsymbol{w}_{k}\}$
\end_inset

, where 
\begin_inset Formula $\boldsymbol{w}_{k}$
\end_inset

 is the eigenvector corresponding to the 
\begin_inset Formula $k^{\text{th}}$
\end_inset

 largest eigenvalue.
 Since 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "wk"
symbol "$\\boldsymbol{w}_{k}$"
description "the $k^{\\text{th}}$ principal component"
literal "true"

\end_inset


\begin_inset Formula $\boldsymbol{w}_{k}$
\end_inset

, the principal components, are arranged such that the first corresponds
 to the most variance, we can turn this into a dimension reduction by only
 considering the first few components, that is, the first few columns of
 
\begin_inset Formula $W$
\end_inset

, we will have lost the least amount of variance possible.
\end_layout

\begin_layout Standard
For applications of this technique in network science, see section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Matrix-decompositions"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/GaussianScatterPCA.svg.png
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
Samples
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA applied to a Gaussian blob
\end_layout

\end_inset

 from a two-dimensional Gaussian distribution centred at (1,3) with the
 principal components plotted: the first component points towards the top
 right following the direction of maximum variance and the second component
 is orthogonal to the first (Nicoguaro, 2016)
\begin_inset CommandInset label
LatexCommand label
name "fig:PCA-applied-to"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
\begin_inset Index idx
status open

\begin_layout Plain Layout
tensor decomposition
\end_layout

\end_inset

Tensor decomposition
\begin_inset CommandInset label
LatexCommand label
name "sec:bg-Tensor-decomposition"

\end_inset


\end_layout

\begin_layout Standard
Tensor decompositions can be seen as an extension of matrix decompositions:
 matrix decompositions factorise a matrix into a product of matrices; tensor
 decompositions factorise a tensor into a tensor product of tensors.
 In this work, we choose to decompose tensors into the tensor product of
 rank-1 tensors (i.e.
 vectors).
 A order-
\begin_inset Formula $N$
\end_inset

 tensor 
\begin_inset Formula $X$
\end_inset

 can be represented as
\begin_inset Index idx
status open

\begin_layout Plain Layout
reduction function
\end_layout

\end_inset


\begin_inset Formula 
\begin{equation}
X=\sum_{r=1}^{R}\text{reduce}_{n=1}^{N}(\otimes,\boldsymbol{a}_{nr})\text{,}\label{eq:tensordecomp}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
\noindent
where 
\begin_inset Formula $\otimes$
\end_inset

 is the tensor product
\begin_inset Index idx
status open

\begin_layout Plain Layout
tensor product
\end_layout

\end_inset

 and each 
\begin_inset Formula $\boldsymbol{a}_{nr}\in\mathbb{R}^{d_{n}}$
\end_inset

 where 
\begin_inset Formula $d_{n}$
\end_inset

 is the dimension of the 
\begin_inset Formula $n^{\text{th}}$
\end_inset

-mode of the tensor 
\begin_inset Formula $X$
\end_inset

.
 If a smaller 
\begin_inset Formula $R$
\end_inset

 than is necessary is chosen, the decomposition becomes an approximation.
 When an order 
\begin_inset Formula $N$
\end_inset

 tensor is decomposed into 
\begin_inset Formula $RN$
\end_inset

 vectors, the total number of dimensions changes from 
\begin_inset Formula $\prod_{n=1}^{N}d_{n}$
\end_inset

 to 
\begin_inset Formula $\sum_{r=1}^{R}\sum_{n=1}^{N}d_{n}$
\end_inset

, which in all sane applications will be smaller.
 Tensor decompositions are also known as canonical polyadic decompositions,
 tensor rank decompositions, tensor factorisations and CANDECOMP/PARAFAC
\begin_inset Index idx
status open

\begin_layout Plain Layout
PARAFAC
\end_layout

\end_inset

 
\begin_inset CommandInset citation
LatexCommand citep
key "harshman1970foundations,carroll1970analysis"
literal "false"

\end_inset

, reflecting the number of times the technique has been independently rediscover
ed.
\end_layout

\begin_layout Standard
To find the tensor decomposition of a tensor 
\begin_inset Formula $X$
\end_inset

, we need to minimise 
\begin_inset Formula 
\[
\lVert X-\sum_{r=1}^{R}\text{reduce}_{n=1}^{N}(\otimes,\boldsymbol{a}_{nr})\rVert_{F}\text{,}
\]

\end_inset

where 
\begin_inset Formula $\boldsymbol{a}_{nr}$
\end_inset

 are parameters to be found (
\begin_inset Formula $R$
\end_inset

 is generally chosen from domain knowledge).
 The most common way to approach this is to use an alternating least squares
 (
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "als"
symbol "ALS"
description "alternating least squares"
literal "false"

\end_inset

ALS) method where one optimises for a single component at a time using least
 squares, holding all other components fixed, and then iterating through
 each component repeatedly until convergence 
\begin_inset CommandInset citation
LatexCommand citep
key "TensorReview"
literal "false"

\end_inset

.
 In this work, we were not especially concerned with using the most computationa
lly efficient method so picked alternating proximal gradient tensor factorisatio
n (APG-TF
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "apgtf"
symbol "APG-TF"
description "alternating proximal gradient tensor factorisation"
literal "false"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand citealp
key "Xu2013b"
literal "false"

\end_inset

) as it was fast enough and had a public implementation.
 For our applications, we impose a further constraint such that 
\begin_inset Formula 
\[
\forall i,n,r:a_{inr}\geq0\text{,}
\]

\end_inset

in order to aid interpretation as all of our adjacency tensors are non-negative.
 Such a decomposition is often referred to as non-negative CANDECOMP/PARAFAC
 (NNCP
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "nncp"
symbol "NNCP"
description "non-negative CANDECOMP/PARAFAC"
literal "false"

\end_inset

) decomposition.
 Tensor decompositions are generally used in network science for community
 detection and characterisation in temporal networks through decomposing
 their adjacency tensors, as we explore in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Community-characterisation"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Tensor-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\begin_inset Note Note
status collapsed

\begin_layout Plain Layout
Viva: I don't think we need to go into any of this.
\end_layout

\begin_layout Itemize
CANDECOMP
\begin_inset Index idx
status open

\begin_layout Plain Layout
CANDECOMP
\end_layout

\end_inset

/PARAFAC
\begin_inset Index idx
status open

\begin_layout Plain Layout
PARAFAC
\end_layout

\end_inset

 vs Tucker decomposition
\begin_inset Index idx
status open

\begin_layout Plain Layout
Tucker decomposition
\end_layout

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
this clearly needs to be finished
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
CP decomposes into 
\begin_inset Formula $N$
\end_inset

 factor matrices
\end_layout

\begin_layout Itemize
TD decomposes into core tensor and 
\begin_inset Formula $N$
\end_inset

 factor matrices (i.e.
\begin_inset space \space{}
\end_inset

superdiagonal core tensor becomes equivalent to CP)
\end_layout

\begin_layout Itemize
Nonnegative variants - all elements are positive, easing intuition and improving
 explainability in many applications
\end_layout

\end_deeper
\begin_layout Itemize
Methods for Tucker:
\end_layout

\begin_deeper
\begin_layout Itemize
ALS, CCD, HONMF, HALS, ANLS (all alternate between core tensor and matrix
 updates)
\end_layout

\begin_layout Itemize
APG updates factor matrices by solving relaxed subproblems with separable
 quadratic objectives; low iteration costs 
\begin_inset Note Note
status open

\begin_layout Plain Layout
ask colin about the others he looked into
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Plain Layout
\begin_inset Note Note
status open

\begin_layout Plain Layout
NRSS
\begin_inset Note Note
status open

\begin_layout Plain Layout
which has another name in the literature - residual error? sum of squares
 error? standard error? can't remember, check our paper
\end_layout

\end_inset

, CONCORDIA 
\begin_inset Note Note
status open

\begin_layout Plain Layout
but then need to explain why we didn't use it! also, is it called concordia?
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Note Note
status open

\begin_layout Plain Layout
HOPCA (higher order PCA) imposes orthogonality on the matrices; i.e.
 Tucker decomp does not! 
\begin_inset Note Note
status open

\begin_layout Plain Layout
sparsity of Tucker might refer to sparsity of core matrices, have we done
 goofed?
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Note Note
status open

\begin_layout Plain Layout
the thread of thought here: characterise graphs with graphlet compositions
 -> characterise nodes by their ego-networks; characterise graphs by comparing
 to other graphs (matrix decomposition) -> characterise dynamic graphs by
 comparing to themselves [temporal clustering (tensor decomposition)]
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Finding locally optimal numbers of components
\begin_inset CommandInset label
LatexCommand label
name "subsec:Finding-locally-optimal"

\end_inset


\end_layout

\begin_layout Standard
In the literature, by far the most popular method for finding the optimal
 number of components, 
\begin_inset Formula $R$
\end_inset

, or communities in network science, to decompose a tensor into is to already
 know the answer from domain knowledge.
 
\begin_inset CommandInset citation
LatexCommand citet
key "Gauvin2014b"
literal "false"

\end_inset

 did this when they chose to split their pupil-interaction network into
 the same number of communities as there were classes, into which the children
 were split.
\end_layout

\begin_layout Standard
In this work, we generally don't know the how many communities there should
 be; in many cases it is not even clear if 
\begin_inset Quotes xld
\end_inset

how many communities are there?
\begin_inset Quotes xrd
\end_inset

 is a meaningful question.
 We therefore must use some metrics to help us determine the best number
 of communities.
\end_layout

\begin_layout Standard
The normalised residual sum of squares
\begin_inset Index idx
status open

\begin_layout Plain Layout
residual sum of squares
\end_layout

\end_inset

 or relative error
\begin_inset Index idx
status open

\begin_layout Plain Layout
relative error|see
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\end_layout

\end_inset

residual sum of squares
\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset

, 
\begin_inset Formula $\text{NRSS}_{R}$
\end_inset

, is a measure of reconstruction error and is one approach used with tensor
 decompositions 
\begin_inset CommandInset citation
LatexCommand citep
key "Papalexakis2012a"
literal "false"

\end_inset

.
 It can be used to inform one's choice of 
\begin_inset Formula $R$
\end_inset

 by looking for the number of components where the rate at which the residual
 sum of squares decreases as the number of components increases becomes
 linear, commonly known as an elbow.
 We define the normalised residual sum of squares as: 
\begin_inset Formula 
\begin{equation}
\text{NRSS}_{R}(X)=\frac{\|X-\sum_{r=1}^{R}\text{reduce}_{n=1}^{N}(\otimes,\boldsymbol{a}_{nr})\|_{F}}{\|X\|_{\text{F}}}\text{,}\label{NRSS-2}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
\noindent
where 
\begin_inset Formula $\|X\|_{\text{F}}$
\end_inset

 is the 
\begin_inset Index idx
status open

\begin_layout Plain Layout
Frobenius norm
\end_layout

\end_inset

Frobenius norm of 
\begin_inset Formula $X$
\end_inset

.
\end_layout

\begin_layout Standard
Another approach in the literature is 
\begin_inset Quotes xld
\end_inset

CORCONDIA
\begin_inset Quotes xrd
\end_inset

, introduced by 
\begin_inset CommandInset citation
LatexCommand citet
key "Bro2003b"
literal "false"

\end_inset

.
 It attempts to measure how closely some assumptions made by CANDECOMP/PARAFAC
 decompositions are followed by the resultant decomposition.
 As with NRSS, one uses it to choose a value for 
\begin_inset Formula $R$
\end_inset

 based on where an 
\begin_inset Quotes xld
\end_inset

elbow
\begin_inset Quotes xrd
\end_inset

 is visible.
\end_layout

\begin_layout Section
Node characterisation
\begin_inset CommandInset label
LatexCommand label
name "sec:Node-characterisation"

\end_inset


\end_layout

\begin_layout Standard
The smallest scale of a network is that which considers nodes.
 They are therefore often the easiest thing to improve in a real-world network,
 for example, one might upgrade traffic lights at a road junction, and so
 being able to describe them is vitally important.
 We apply node characterisation extensively in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Career-length-in"
plural "false"
caps "false"
noprefix "false"

\end_inset

 where we consider how academic researchers might change the way they choose
 co-authors for their papers with a view to increasing the length of their
 academic careers.
\end_layout

\begin_layout Standard
Functions which characterise the importance of nodes are commonly called
 
\begin_inset Quotes xld
\end_inset

centralities
\begin_inset Quotes xrd
\end_inset

.
\end_layout

\begin_layout Subsection
Characterisation from local topology
\end_layout

\begin_layout Standard
The most obvious and most commonly used node centrality is degree centrality.
 It is simply the degree of a node and obviously gives some measure of a
 nodes importance.
 Other, more involved centralities are discussed in this section.
\end_layout

\begin_layout Subsubsection
Local clustering coefficient
\begin_inset Index idx
status open

\begin_layout Plain Layout
centralities!local clustering coefficient
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "ci"
symbol "$C_i$"
description "local clustering coefficient of the node $i$"
literal "true"

\end_inset

immediate neighbourhood of a node 
\begin_inset Formula $v_{i}$
\end_inset

 is 
\begin_inset Formula $N_{i}=\{v_{j}\vert e_{ij}\in E\lor e_{ji}\in E\}$
\end_inset

.
 Note that 
\begin_inset Formula $v_{i}\notin N_{i}$
\end_inset

 unless self-loops are permitted.
 The local clustering coefficient for undirected graphs 
\begin_inset CommandInset citation
LatexCommand citep
key "newman_networks:_2010"
literal "false"

\end_inset

 is then
\begin_inset Formula 
\begin{align*}
C_{i} & =\frac{\text{number of links in subgraph induced by neighbourhood}}{\text{number of possible links in neighbourhood}}\\
 & =\frac{2\lvert\{e_{jk}\vert(v_{j},v_{k}\in N_{i})\land(e_{jk}\in E)\}\rvert}{k_{i}(k_{i}-1)}\text{.}
\end{align*}

\end_inset


\end_layout

\begin_layout Standard
Note that this is equivalent to the global clustering coefficient, introduced
 in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Global-clustering-coefficient"
plural "false"
caps "false"
noprefix "false"

\end_inset

, on the 1-ego-network of the node 
\begin_inset Formula $v_{i}$
\end_inset

.
\end_layout

\begin_layout Subsubsection
Ego-networks
\end_layout

\begin_layout Standard
As was just touched on, nodes can be characterised by the characterisation
 of their ego-networks.
 A number of authors have explored this possibility.
 
\begin_inset CommandInset citation
LatexCommand citet
key "akoglu2010oddball"
literal "false"

\end_inset

 use simple criteria—number of edges, etc.—from a node's ego-network to character
ise it.
 A similar approach is employed by others who use graphlet profiles (explored
 in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Graphlets"
plural "false"
caps "false"
noprefix "false"

\end_inset

) on ego-networks to characterise nodes 
\begin_inset CommandInset citation
LatexCommand citep
key "OCallaghan2012,Cunningham2013"
literal "false"

\end_inset

.
 For more detail on the characterisation of networks see section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Network-characterisation"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Other approaches applying machine learning techniques exist 
\begin_inset CommandInset citation
LatexCommand citep
key "Rossi2017"
literal "false"

\end_inset

.
\end_layout

\begin_layout Subsection
Characterisation from global topology
\end_layout

\begin_layout Standard
A node's position within the whole network is also often worth considering.
 Imagine a very close-knit group of friends who do not interact much with
 people outside of that group of friends.
 From local topological measures, people within that group of friends may
 appear very important when they are globally very isolated.
\end_layout

\begin_layout Subsubsection
Betweenness centrality
\end_layout

\begin_layout Standard
The betweenness centrality of a node 
\begin_inset Formula $v_{i}$
\end_inset

 is the normalised sum of the ratios of the number of shortest paths that
 pass through the node,
\begin_inset Formula 
\[
g(v_{i})=\frac{2\sum_{i\neq j\neq k}\sigma_{ij}(v_{k})}{(N-1)(N-2)\sum_{i\neq j\neq k}(\sum_{k}\sigma_{ij}(v_{k}))}\text{,}
\]

\end_inset

where 
\begin_inset Formula $\sigma_{i\leftarrow j}(v_{k})$
\end_inset

 is the number of geodesic paths from node 
\begin_inset Formula $v_{j}$
\end_inset

 to node 
\begin_inset Formula $v_{i}$
\end_inset

 which pass through node 
\begin_inset Formula $v_{k}$
\end_inset

 and 
\begin_inset Formula $N$
\end_inset

 is the number of nodes in the graph 
\begin_inset CommandInset citation
LatexCommand citep
key "freeman1978centrality"
literal "false"

\end_inset

.
\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: the scaling factor is the number of pairs of nodes that aren't v_i,
 hence the extra -1
\end_layout

\end_inset

 Intuitively, it is a measure of how many short paths a node lies on, so
 
\begin_inset Quotes xld
\end_inset

bridges
\begin_inset Quotes xrd
\end_inset

 that connect two clusters have a high betweenness centrality.
\end_layout

\begin_layout Subsubsection
Closeness centrality
\end_layout

\begin_layout Standard
Closeness centrality 
\begin_inset CommandInset citation
LatexCommand citep
key "rochat2009closeness"
literal "false"

\end_inset

, 
\begin_inset Formula $H$
\end_inset

, is a measure of how close a node is to all other nodes.
 It is defined as
\begin_inset Formula 
\[
H(v_{i})=N\sum_{j}(d(v_{i},v_{j})^{-1})\text{,}
\]

\end_inset

where
\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: the N normalisation is the average distance to the nodes
\end_layout

\end_inset


\begin_inset Formula 
\begin{align*}
d(v_{i},v_{j})^{-1} & =\begin{cases}
(\text{the distance between }v_{i}\text{ and }v_{j})^{-1} & \text{if }v_{i}\text{ is connected to }v_{j}\text{,}\\
0 & \text{otherwise.}
\end{cases}
\end{align*}

\end_inset


\end_layout

\begin_layout Subsubsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
trophic level
\end_layout

\end_inset

Trophic level
\end_layout

\begin_layout Standard
The trophic level of a node in a directed graph 
\begin_inset CommandInset citation
LatexCommand citep
key "levine1980several"
literal "false"

\end_inset

, 
\begin_inset Formula $s(v_{i})$
\end_inset

, is defined such that
\begin_inset Formula 
\[
s(v_{i})-1=\begin{cases}
0 & \text{if }k_{i}^{\text{in}}=0,\\
\frac{\sum_{j}a_{ij}s(v_{j})}{k_{i}^{\text{in}}} & \text{otherwise.}
\end{cases}
\]

\end_inset

It is predominantly used in ecological contexts where it denotes how 
\begin_inset Quotes xld
\end_inset

far up
\begin_inset Quotes xrd
\end_inset

 a food chain a node is.
\end_layout

\begin_layout Subsection
Graph embeddings and representation learning
\end_layout

\begin_layout Standard
A graph embedding maps nodes to a low dimensional space.
 This is a very active area of research because such embeddings are ideal
 as inputs to machine learning techniques where one is trying to predict
 things about nodes from their position within the topology.
 
\begin_inset CommandInset citation
LatexCommand citet
key "hamilton2017representation"
literal "false"

\end_inset

 and 
\begin_inset CommandInset citation
LatexCommand citet
key "goyal2018graph"
literal "false"

\end_inset

 provide thorough reviews of the subject.
 It is not an area we have time to explore deeply in this thesis, but it
 would be remiss of us if we failed to mention the similarities between
 graph embedding and the work we perform in part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Matrix-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

 where we map entire graphs to a low dimensional space.
\end_layout

\begin_layout Section
Network characterisation
\begin_inset CommandInset label
LatexCommand label
name "sec:Network-characterisation"

\end_inset


\end_layout

\begin_layout Standard
Network characterisation is the task of describing a network mathematically
 such that it can be compared with other networks.
 For a good review of the subject, see 
\begin_inset CommandInset citation
LatexCommand citet
key "Costa2007"
literal "false"

\end_inset

.
 Many network characteristics are termed metrics in the sense that 
\begin_inset Quotes xld
\end_inset

they record something about the network
\begin_inset Quotes xrd
\end_inset

; this does not mean that they satisfy the identity of indiscernibles (i.e.
\begin_inset space ~
\end_inset


\begin_inset Formula $d(x,y)=0\iff x=y$
\end_inset

 for any metric 
\begin_inset Formula $d$
\end_inset

) or any of the other conditions for a formal mathematical metric.
\end_layout

\begin_layout Standard
Here, we will only present the work most relevant to our own.
 First, we start with the simplest: networks are often characterised by
 the number of nodes, 
\begin_inset Formula $N=\rVert\mathcal{V}\rVert$
\end_inset

, and the number of edges, 
\begin_inset Formula $\rVert\mathcal{E}\rVert$
\end_inset

.
 The density of an undirected graph is 
\begin_inset Formula $\rho=\frac{2\rVert\mathcal{E}\rVert}{N(N-1)}$
\end_inset

.
\end_layout

\begin_layout Subsection
Laplacian
\begin_inset Index idx
status open

\begin_layout Plain Layout
network representations!Laplacian matrix
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "subsec:Laplacian"

\end_inset


\end_layout

\begin_layout Standard
The Laplacian
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "L"
symbol "$L$"
description "Laplacian matrix"
literal "true"

\end_inset

 of a network is defined as
\begin_inset Formula 
\[
\boldsymbol{L}=\boldsymbol{D}-\boldsymbol{A}\text{,}
\]

\end_inset

where 
\begin_inset Formula $\boldsymbol{D}$
\end_inset

 is a diagonal matrix of the degrees of each node, 
\begin_inset Formula $\boldsymbol{D}=\{\delta_{ij}k_{i}\}=\{\delta_{ij}\sum_{k}a_{ik}\}$
\end_inset


\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "d"
symbol "$\\delta_{ij}$"
description "Kronecker delta"
literal "true"

\end_inset

, and 
\begin_inset Formula $\boldsymbol{A}$
\end_inset

 is the adjacency matrix 
\begin_inset CommandInset citation
LatexCommand citep
after "p. 152"
key "newman_networks:_2010"
literal "false"

\end_inset

.
\end_layout

\begin_layout Subsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!shortest path length
\end_layout

\end_inset

Average path length
\end_layout

\begin_layout Standard
The path length between two nodes is the length of the route taken between
 the two nodes.
 For an unweighted network, this is the number of edges such a path would
 touch; for a weighted network it would be the sum of the weights of those
 edges.
 The shortest, or geodesic, path length is the shortest possible route between
 two nodes; often this is just abbreviated to 
\begin_inset Quotes xld
\end_inset

path length
\begin_inset Quotes xrd
\end_inset

 in the literature.
 A key network characteristic is the average length of all shortest paths.
\end_layout

\begin_layout Subsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!diameter
\end_layout

\end_inset

Diameter
\end_layout

\begin_layout Standard
The diameter of a network, 
\begin_inset Formula $D$
\end_inset


\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "D"
symbol "$D$"
description "network diameter"
literal "true"

\end_inset

, is the length of the longest geodesic path in the network.
 In very large networks, one might instead take the 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
liningnums{90}
\end_layout

\end_inset


\begin_inset Formula $^{\text{th}}$
\end_inset

 percentile of the path-length distribution to ensure a more representative
 picture is gained.
\end_layout

\begin_layout Subsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!path efficiency
\end_layout

\end_inset

Path efficiency
\end_layout

\begin_layout Standard
The path efficiency, 
\begin_inset Formula $E$
\end_inset

, of a graph is
\begin_inset Formula 
\[
E(\mathcal{G})=\frac{1}{N(N-1)}\sum_{i\neq j}\frac{1}{d(v_{i},v_{j})}\text{,}
\]

\end_inset


\end_layout

\begin_layout Standard
where 
\begin_inset Formula $d(v_{i},v_{j})$
\end_inset

 is the distance between node 
\begin_inset Formula $v_{i}$
\end_inset

 and node 
\begin_inset Formula $v_{j}$
\end_inset

, where 
\begin_inset Formula $d(v_{i},v_{j})^{-1}=0\iff\text{there is no path between }v_{i}\text{ and }v_{j}$
\end_inset

.
 It is closely related to the average path length and has the added benefit
 of naturally handling disconnected components.
\end_layout

\begin_layout Subsection
Degree correlation matrices
\end_layout

\begin_layout Standard
The degree mixing matrix
\begin_inset Index idx
status open

\begin_layout Plain Layout
degree mixing matrix|see
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\end_layout

\end_inset

degree correlation matrix
\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset

 or degree correlation matrix
\begin_inset Index idx
status open

\begin_layout Plain Layout
degree correlation matrix
\end_layout

\end_inset

, 
\begin_inset Formula $D$
\end_inset


\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "D"
symbol "$D$"
description "degree correlation matrix"
literal "true"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand citep
key "newmanmix2003"
literal "false"

\end_inset

 is defined such that each entry
\begin_inset Formula 
\[
d_{ij}=P(\text{edge chosen at random connects a node of degree }i\text{ to a node of degree }j)\text{.}
\]

\end_inset

Examples can be found in figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:er-deg-dist-mix"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:ws-deg-dist-mix"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:ba-deg-dist-mix"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Entries in the off-diagonals correspond to disassortative mixing—nodes
 linking to nodes unlike themselves (at least in terms of degree)—and diagonal
 entries (or close to diagonal) correspond to assortative mixing.
\end_layout

\begin_layout Subsection
Global clustering coefficient
\begin_inset CommandInset label
LatexCommand label
name "subsec:Global-clustering-coefficient"

\end_inset


\end_layout

\begin_layout Standard
The global clustering coefficient (GCC
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "gcc"
symbol "GCC"
description "global clustering coefficient"
literal "false"

\end_inset


\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "c"
symbol "$C$"
description "global clustering coefficient"
literal "true"

\end_inset

), 
\begin_inset Formula $C$
\end_inset

, is a measure of how many triangles there are in a network and is closely
 related to transitivity, i.e.
 for an undirected graph,
\begin_inset space ~
\end_inset


\begin_inset Formula $(v_{i},v_{j})\in\mathcal{E}\land(v_{j},v_{k})\in\mathcal{E}\implies(v_{k},v_{i})\in\mathcal{E}$
\end_inset

.
 Precisely,
\begin_inset Formula 
\[
C=\frac{\text{number of closed paths of length 2}}{\text{number of paths of length 2}}\text{;}
\]

\end_inset

e.g.
\begin_inset space \space{}
\end_inset

in a network of social connections, it measures the probability that if
 Alice knows Bob and Bob knows Charlie, Charlie knows Alice 
\begin_inset CommandInset citation
LatexCommand citep
key "newman_networks:_2010"
literal "false"

\end_inset

.
\end_layout

\begin_layout Subsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!motifs
\end_layout

\end_inset

Graphlets and motifs
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlets
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "subsec:Graphlets"

\end_inset


\end_layout

\begin_layout Standard
Motifs are characteristic subgraphs, tallied by automorphism group—that
 is, graphs whose nodes can be relabelled to produce an identical adjacency
 matrix—of a graph.
 The 
\begin_inset Quotes xld
\end_inset

characteristic
\begin_inset Quotes xrd
\end_inset

 refers to subgraphs that appear more often than one would expect from a
 null model such as a configuration model 
\begin_inset CommandInset citation
LatexCommand citep
key "milo_network_2002"
literal "false"

\end_inset

.
 They described them as:
\end_layout

\begin_layout Quote
patterns of inter-connections occurring in complex networks at numbers that
 are significantly higher than those in randomized networks.
\end_layout

\begin_layout Standard
Graphlets are a similar concept, originally with the distinction that the
 subgraphs must be induced and that no significance test or normalisation
 is required 
\begin_inset CommandInset citation
LatexCommand citep
key "przulj_biological_2007"
literal "false"

\end_inset

.
 Many authors now use motifs to refer to induced subgraphs, i.e.
\begin_inset space ~
\end_inset

subgraphs defined by a subset of nodes and the edges between those nodes.
 Graphlets have also been used to characterise nodes through their ego-networks
 
\begin_inset CommandInset citation
LatexCommand citep
key "Cunningham2013,OCallaghan2012"
literal "false"

\end_inset

 through choosing a few graphlets to track over time and counting the frequencie
s of their appearances.
\end_layout

\begin_layout Subsubsection
Coloured graphlets
\end_layout

\begin_layout Standard
The idea of graphlets can be expanded to match two coloured subgraphs if
 they can have their nodes relabelled so that they have the same adjacency
 matrix
\begin_inset Index idx
status open

\begin_layout Plain Layout
network representations!adjacency matrix
\end_layout

\end_inset

 and each node must have the same colour as its counterpart in the other
 subgraph 
\begin_inset CommandInset citation
LatexCommand citep
key "Wernicke2006"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard
We believe this particularly pertinent for exploring networks such as ego-networ
ks, where there are two distinct classes of node, the ego
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego
\end_layout

\end_inset

 and the alters
\begin_inset Index idx
status open

\begin_layout Plain Layout
alters
\end_layout

\end_inset

.
\end_layout

\begin_layout Subsubsection
Graph isomorphism
\begin_inset CommandInset label
LatexCommand label
name "subsec:Graph-isomorphism"

\end_inset


\end_layout

\begin_layout Standard
Two graphs are isomorphic if their adjacency matrices can be made identical
 after relabelling their nodes.
 A stricter form of isomorphism demands that after this relabelling the
 graph colourings match up, i.e.
\begin_inset space \space{}
\end_inset

for two relabelled graphs 
\begin_inset Formula $\mathcal{G}$
\end_inset

 and 
\begin_inset Formula $\mathcal{G}'$
\end_inset

, 
\begin_inset Formula $\pi_{\mathcal{G}}(i)=\pi_{\mathcal{G}'}(i)\text{ }\forall i$
\end_inset

.
 There are a handful of routines for determining whether graphs are isomorphic—s
ee 
\begin_inset CommandInset citation
LatexCommand citet
key "McKay2014"
literal "false"

\end_inset

 for a detailed review—but for small graphs they have broadly the same character
istics.
 Throughout this work, we use nauty 
\begin_inset CommandInset citation
LatexCommand citep
key "McKay2014"
literal "false"

\end_inset

 as it supports colour.
 Nauty takes a graph and returns a 
\begin_inset Quotes xld
\end_inset

canonical representation
\begin_inset Quotes xrd
\end_inset

 string, canon, such that 
\begin_inset Formula 
\[
\text{canon}(\mathcal{G})=\text{canon}(\mathcal{G}')\iff\mathcal{G}\text{ and }\mathcal{G}'\text{ are isomorphic.}
\]

\end_inset


\end_layout

\begin_layout Subsubsection
Graphlet enumeration
\begin_inset CommandInset label
LatexCommand label
name "subsec:Graphlet-enumeration"

\end_inset


\end_layout

\begin_layout Standard
There are many methods for counting graphlets and motifs 
\begin_inset CommandInset citation
LatexCommand citep
key "Masoudi-Nejad2012"
literal "false"

\end_inset

.
 Since that review, some new faster techniques have emerged: 
\begin_inset CommandInset citation
LatexCommand citep
key "meira_acc-motif:_2014,ahmed2016estimation"
literal "false"

\end_inset

.
 Our main criterion was that the the routine should consider node colour
 when determining isomorphism.
 FANMOD 
\begin_inset CommandInset citation
LatexCommand citep
key "Wernicke2006"
literal "false"

\end_inset

 is the only existing algorithm we are aware of which meets this demand.
 Kavosh 
\begin_inset CommandInset citation
LatexCommand citep
key "kashani_kavosh:_2009"
literal "false"

\end_inset

 does not support colour out of the box but uses nauty which does.
 Faster methods exist but do not support graph colour which we explore in
 chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Coloured-graphlet-profiles"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Subsubsection
Graphlet profile
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlet profile
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "subsec:Graphlet-profile"

\end_inset


\end_layout

\begin_layout Standard
The result of graphlet enumeration is a graphlet profile: a mapping from
 graphlets to frequencies.
 A graphlet profile of graphlets of a certain size is a vector 
\begin_inset Formula $\boldsymbol{g}_{n}$
\end_inset

.
 There are many schools of thought on how one ought to normalise graphlet
 profiles.
 In this work, we solely use graphlet ratio profiles 
\begin_inset CommandInset citation
LatexCommand citep
key "Cunningham2013"
literal "false"

\end_inset

 such that
\begin_inset Formula 
\[
\hat{\boldsymbol{g}}_{n}=\frac{\boldsymbol{g}_{n}}{\lvert\boldsymbol{g}_{n}\rvert}\text{.}
\]

\end_inset

Other methods include z-scores 
\begin_inset CommandInset citation
LatexCommand citep
key "milo_network_2002"
literal "false"

\end_inset

 which attempt to create 
\begin_inset Quotes xld
\end_inset

significance
\begin_inset Quotes xrd
\end_inset

 profiles where graphlets which occur more often than random chance would
 suggest receive higher scores.
 We do not apply this in our work as it is computationally expensive because
 many configuration models for each graph must be generated and because
 we compare our graphs with many other graphs, as we discuss in chapters
 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Coloured-graphlet-profiles"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Construction-of-data"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Subsection
Matrix decompositions for network characterisation
\begin_inset CommandInset label
LatexCommand label
name "subsec:Matrix-decompositions"

\end_inset


\end_layout

\begin_layout Standard
As can be seen merely from the number of subsections in the network characterisa
tion section, network characterisation is a potentially very high-dimensional
 problem.
 As was introduced in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Matrix-decomposition-theory"
plural "false"
caps "false"
noprefix "false"

\end_inset

, one approach to making high-dimensional problems tractable is to apply
 matrix decompositions.
 
\begin_inset CommandInset citation
LatexCommand citet
after "pp. 221-222"
key "Costa2007"
literal "false"

\end_inset

 and 
\begin_inset CommandInset citation
LatexCommand citet
key "bounova_overview_2012"
literal "false"

\end_inset

 explore the use of PCA
\begin_inset Index idx
status open

\begin_layout Plain Layout
principal component analysis
\end_layout

\end_inset

 on data matrices constructed from standard network metrics such as the
 clustering coefficient.
 
\begin_inset CommandInset citation
LatexCommand citeauthor
key "Costa2007"
literal "false"

\end_inset

 find that they cannot distinguish between networks with the few features
 they consider and instead resort to canonical variable analysis, a close
 variant of PCA which uses labelled networks (i.e.
\begin_inset space \space{}
\end_inset

network A is type 1, network B is type 2, network C is type 1, etc.) to maximise
 the distance between different labels.
 
\begin_inset CommandInset citation
LatexCommand citeauthor
key "bounova_overview_2012"
literal "false"

\end_inset

 use rather more features and find that they can distinguish between different
 kinds of networks without needing to label them.
 
\begin_inset CommandInset citation
LatexCommand citet
key "Ronqui2015"
literal "false"

\end_inset

 take an interesting approach where they construct features from the pairwise
 correlation of node centralities within a network (e.g.
\begin_inset space \space{}
\end_inset

the correlation between node degree and local clustering coefficient) and
 they also find that they can distinguish between types of network after
 applying PCA to a data or feature matrix formed of these features.
\end_layout

\begin_layout Standard
However, our view is that these authors have somewhat missed the point of
 PCA: it can be used for feature selection and extraction, i.e.
\begin_inset space \space{}
\end_inset

an unbiased selection of the best features to explain the variance in some
 data.
 In our view, one should therefore aim to give PCA the highest-dimensional
 representation of the data that can be reasonably found so that it may
 choose the best features itself.
 Instead, the authors cited have fed PCA a selection of 
\begin_inset Quotes xld
\end_inset

human-crafted
\begin_inset Quotes xrd
\end_inset

 features.
 In the field of machine learning, this practice is termed 
\begin_inset Quotes xld
\end_inset

feature engineering
\begin_inset Quotes xrd
\end_inset

 and has been found to lead to worse results, most notably in the field
 of computer vision to the point where feature engineering is deemed 
\begin_inset Quotes xld
\end_inset

classical computer vision
\begin_inset Quotes xrd
\end_inset

 
\begin_inset CommandInset citation
LatexCommand citep
key "lee2015comparing"
literal "false"

\end_inset

.
 In part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Matrix-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

, we explore simple high-dimensional descriptions of networks that can be
 used in data matrices and see whether they still have the power to distinguish
 between types of network.
\end_layout

\begin_layout Subsection
Information-theoretic measures
\end_layout

\begin_layout Standard
There has been some progress in using metrics from information theory—a
 field which largely deals with the differences between distributions—to
 characterise networks 
\begin_inset CommandInset citation
LatexCommand citep
key "carpi_analyzing_2011,Schieber2016,schieber2017quantification"
literal "false"

\end_inset

.
 They are broadly based on the Shannon entropy
\begin_inset Index idx
status open

\begin_layout Plain Layout
Shannon entropy
\end_layout

\end_inset

, 
\begin_inset Formula $H$
\end_inset

, of a discrete probability distribution 
\begin_inset Formula $P(k)$
\end_inset

,
\begin_inset Formula 
\[
H=-\sum_{k}P(k)\text{log}_{2}P(k)\text{,}
\]

\end_inset

where 
\begin_inset Formula $P(k)$
\end_inset

 in network science is usually the degree distribution 
\begin_inset CommandInset citation
LatexCommand citep
after "p. 167"
key "Costa2007"
literal "false"

\end_inset

.
 A common metric on this space is the square root of the Jensen-Shannon
 divergence,
\begin_inset Formula 
\[
\mathcal{J}^{1/2}[P_{1},P_{2}]=\sqrt{H\left[\frac{P_{1}+P_{2}}{2}\right]-\frac{H[P_{1}]+H[P_{2}]}{2}}\text{.}
\]

\end_inset


\begin_inset CommandInset citation
LatexCommand citet
key "schieber2017quantification"
literal "false"

\end_inset

 take an approach which has a nice symmetry to the approach we take in parts
 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Graphlets"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Matrix-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

: they characterise whole networks by first characterising nodes and then
 comparing the distributions of characterised nodes using 
\begin_inset Formula $\mathcal{J}$
\end_inset

.
 
\begin_inset CommandInset citation
LatexCommand citet
key "bagrow2019information"
literal "false"

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: if you have time, look through introduction of this paper as it probably
 contains relevant papers
\end_layout

\end_inset

 apply 
\begin_inset Formula $\mathcal{J}$
\end_inset

 to 
\begin_inset Formula 
\[
b_{lk}=\text{the number of nodes who have }k\text{ nodes at distance }l\text{,}
\]

\end_inset

the network portrait 
\begin_inset CommandInset citation
LatexCommand citep
key "bagrow2008portraits"
literal "false"

\end_inset

.
 They show that small instances of Erdős-Rényi and Barabási-Albert models
 are distinguishable with intra-model distances smaller than inter-model
 distances.
 They also provide an example of it measuring changes in temporal networks,
 highlighting a key issue with 
\begin_inset Formula $\mathcal{J}$
\end_inset

: as it is always positive, there is no indication of the 
\begin_inset Quotes xld
\end_inset

direction
\begin_inset Quotes xrd
\end_inset

 of change.
 In the authors' temporal network example, the network clearly takes a 
\begin_inset Quotes xld
\end_inset

step back
\begin_inset Quotes xrd
\end_inset

 at one time step to a simpler, more regular network, before it becomes
 more complex again, which 
\begin_inset Formula $\mathcal{J}$
\end_inset

 cannot represent.
\end_layout

\begin_layout Standard
We are unaware of any attempt to compare these various techniques in a quantitat
ive manner.
\end_layout

\begin_layout Section
Community characterisation
\begin_inset CommandInset label
LatexCommand label
name "sec:Community-characterisation"

\end_inset


\end_layout

\begin_layout Standard
Many of the techniques described in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Network-characterisation"
plural "false"
caps "false"
noprefix "false"

\end_inset

 can be trivially applied to communities, that is, induced subgraphs whose
 nodes are related in some manner, by considering the subgraphs induced
 by those subsets of nodes.
 In this work, we predominantly focus on community characterisation in temporal
 networks, although many of the techniques could be applied to multilayer
\begin_inset Index idx
status open

\begin_layout Plain Layout
multilayer network
\end_layout

\end_inset

 or static networks.
\end_layout

\begin_layout Standard
The first step in community characterisation is to determine the composition
 of communities.
 There are countless routines to determine these, for example: the Louvain
 algorithm 
\begin_inset CommandInset citation
LatexCommand citep
key "Aynaud2010b"
literal "false"

\end_inset

, statistical null models 
\begin_inset CommandInset citation
LatexCommand citep
key "Bassett2013b"
literal "false"

\end_inset

, algorithms which exploit the historic community structure of the network
 
\begin_inset CommandInset citation
LatexCommand citep
key "He2015b,He2017b"
literal "false"

\end_inset

, Markov models 
\begin_inset CommandInset citation
LatexCommand citep
key "Rosvall2014b"
literal "false"

\end_inset

, semi-definite programming 
\begin_inset CommandInset citation
LatexCommand citep
key "Tantipathananandh2011b"
literal "false"

\end_inset

, gravitational relationship between nodes 
\begin_inset CommandInset citation
LatexCommand citep
key "Yin2017b"
literal "false"

\end_inset

 and temporal matrix factorisation 
\begin_inset CommandInset citation
LatexCommand citep
key "Yu2017b"
literal "false"

\end_inset

, amongst others.
 Machine learning techniques 
\begin_inset CommandInset citation
LatexCommand citep
key "Savic2019h"
literal "false"

\end_inset

, genetic algorithms 
\begin_inset CommandInset citation
LatexCommand citep
key "Folino2014b"
literal "false"

\end_inset

, consensus clustering 
\begin_inset CommandInset citation
LatexCommand citep
key "Aynaud2010b"
literal "false"

\end_inset

 and tensor factorisation 
\begin_inset CommandInset citation
LatexCommand citep
key "Gauvin2014b,hung2016distributed,Verma2017"
literal "false"

\end_inset

 have only recently been used for the detection of communities in temporal
 networks.
\end_layout

\begin_layout Standard
Of these, tensor factorisation has a number of advantages.
 It is a kind of soft clustering, where nodes have membership strengths
 to multiple communities, and so it lends itself naturally to situations
 where nodes are in multiple communities or community membership is not
 well-defined.
 It can deal with clustering across multilayer networks and is especially
 useful at characterising the communities of temporal networks, where it
 can produce measures of the activity level of each community over time
 
\begin_inset CommandInset citation
LatexCommand citep
key "Gauvin2014b"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard
In part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Tensor-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

, we apply tensor decomposition or factorisation, as was introduced in section
 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:bg-Tensor-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 We chose this approach above all the others because it simultaneously performs
 soft clustering of the network—i.e.
 nodes are given 
\begin_inset Quotes xld
\end_inset

strengths
\begin_inset Quotes xrd
\end_inset

 of membership to all communities rather than only being in a single community—a
nd characterises these communities by giving their activity over time, which
 no other technique can do.
\end_layout

\begin_layout Standard
Previous works applying tensor decompositions to temporal networks have
 had very clearly defined 
\begin_inset Quotes xld
\end_inset

real
\begin_inset Quotes xrd
\end_inset

 communities—such as pupils separated into classes at school 
\begin_inset CommandInset citation
LatexCommand citep
key "Gauvin2014b"
literal "false"

\end_inset

—or have not been concerned with picking an optimal number of components
 
\begin_inset CommandInset citation
LatexCommand citep
key "kang2012gigatensor"
literal "false"

\end_inset

.
 Tensor decompositions have also been used for community detection in static
 multilayer networks.
 Generally in these works, the 
\begin_inset Quotes xld
\end_inset

ground truth
\begin_inset Quotes xrd
\end_inset

 (i.e.
\begin_inset space \space{}
\end_inset


\begin_inset Quotes xld
\end_inset

real
\begin_inset Quotes xrd
\end_inset

) communities have been known.
 We consider how we can extend this technique to areas where there are no
 ground truth communities in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Locally-optimal-numbers"
plural "false"
caps "false"
noprefix "false"

\end_inset

, which we apply in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Software"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Section
Edge characterisation
\end_layout

\begin_layout Standard
We do not consider the characterisation of edges in this thesis, but we
 mention it here for completeness.
 Edges share many similarities with nodes: edge betweenness centrality follows
 readily from node betweenness centrality and the ego-network of an edge
 can be defined with ease.
 Other more exotic measures have been explored, such as those based on 
\begin_inset Formula $\kappa$
\end_inset

-path centrality, an extension of betweenness centrality to 
\begin_inset Quotes xld
\end_inset

fairly short
\begin_inset Quotes xrd
\end_inset

 rather than shortest paths 
\begin_inset CommandInset citation
LatexCommand citep
key "de2012novel"
literal "false"

\end_inset

.
\end_layout

\begin_layout Section
Complex networks
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:Complex-networks"

\end_inset


\end_layout

\begin_layout Standard
Complex networks are large networks with non-trivial topologies: e.g.
 with degree mixing matrices that exhibit assortativity or disassortativity,
 community structure where there are groups of nodes within which nodes
 more densely linked to each other than outside the groups, and average
 path lengths that are not functions of the number of nodes in the network,
 or at most, grow very slowly with the number of nodes in the network.
 A complex networked system consists of many interacting agents that communicate
 with each other over a web of complex interconnections.
 In the formalism of network science, the agents, or 
\begin_inset Quotes xld
\end_inset

things
\begin_inset Quotes xrd
\end_inset

, are called nodes, which are connected together by links or edges; other
 formalisms such as multi-agent systems could be applied but are not explored
 in this work.
\end_layout

\begin_layout Subsection
Dynamics (of topology)
\end_layout

\begin_layout Standard
Complex networks are often formed by dynamical processes; they grow and
 change over time.
 Complex networks are generally created by many agents making small decisions,
 such as websites choosing which other websites to link to, or people choosing
 which other people to send emails to, or neurons making links with nearby
 neurons.
 
\begin_inset CommandInset citation
LatexCommand citet
key "Strogatz2001"
literal "true"

\end_inset

 and 
\begin_inset CommandInset citation
LatexCommand citet
key "gross_adaptive_2009"
literal "false"

\end_inset

 give a very good introduction to the field for the uninitiated reader.
\end_layout

\begin_layout Standard
Complex networks often have complex dynamics on each node, influenced by
 the topology—for example, in a social network, each person could be considered
 
\begin_inset Quotes xld
\end_inset

ill
\begin_inset Quotes xrd
\end_inset

 or 
\begin_inset Quotes xld
\end_inset

healthy
\begin_inset Quotes xrd
\end_inset

 and the likelihood of each state depends on the number of 
\begin_inset Quotes xld
\end_inset

ill
\begin_inset Quotes xrd
\end_inset

 and 
\begin_inset Quotes xld
\end_inset

healthy
\begin_inset Quotes xrd
\end_inset

 friends they are in contact with; this is the SI (susceptible-infected)
 model.
 Interested readers should see 
\begin_inset CommandInset citation
LatexCommand citet
key "danon2011networks"
literal "false"

\end_inset

 and 
\begin_inset CommandInset citation
LatexCommand citet
key "Pastor2015"
literal "false"

\end_inset

 for detailed reviews.
 This is not something we have room to consider explicitly in this work,
 although it is of course considered implicitly in many chapters where we
 consider the real-world characteristics of nodes and theorise about how
 the topology might affect these characteristics.
\end_layout

\begin_layout Subsection
Network models
\begin_inset CommandInset label
LatexCommand label
name "subsec:Network-models"

\end_inset


\end_layout

\begin_layout Standard
Dealing with real-world networks poses a number of issues, such as being
 inflexible in the number of nodes there are or poor labelling.
 Network models are convenient for the generation of synthetic networks
 with specific parameters and we use them extensively in part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Matrix-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

 as paragons by which we can test the quality of our characterisation.
\end_layout

\begin_layout Subsubsection
Random networks
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!random networks
\end_layout

\end_inset


\end_layout

\begin_layout Standard
A random network is created by choosing a random subset of all of the possible
 links between nodes.
 The links are each present with some probability, 
\begin_inset Formula $p$
\end_inset

.
 This model therefore has two parameters: 
\begin_inset Formula $G(n,p)$
\end_inset

, where 
\begin_inset Formula $n$
\end_inset

 is the number of nodes
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
It should be noted that Erdős and Rényi originally formulated their model
 with a number of nodes and a number of edges as its parameters.
 This is equivalent to having an edge probability which we feel captures
 a more important aspect of the model.
\end_layout

\end_inset

.
 This type of network was first introduced by 
\begin_inset CommandInset citation
LatexCommand citet
key "Erdos1959"
literal "true"

\end_inset

 and can be considered the 
\begin_inset Quotes xld
\end_inset

mother
\begin_inset Quotes xrd
\end_inset

 of all research into complex networks, as they were the first people to
 consider large, random networks, as opposed to the traditional, regular
 graphs that were studied before.
 These networks have many interesting properties, such as the sudden appearance
 of a giant component which contains a large fraction of the nodes as 
\begin_inset Formula $p$
\end_inset

 is increased.
 An example implementation of the model is given in algorithm 
\begin_inset CommandInset ref
LatexCommand ref
reference "alg:Erdos-Renyi-random-graph"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 The degree distribution and degree mixing matrix of an instance of an Erdős-Rén
yi model are shown in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:er-deg-dist-mix"
plural "false"
caps "false"
noprefix "false"

\end_inset

 where the Poisonnian nature of both can be seen.
\end_layout

\begin_layout Standard
\begin_inset Float algorithm
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{minted}[linenos=false,mathescape=true]{python}
\end_layout

\begin_layout Plain Layout

# $G(n,p)$
\end_layout

\begin_layout Plain Layout

def Erdős_Rényi_graph(n, p, seed=None, directed=False):
\end_layout

\begin_layout Plain Layout

    if directed:
\end_layout

\begin_layout Plain Layout

        edges = itertools.permutations(range(n), 2)
\end_layout

\begin_layout Plain Layout

        G = nx.DiGraph()
\end_layout

\begin_layout Plain Layout

    else:
\end_layout

\begin_layout Plain Layout

        edges = itertools.combinations(range(n), 2)
\end_layout

\begin_layout Plain Layout

        G = nx.Graph()
\end_layout

\begin_layout Plain Layout

    G.add_nodes_from(range(n))
\end_layout

\begin_layout Plain Layout

    if p <= 0:
\end_layout

\begin_layout Plain Layout

        return G
\end_layout

\begin_layout Plain Layout

    if p >= 1:
\end_layout

\begin_layout Plain Layout

        return complete_graph(n, create_using=G)
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    for e in edges:
\end_layout

\begin_layout Plain Layout

        if seed.random() < p:
\end_layout

\begin_layout Plain Layout

            G.add_edge(*e)
\end_layout

\begin_layout Plain Layout

    return G
\end_layout

\begin_layout Plain Layout


\backslash
end{minted}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!Erdős-Rényi
\end_layout

\end_inset


\begin_inset Argument 1
status open

\begin_layout Plain Layout
Erdős-Rényi random graph model implementation in Python 3
\end_layout

\end_inset

Erdős-Rényi random graph model implementation in Python 3 
\begin_inset CommandInset citation
LatexCommand citep
key "schult_exploring_2008"
literal "false"

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "alg:Erdos-Renyi-random-graph"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/background/er_10k_4bps_deg_dist_and_hist.pdf
	display false
	width 100text%
	groupId toowide

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:er-deg-dist-mix"

\end_inset

The degree distribution (left) and the degree mixing matrix (right) of an
 instance of an Erdős-Rényi
\begin_inset Argument 1
status open

\begin_layout Plain Layout
The degree distribution and the degree mixing matrix of an Erdős-Rényi model
 network
\end_layout

\end_inset

 model with 10,000 nodes and 20,000 edges
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!small-world
\end_layout

\end_inset

Small-world
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!small-world|seealso
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\end_layout

\end_inset

network models!Watts-Strogatz
\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
A small-world network is one whose average path length, 
\begin_inset Formula $L$
\end_inset

, grows with the network such that 
\begin_inset Formula $L\propto\log N$
\end_inset

, whilst maintaining a high clustering coefficient.
 The first people to propose a model that could generate such a network
 were 
\begin_inset CommandInset citation
LatexCommand citet
key "watts_collective_1998"
literal "true"

\end_inset

, whose network starts out as a nearest-neighbour ring network.
 Then, a random shortcut is created from each node with a set probability.
 This then also has three parameters: 
\begin_inset Formula $G(n,p,m)$
\end_inset

 where 
\begin_inset Formula $n$
\end_inset

 is the number of nodes, 
\begin_inset Formula $p$
\end_inset

 is the probability of each edge being rewired a random node and 
\begin_inset Formula $m$
\end_inset

 is the initial degree of the nodes.
 These networks are called small-world because their path efficiencies are
 much greater than the corresponding nearest-neighbour network.
 An example implementation of the model is given in algorithm 
\begin_inset CommandInset ref
LatexCommand ref
reference "alg:watts-strogatz"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 The degree distribution and degree mixing matrix of an instance of a Watts-Stro
gatz model are shown in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:ws-deg-dist-mix"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 The difference with figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:er-deg-dist-mix"
plural "false"
caps "false"
noprefix "false"

\end_inset

 is striking; the distributions are very tight.
\end_layout

\begin_layout Standard
\begin_inset Float algorithm
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{minted}[linenos=false,mathescape=true]{python}
\end_layout

\begin_layout Plain Layout

# $G(n,p,m)$=watts_strogatz_graph(n,m,p)
\end_layout

\begin_layout Plain Layout

def watts_strogatz_graph(n, k, p, seed=None):
\end_layout

\begin_layout Plain Layout

    if k >= n:
\end_layout

\begin_layout Plain Layout

        raise NetworkXError("k>=n, choose smaller k or larger n")
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    G = Graph()
\end_layout

\begin_layout Plain Layout

    nodes = list(range(n))  # nodes are labeled 0 to n-1
\end_layout

\begin_layout Plain Layout

    # connect each node to k/2 neighbors
\end_layout

\begin_layout Plain Layout

    for j in range(1, k // 2 + 1):
\end_layout

\begin_layout Plain Layout

        targets = nodes[j:] + nodes[0:j]
\end_layout

\begin_layout Plain Layout

        G.add_edges_from(zip(nodes, targets))
\end_layout

\begin_layout Plain Layout

    # rewire edges from each node
\end_layout

\begin_layout Plain Layout

    # loop over all nodes in order (label) and neighbors in order (distance)
\end_layout

\begin_layout Plain Layout

    # no self loops or multiple edges allowed
\end_layout

\begin_layout Plain Layout

    for j in range(1, k // 2 + 1):  # outer loop is neighbors
\end_layout

\begin_layout Plain Layout

        targets = nodes[j:] + nodes[0:j] 
\end_layout

\begin_layout Plain Layout

        # inner loop in node order
\end_layout

\begin_layout Plain Layout

        for u, v in zip(nodes, targets):
\end_layout

\begin_layout Plain Layout

            if seed.random() < p:
\end_layout

\begin_layout Plain Layout

                w = seed.choice(nodes)
\end_layout

\begin_layout Plain Layout

                # Enforce no self-loops or multiple edges
\end_layout

\begin_layout Plain Layout

                while w == u or G.has_edge(u, w):
\end_layout

\begin_layout Plain Layout

                    w = seed.choice(nodes)
\end_layout

\begin_layout Plain Layout

                    if G.degree(u) >= n - 1:
\end_layout

\begin_layout Plain Layout

                        break  # skip this rewiring
\end_layout

\begin_layout Plain Layout

                else:
\end_layout

\begin_layout Plain Layout

                    G.remove_edge(u, v)
\end_layout

\begin_layout Plain Layout

                    G.add_edge(u, w)
\end_layout

\begin_layout Plain Layout

    return G
\end_layout

\begin_layout Plain Layout


\backslash
end{minted}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!Watts-Strogatz
\end_layout

\end_inset

Watts-Strogatz small-world graph model implementation in Python 3
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Watts-Strogatz small-world graph model implementation in Python 3
\end_layout

\end_inset

 
\begin_inset CommandInset citation
LatexCommand citep
key "schult_exploring_2008"
literal "false"

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "alg:watts-strogatz"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/background/ws_10k_4_deg_dist_and_hist.pdf
	display false
	width 100text%
	groupId toowide

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:ws-deg-dist-mix"

\end_inset


\begin_inset Argument 1
status open

\begin_layout Plain Layout
The degree distribution and the degree mixing matrix of a Watts-Strogatz
 model network
\end_layout

\end_inset

The degree distribution (left) and the degree mixing matrix (right) of an
 instance of a Watts-Strogatz model with 10,000 nodes and 20,000 edges
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!scale-free
\end_layout

\end_inset

Scale-free
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!scale-free|seealso
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\end_layout

\end_inset

network models!Barabási-Albert
\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Scale-free networks are networks with power-law degree distributions, 
\begin_inset Formula $P(k)\propto k^{-\alpha}$
\end_inset

, where 
\begin_inset Formula $\alpha$
\end_inset

 is a constant and generally lies such that 
\begin_inset Formula $2<\alpha<3$
\end_inset

, with so-called 
\begin_inset Quotes xld
\end_inset

fat tails
\begin_inset Quotes xrd
\end_inset

 where a small number of nodes have a very large number of links 
\begin_inset CommandInset citation
LatexCommand citep
key "barabasi1999emergence"
literal "false"

\end_inset

.
 They are called scale-free because their degree distributions look the
 same at different scales—e.g.
 if one takes the 
\begin_inset Formula $k$
\end_inset

-core of a scale-free network, that is, a subgraph where every node has
 at least degree 
\begin_inset Formula $k$
\end_inset

, the resultant network is also a scale-free network.
 The most famous model of these networks, and the only scale-free model
 we consider in this work, is the Barabási-Albert model.
 This model grows a network from 
\begin_inset Formula $m$
\end_inset

 unconnected nodes.
 Nodes are added and connected to 
\begin_inset Formula $m$
\end_inset

 pre-existing nodes chosen with probability proportional to the degree of
 those nodes.
 These models therefore have two main parameters: 
\begin_inset Formula $G(n,m)$
\end_inset

 where 
\begin_inset Formula $n$
\end_inset

 is the number of nodes and 
\begin_inset Formula $m$
\end_inset

 is the number of links to add with each node.
 An implementation is shown in algorithm 
\begin_inset CommandInset ref
LatexCommand ref
reference "alg:barabasi-albert"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 The degree distribution and degree mixing matrix of an instance of a Barabási-A
lbert 
\begin_inset CommandInset citation
LatexCommand citep
key "barabasi1999emergence"
literal "false"

\end_inset

 model are shown in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:ba-deg-dist-mix"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 The contrast with figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:er-deg-dist-mix"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:ws-deg-dist-mix"
plural "false"
caps "false"
noprefix "false"

\end_inset

 is immediate—there are very clear 
\begin_inset Quotes xld
\end_inset

fat tails
\begin_inset Quotes xrd
\end_inset

 where a handful of nodes have a very large number of links, and are fairly
 likely to be linked to by nodes of all degrees despite their small number.
 Recent research has suggested that true scale-free networks are rarer than
 once thought 
\begin_inset CommandInset citation
LatexCommand citep
key "house2015testing,broido2019scale"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float algorithm
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{minted}[linenos=false,mathescape=true]{python}
\end_layout

\begin_layout Plain Layout

# $G(n,m)$
\end_layout

\begin_layout Plain Layout

def Barabási_albert_graph(n, m, seed=None):
\end_layout

\begin_layout Plain Layout

    # Add m initial nodes (m0 in Barabási-speak)
\end_layout

\begin_layout Plain Layout

    G = empty_graph(m)
\end_layout

\begin_layout Plain Layout

    # Target nodes for new edges
\end_layout

\begin_layout Plain Layout

    targets = list(range(m))
\end_layout

\begin_layout Plain Layout

    # List of existing nodes, with nodes repeated
\end_layout

\begin_layout Plain Layout

    # once for each adjacent edge
\end_layout

\begin_layout Plain Layout

    repeated_nodes = []
\end_layout

\begin_layout Plain Layout

    # Start adding the other n-m nodes.
 The first node is m.
\end_layout

\begin_layout Plain Layout

    source = m
\end_layout

\begin_layout Plain Layout

    while source < n:
\end_layout

\begin_layout Plain Layout

        # Add edges to m nodes from the source.
\end_layout

\begin_layout Plain Layout

        G.add_edges_from(zip([source] * m, targets))
\end_layout

\begin_layout Plain Layout

        # Add one node to the list for each new edge just created.
\end_layout

\begin_layout Plain Layout

        repeated_nodes.extend(targets)
\end_layout

\begin_layout Plain Layout

        # And the new node "source" has m edges to add to the list.
\end_layout

\begin_layout Plain Layout

        repeated_nodes.extend([source] * m)
\end_layout

\begin_layout Plain Layout

        # Now choose m unique nodes from the existing nodes
\end_layout

\begin_layout Plain Layout

        # Pick uniformly from repeated_nodes
\end_layout

\begin_layout Plain Layout

        # (preferential attachment)
\end_layout

\begin_layout Plain Layout

        targets = _random_subset(repeated_nodes, m, seed)
\end_layout

\begin_layout Plain Layout

        source += 1
\end_layout

\begin_layout Plain Layout

    return G
\end_layout

\begin_layout Plain Layout


\backslash
end{minted}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Barabási-Albert scale-free graph model implementation in Python 3
\end_layout

\end_inset

Barabási-Albert
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!Barabási-Albert
\end_layout

\end_inset

 scale-free graph model implementation in Python 3 
\begin_inset CommandInset citation
LatexCommand citep
key "schult_exploring_2008"
literal "false"

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "alg:barabasi-albert"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/background/ba_10k_2_deg_dist_and_hist.pdf
	display false
	width 100text%
	groupId toowide

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:ba-deg-dist-mix"

\end_inset

The
\begin_inset Argument 1
status open

\begin_layout Plain Layout
The degree distribution and the degree mixing matrix of a Barabási-Albert
 model network
\end_layout

\end_inset

 degree distribution (left) and the degree mixing matrix (right) of an instance
 of a Barabási-Albert model with 10,000 nodes and 20,000 edges.
 As this is a scale-free network, the majority of the nodes are in the bottom
 left of each subfigure, especially with the degree mixing matrix
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
Configuration model
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!configuration model
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "subsec:Configuration-model"

\end_inset


\end_layout

\begin_layout Standard
The configuration model 
\begin_inset CommandInset citation
LatexCommand citep
key "molloy_critical_1995"
literal "false"

\end_inset

 takes a degree sequence, 
\begin_inset Formula $\{k_{i}|i\in\mathcal{E}\}$
\end_inset

, and returns a randomly rewired graph with the same degree sequence.
 These models are predominantly used to generate random graphs similar to
 other graphs with a given degree distribution.
\end_layout

\begin_layout Subsection
Real-world networks
\begin_inset Index idx
status open

\begin_layout Plain Layout
real-world networks
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "subsec:Real-world-complex"

\end_inset


\end_layout

\begin_layout Standard
Real-world networks are a key component of this thesis: complex networks
 are predominantly of interest to us because they represent real-world systems.
 We refer back to them in every part: part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Graphlets"
plural "false"
caps "false"
noprefix "false"

\end_inset

 explores an academic co-authorship network, part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Matrix-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

 characterises a covert collaboration network and a selection of food webs
 and part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Tensor-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

 analyses a number of temporal software dependency networks and a temporal
 linguistic network.
\end_layout

\begin_layout Subsubsection
Collaboration networks
\end_layout

\begin_layout Standard
There are many types of social collaboration network.
 In this thesis, we only consider three.
\end_layout

\begin_layout Paragraph
Academic co-authorship networks
\begin_inset CommandInset label
LatexCommand label
name "par:Academic-coauthorship-networks"

\end_inset


\end_layout

\begin_layout Standard
describe academics who have written papers or other publications together.
 In a co-authorship network, the nodes are individual authors and the links
 between the authors represent collaborations.
 
\end_layout

\begin_layout Standard
\begin_inset Index idx
status open

\begin_layout Plain Layout
real-world networks!co-authorship network
\end_layout

\end_inset

One of the main challenges with real-world co-authorship networks is disambiguat
ion: the ability to identify when two authors with the same name are different
 people.
 Datasets exist where this procedure has been completed 
\begin_inset CommandInset citation
LatexCommand citep
key "Zhuang2013"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard
As with other temporal networks made from data where events are recorded
 as instantaneous but in reality have some unknown duration, a window size
 must be chosen for which the events are extended.
 For example, a 3-year window size means that a co-authorship registered
 in a certain year is extended to also include the two years before it;
 many collaboration network datasets effectively set the window size to
 infinity and consider the network as if it were static.
 This author's own co-authorship network is shown in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:This-author's-co-authorship"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
For a detailed overview, see 
\begin_inset CommandInset citation
LatexCommand cite
key "Kumar2015"
literal "false"

\end_inset

.
 Some work has been conducted with graphlets and motifs on co-authorship
 networks 
\begin_inset CommandInset citation
LatexCommand citep
key "Krumov2011,Choobdar2012,Chakraborty2015"
literal "false"

\end_inset

, but coloured graphlets have been neglected; some authors have considered
 the link between node centralities, career length and citation counts 
\begin_inset CommandInset citation
LatexCommand citep
key "Li2013"
literal "false"

\end_inset

.
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/my_coauthor_network.png
	display false
	height 30pheight%

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
This author's co-authorship 2-ego-network
\begin_inset Argument 1
status open

\begin_layout Plain Layout
This author's co-authorship 2-ego-network
\end_layout

\end_inset

 with a window size of 
\begin_inset Formula $\infty$
\end_inset

 visualised using VOSViewer 
\begin_inset CommandInset citation
LatexCommand citep
key "van2009software"
literal "false"

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:This-author's-co-authorship"

\end_inset

.
 Node size is proportional to number of papers published with people in
 the network shown
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Paragraph
Clandestine collaboration networks
\begin_inset Index idx
status open

\begin_layout Plain Layout
real-world networks!Suffragette
\end_layout

\end_inset


\end_layout

\begin_layout Standard
are networks of human collaboration operating in a covert fashion.
 In this work, we briefly consider the network of Suffragettes, women who
 campaigned for universal suffrage in the UK 
\begin_inset CommandInset citation
LatexCommand citep
key "edwards_measures_2009"
literal "false"

\end_inset

 in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Application-of-PCA"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 See 
\begin_inset CommandInset citation
LatexCommand citet
key "broccatelli_temporal_2016"
literal "false"

\end_inset

 for a thorough treatment.
\end_layout

\begin_layout Paragraph
Software dependency networks
\begin_inset Index idx
status open

\begin_layout Plain Layout
real-world networks!software dependency network
\end_layout

\end_inset


\end_layout

\begin_layout Standard
are directed networks of collaboration between computer programmes 
\begin_inset CommandInset citation
LatexCommand citep
key "Decan2018h"
literal "false"

\end_inset

.
 Modern software development is often a collaborative effort involving many
 authors through the re-use and sharing of code through software libraries.
 Many of these libraries and software packages are open-source and developed
 in the open on sites such as 
\family typewriter
GitHub
\family default
, so there is a large amount of data available about these networks.
\end_layout

\begin_layout Standard
We explore these networks in detail in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Linguistic-evolution"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Subsubsection
Semantic networks
\begin_inset CommandInset label
LatexCommand label
name "subsec:Semantic-networks"

\end_inset


\end_layout

\begin_layout Standard
A semantic network is one where the nodes are individual meanings of a word—lemm
as, in the jargon.
 For example, here are two of the lemmas relating to 
\begin_inset Quotes xld
\end_inset

mean
\begin_inset Quotes xrd
\end_inset

 from the Oxford English Dictionary (OED, 
\begin_inset CommandInset citation
LatexCommand citealp
key "Coleridge2019"
literal "false"

\end_inset

):
\end_layout

\begin_layout Quote
mean: The average of a set of numerical values, as calculated by adding
 them together and dividing by the number of terms in the set.
\end_layout

\begin_layout Quote
mean: Inferior in rank or quality; unpleasant.
\end_layout

\begin_layout Standard
which leads to one of this author's favourite jokes:
\end_layout

\begin_layout Quote
The University of California Statistics Department: where mean is normal
 and deviation standard.
\end_layout

\begin_layout Standard
A hierarchical linguistic network links lemmas in a categorical fashion;
 for example, 
\begin_inset Quotes xld
\end_inset

the world
\begin_inset Quotes xrd
\end_inset

 would have an outgoing link to 
\begin_inset Quotes xld
\end_inset

Asia
\begin_inset Quotes xrd
\end_inset

 which in turn would have an outgoing link to 
\begin_inset Quotes xld
\end_inset

China
\begin_inset Quotes xrd
\end_inset

 and 
\begin_inset Quotes xld
\end_inset

brick
\begin_inset Quotes xrd
\end_inset

 may be pointed to by 
\begin_inset Quotes xld
\end_inset

house
\begin_inset Quotes xrd
\end_inset

 which might be pointed to by 
\begin_inset Quotes xld
\end_inset

shelter
\begin_inset Quotes xrd
\end_inset

.
 See 
\begin_inset CommandInset citation
LatexCommand citet
key "Sole2010"
literal "false"

\end_inset

 for a good introduction to language networks.
 We consider a temporal linguistic network in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Linguistic-evolution"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
The majority of work studying semantic networks is concerned with improving
 how machines can 
\begin_inset Quotes xld
\end_inset

understand
\begin_inset Quotes xrd
\end_inset

 it by changing the way semantic networks are presented to machine learning
 techniques 
\begin_inset CommandInset citation
LatexCommand citep
key "Anisimov2014,Nakatsuji2016,Piao2017"
literal "false"

\end_inset

.
 
\begin_inset CommandInset citation
LatexCommand citet
key "kay2009historical"
literal "false"

\end_inset

 provide similar data to that which we use in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Linguistic-evolution"
plural "false"
caps "false"
noprefix "false"

\end_inset

 
\begin_inset CommandInset citation
LatexCommand citep
key "Coleridge2019"
literal "false"

\end_inset

.
 Some analysis has been performed on this with the intent of uncovering
 interesting patterns in the language for humans rather than machines 
\begin_inset CommandInset citation
LatexCommand citep
key "alexander2012patchworks,alexander2014spread"
literal "false"

\end_inset

, but no-one has really applied state-of-the-art network science techniques
 to this data in an attempt to understand its evolution.
\end_layout

\begin_layout Standard
For a detailed survey on detecting semantic change, such that temporal semantic
 networks can be created, see 
\begin_inset CommandInset citation
LatexCommand citet
key "Tahmasebi2018"
literal "false"

\end_inset

.
\end_layout

\begin_layout Subsubsection
Ecological networks
\end_layout

\begin_layout Standard
Ecological networks represent the interactions between species.
 In this work, we predominantly consider food webs where 
\begin_inset Formula $a_{i\leftarrow j}=1\iff\text{species }i\text{ predates species }j$
\end_inset

; the direction of a link shows the flow of energy in the food web 
\begin_inset CommandInset citation
LatexCommand citep
key "montoya2006ecological"
literal "false"

\end_inset

.
 This leads to several interesting properties: cycles are very rare and
 the diameters of the networks, closely related to the maximum trophic level
\begin_inset Index idx
status open

\begin_layout Plain Layout
trophic level
\end_layout

\end_inset

, tend to be small due to the unavoidable loss of energy.
 Recent work has suggested that the lack of cycles could be linked to trophic
 levels 
\begin_inset CommandInset citation
LatexCommand citep
key "johnson2017looplessness"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard
In this work, the real-world food webs we consider are provided by 
\begin_inset CommandInset citation
LatexCommand citet
key "Thompson"
literal "false"

\end_inset

.
 We explore similarities between these and other networks in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Application-of-PCA"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Section
Statistical methods
\begin_inset CommandInset label
LatexCommand label
name "sec:Statistical-methods"

\end_inset


\end_layout

\begin_layout Standard
When dealing with noisy real-world data and many parameters, it is often
 beneficial to apply statistical techniques to ensure that what appear to
 be firm results are unlikely to be due to random chance.
 The methods from this section are applied particularly in part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Graphlets"
plural "false"
caps "false"
noprefix "false"

\end_inset

 where incautious analysis of our high-dimensional graphlet profile
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlet profile
\end_layout

\end_inset

s would put us at a high-risk of finding spurious correlations.
\end_layout

\begin_layout Subsection
Multiple linear regression
\begin_inset CommandInset label
LatexCommand label
name "subsec:Multiple-linear-regression"

\end_inset


\end_layout

\begin_layout Standard
Multiple linear or multilinear regression
\begin_inset Index idx
status open

\begin_layout Plain Layout
multiple linear regression
\end_layout

\end_inset

 is a simple method for investigating the relationship between many continuous
 so-called explanatory variables 
\begin_inset Formula $x_{i}$
\end_inset

 and a single dependent variable, 
\begin_inset Formula $y$
\end_inset

, with
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
y=\Sigma_{i}\beta_{i}x_{i}\text{,}
\]

\end_inset


\end_layout

\begin_layout Standard
where 
\begin_inset Formula $\beta_{i}$
\end_inset

 are coefficients to be determined, commonly by the least-squares method
 
\begin_inset CommandInset citation
LatexCommand citep
key "montgomery2012"
literal "false"

\end_inset

.
 
\begin_inset Formula $\beta_{\text{reduced}}$
\end_inset

 coefficients, the coefficients multiplied by the standard deviation of
 each explanatory variable, are often reported to ease interpretation of
 the coefficients.
 Each reduced coefficient corresponds to the change in the dependent variable
 one could reasonably expect as a result of increasing the corresponding
 explanatory variable from a figure representative of the samples with a
 low amount of that variable, to a figure representative of samples with
 a high amount of that variable.
 
\end_layout

\begin_layout Subsection
Statistical significance
\end_layout

\begin_layout Standard
When sampling data from a population—i.e.
\begin_inset space ~
\end_inset

choosing a subset of that population—and performing an experiment on that
 sample, it is possible that any conclusions drawn from that sample are
 simply due to the composition of the sample and are not representative
 of the population as a whole.
 For example, assume we have a population represented by a matrix
\begin_inset Formula 
\[
P=[p_{ij}]=[\delta_{ij}]\text{,}
\]

\end_inset

where 
\begin_inset Formula $\delta_{ij}$
\end_inset

 is the Kronecker delta, and we happen to sample the diagonal of the matrix.
 We would be led to believe that 
\begin_inset Formula $p_{ij}=1$
\end_inset

.
 Statistical significance tests aim to quantify the likelihood of us being
 misled in such a manner 
\begin_inset CommandInset citation
LatexCommand citep
key "fisher1925statistical"
literal "false"

\end_inset

.
 More rigorously, a statistical significance test calculates the probability
 that an experiment would give a result at least as extreme at the one received
 given that the null hypothesis is true; this probability is called the
 
\begin_inset Quotes xld
\end_inset

p-value
\begin_inset Quotes xrd
\end_inset

.
 The null hypothesis will vary according to the experiment performed but
 generally corresponds to a 
\begin_inset Quotes xld
\end_inset

negative
\begin_inset Quotes xrd
\end_inset

 result.
 For example, if we propose that two variables are linked such that 
\begin_inset Formula $y=\beta x+c$
\end_inset

, the null hypothesis is that 
\begin_inset Formula $\beta=0$
\end_inset

, that there is no link.
 One generally chooses a threshold for the p-value
\begin_inset Index idx
status open

\begin_layout Plain Layout
p-value
\end_layout

\end_inset

 below which results are accepted or rejected; such a value is called the
 significance level.
 A significance level of 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
liningnums{5}
\end_layout

\end_inset

% is generally used for preliminary studies, and corresponds to a confidence
 level of 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
liningnums{95}
\end_layout

\end_inset

%, meaning that 19 times out of 20 a result is real and not due to sampling
 error.
\end_layout

\begin_layout Standard
We apply a specific implementation of a significance test, the 
\begin_inset Index idx
status open

\begin_layout Plain Layout
F-test
\end_layout

\end_inset

F-test 
\begin_inset CommandInset citation
LatexCommand citep
key "bates_2019"
literal "false"

\end_inset

, to test whether the coefficients in our linear regression are statistically
 significant.
 Its p-values are sometimes referred to as 
\begin_inset Formula $Pr(>|t|)$
\end_inset

.
\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: read up on this
\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Bonferroni correction
\begin_inset Index idx
status open

\begin_layout Plain Layout
Bonferroni correction
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "subsec:Bonferroni-correction"

\end_inset


\end_layout

\begin_layout Standard
The significance levels mentioned above are only valid in the context of
 a single, isolated, hypothesis test (where the null hypothesis is that
 the coefficient is zero).
 Testing many hypotheses simultaneously, for example with a multilinear
 regression, greatly increases the probability of false positives.
 We therefore apply a Bonferroni correction
\begin_inset Index idx
status open

\begin_layout Plain Layout
Bonferroni correction
\end_layout

\end_inset

 to guard against this—we check the p-values against a corrected significance
 level of 
\begin_inset Formula $\frac{\alpha}{n}$
\end_inset

, where 
\begin_inset Formula $\alpha$
\end_inset

 is the desired p-value and 
\begin_inset Formula $n$
\end_inset

 is the number of hypotheses tested 
\begin_inset CommandInset citation
LatexCommand citep
key "Bonferroni1950"
literal "false"

\end_inset

.
\end_layout

\begin_layout Section
Software used
\begin_inset CommandInset label
LatexCommand label
name "sec:Software-used"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Tabular
<lyxtabular version="3" rows="7" columns="2">
<features booktabs="true" tabularvalignment="middle" tabularwidth="98text%">
<column alignment="center" valignment="middle" width="35text%">
<column alignment="center" valignment="middle" width="60text%">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Activity
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Software used
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
General programming
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Julia 
\begin_inset CommandInset citation
LatexCommand citep
key "bezanson2012julia"
literal "false"

\end_inset

, Python 
\begin_inset CommandInset citation
LatexCommand citep
key "van1995python"
literal "false"

\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Exploratory data analysis
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Jupyter notebook 
\begin_inset CommandInset citation
LatexCommand citep
key "kluyver2016jupyter"
literal "false"

\end_inset

, Julia
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Communication
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\SpecialChar LaTeX
, \SpecialChar LyX

\begin_inset Foot
status open

\begin_layout Plain Layout
\begin_inset CommandInset href
LatexCommand href
target "https://www.lyx.org/"
literal "false"

\end_inset


\end_layout

\end_inset

, natbib, ebgaramond, fancyhdr, beamer, microtype, pandoc
\begin_inset Foot
status open

\begin_layout Plain Layout
\begin_inset CommandInset href
LatexCommand href
target "https://pandoc.org/"
literal "false"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Visualisation
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Plots.jl
\begin_inset Foot
status open

\begin_layout Plain Layout
\begin_inset CommandInset href
LatexCommand href
target "https://github.com/JuliaPlots/Plots.jl"

\end_inset


\end_layout

\end_inset

, GR
\begin_inset Foot
status open

\begin_layout Plain Layout
\begin_inset CommandInset href
LatexCommand href
target "https://github.com/sciapp/gr"

\end_inset


\end_layout

\end_inset

, Gadfly.jl
\begin_inset Foot
status open

\begin_layout Plain Layout
\begin_inset CommandInset href
LatexCommand href
target "https://github.com/GiovineItalia/Gadfly.jl"
literal "false"

\end_inset


\end_layout

\end_inset

, Gephi 
\begin_inset CommandInset citation
LatexCommand citep
key "bastian2009gephi"
literal "false"

\end_inset

, VOSViewer 
\begin_inset CommandInset citation
LatexCommand citep
key "van2009software"
literal "false"

\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Final data analysis
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Julia, MATLAB, Python,
\begin_inset Newline newline
\end_inset

LightGraphs.jl 
\begin_inset CommandInset citation
LatexCommand citep
key "seth_bromberger_2017"
literal "false"

\end_inset

, ScikitLearn.jl
\begin_inset Foot
status open

\begin_layout Plain Layout
\begin_inset CommandInset href
LatexCommand href
target "https://github.com/cstjean/ScikitLearn.jl"
literal "false"

\end_inset


\end_layout

\end_inset

 
\begin_inset CommandInset citation
LatexCommand citep
key "Pedregosa2011"
literal "false"

\end_inset

, JuliaDB
\begin_inset Foot
status open

\begin_layout Plain Layout
\begin_inset CommandInset href
LatexCommand href
target "https://github.com/JuliaComputing/JuliaDB.jl"
literal "false"

\end_inset


\end_layout

\end_inset

, Query.jl
\begin_inset Foot
status open

\begin_layout Plain Layout
\begin_inset CommandInset href
LatexCommand href
target "https://github.com/queryverse/Query.jl"
literal "false"

\end_inset


\end_layout

\end_inset

, MATLAB.jl
\begin_inset Foot
status open

\begin_layout Plain Layout
\begin_inset CommandInset href
LatexCommand href
target "https://github.com/JuliaInterop/MATLAB.jl"
literal "false"

\end_inset


\end_layout

\end_inset

,
\begin_inset Newline newline
\end_inset

TensorToolbox 
\begin_inset CommandInset citation
LatexCommand citep
key "TTB_Sparse"
literal "false"

\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Literature management
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Mendeley, Zotero
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
List of software used
\begin_inset CommandInset label
LatexCommand label
name "tab:List-of-software"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
This thesis was made possible through the efforts of many software developers.
 Some of the software used in the course of creating this work is listed
 in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:List-of-software"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Newpage newpage
\end_inset


\end_layout

\begin_layout Section*
Bibliography
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "monorefs"
options "authordate4"

\end_inset


\end_layout

\begin_layout Part
Node characterisation from local topology
\begin_inset CommandInset label
LatexCommand label
name "part:Graphlets"

\end_inset


\end_layout

\begin_layout Chapter
Graphlets: colour, profiles and node characterisation
\begin_inset CommandInset label
LatexCommand label
name "chap:Graphlets:-colour,-profiles"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
epigraph{``If I could remember the names of all these particles, I'd be
 a botanist"}{--- 
\backslash
textup{Enrico Fermi}}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
MyLettrine{N}{ode characterisation}
\end_layout

\end_inset

, as introduced in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Node-characterisation"
plural "false"
caps "false"
noprefix "false"

\end_inset

, describes single nodes such that they can be compared with other nodes
 in a network or, perhaps, with other nodes in other networks.
 Here, we focus on characterising nodes from their local topology as opposed
 to being concerned with any dynamical equations governing state vectors
 associated with those nodes.
\end_layout

\begin_layout Standard
Of the works summarised in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Node-characterisation"
plural "false"
caps "false"
noprefix "false"

\end_inset

, we believed that graphlet profile
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlet profile
\end_layout

\end_inset

s 
\begin_inset CommandInset citation
LatexCommand citep
key "Cunningham2013"
literal "false"

\end_inset

 showed the most promise as their technique could be extended to use graph
 colour to denote the ego 
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego
\end_layout

\end_inset

and thus retain the information about the position of a node under consideration
 within its local topology.
 Additionally, this technique could serve as an input for the decompositions
 discussed in part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Matrix-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
This part is a discussion of our contributions to this technique and includes
 an example application using coloured graphlet profiles to investigate
 the relationship between local topology on academic collaboration networks
 and academic career length.
 A paper based on this work was presented at a conference for which proceedings
 are available 
\begin_inset CommandInset citation
LatexCommand citep
key "BlanthornMorocco2019"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard
The rest of this chapter and part is organised as follows:
\end_layout

\begin_layout Itemize
section 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Coloured-graphlet-profiles"
plural "false"
caps "false"
noprefix "false"

\end_inset

 provides details on extensions to the algorithms used to create the coloured
 graphlet profiles of a network,
\end_layout

\begin_layout Itemize
section 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Node-characterisation-graphlet"
plural "false"
caps "false"
noprefix "false"

\end_inset

 describes how we apply graphlet profiles to the ego-networks of nodes to
 characterise them,
\end_layout

\begin_layout Itemize
and chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Career-length-in"
plural "false"
caps "false"
noprefix "false"

\end_inset

 shows how this technique can be used to explore a real-word collaboration
 network.
 Importantly, we find that our graphlet profiles are not degenerate to the
 standard node centralities or monochromatic graphlet profiles—i.e.
\begin_inset space \space{}
\end_inset

the new technique is worthwhile.
\end_layout

\begin_layout Section
\begin_inset CommandInset label
LatexCommand label
name "chap:Coloured-graphlet-profiles"

\end_inset

Coloured graphlet profiles
\end_layout

\begin_layout Standard
As described in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Graphlet-profile"
plural "false"
caps "false"
noprefix "false"

\end_inset

, the 
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlet profile
\end_layout

\end_inset

graphlet profile of a graph is a mapping from automorphism groups corresponding
 to a certain number of nodes to the number of times subgraphs belonging
 to those automorphism groups appear in a graph.
\end_layout

\begin_layout Standard
Coloured graphlet profiles are much less well supported in software than
 monochromatic graphlet profiles.
 Of the algorithms introduced in section 
\begin_inset CommandInset ref
LatexCommand vref
reference "subsec:Graphlet-enumeration"
plural "false"
caps "false"
noprefix "false"

\end_inset

, only FANMOD 
\begin_inset CommandInset citation
LatexCommand citep
key "Wernicke2006"
literal "false"

\end_inset

 supports coloured graphlets.
 However, FANMOD is far from the state-of-the-art in graphlet enumeration;
 it is many times slower than more modern routines 
\begin_inset CommandInset citation
LatexCommand citep
key "Masoudi-Nejad2012"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard
Kavosh is substantially faster than FANMOD 
\begin_inset CommandInset citation
LatexCommand citep
key "kashani_kavosh:_2009"
literal "false"

\end_inset

 and uses nauty 
\begin_inset CommandInset citation
LatexCommand citep
key "McKay2014"
literal "false"

\end_inset

 to tally graphlets by their automorphism group.
 Nauty has built-in support for graph colour.
 There are other graphlet enumeration routines which are quicker for monochromat
ic graphs 
\begin_inset CommandInset citation
LatexCommand citep
key "Masoudi-Nejad2012"
literal "false"

\end_inset

, but adapting these for coloured graphs seemed like a far larger undertaking.
 It was therefore decided that adapting Kavosh to work with coloured graphs
 struck a good balance between being a useful contribution to the field
 and the amount of time required to design and build.
 In our testing, the original Kavosh implementation suffered from memory
 leaks.
 As the algorithm is quite short, we re-implemented it from scratch in Julia,
 which required a number of changes which we will enumerate in this chapter.
\end_layout

\begin_layout Subsection
Calling nauty from Julia
\end_layout

\begin_layout Standard
We wrote a small package to call nauty from Julia and a small type for coloured
 graphs as the first step before Kavosh could be reimplemented.
 The most important elements of these packages are enumerated here:
\end_layout

\begin_layout Itemize
Creation of a partition array of node labels such that all nodes of the
 same colour are in a single contiguous block, as shown in algorithm 
\begin_inset CommandInset ref
LatexCommand ref
reference "alg:nauty-labelpart"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Itemize
Creation of labels for nauty describing where the changes in colour are—an
 array where zeroes denote that a node is the same colour as the preceding
 node in the partition, and non-zeroes denote a change in colour, as shown
 in algorithm 
\begin_inset CommandInset ref
LatexCommand ref
reference "alg:nauty-labelpart"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float algorithm
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{minted}[linenos=false,mathescape=true]{julia}
\end_layout

\begin_layout Plain Layout

function nautylabelspartition(g;implicit_colour=false,ego=0)
\end_layout

\begin_layout Plain Layout

    colorsarray::Array{Array{Int64,1},1} = !implicit_colour?
\end_layout

\begin_layout Plain Layout

                   # if the colours are explicitly specified
\end_layout

\begin_layout Plain Layout

                   # create the array of colours from that
\end_layout

\begin_layout Plain Layout

                   [collect(filter_vertices(
\end_layout

\begin_layout Plain Layout

                        g,(graph, vertex) -> begin
\end_layout

\begin_layout Plain Layout

                            get(
\end_layout

\begin_layout Plain Layout

                                props(graph, vertex), :color, ""
\end_layout

\begin_layout Plain Layout

                            ) == c
\end_layout

\begin_layout Plain Layout

                        end
\end_layout

\begin_layout Plain Layout

                   )) for c in colors(g)]:
\end_layout

\begin_layout Plain Layout

        # otherwise denote the ego as a different colour to
\end_layout

\begin_layout Plain Layout

        # the alters
\end_layout

\begin_layout Plain Layout

        ego==1 ? [[1],collect(2:number_of_vertices(g))]:
\end_layout

\begin_layout Plain Layout

        ego==0 ? [collect(1:number_of_vertices(g))]:
\end_layout

\begin_layout Plain Layout

        ego==number_of_vertices(g) ? [collect(1:ego-1),[ego]]:
\end_layout

\begin_layout Plain Layout

        [[ego],[1:ego-1..., ego+1:number_of_vertices(g)...]]
\end_layout

\begin_layout Plain Layout

    
\end_layout

\begin_layout Plain Layout

    # flatten the array and subtract one from all indices
\end_layout

\begin_layout Plain Layout

    # as nauty makes the unfortunate choice of numbering
\end_layout

\begin_layout Plain Layout

    # nodes from 0.
\end_layout

\begin_layout Plain Layout

    labels = vcat(colorsarray.-1...)
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

    # set the last element of each colour block to be zero
\end_layout

\begin_layout Plain Layout

    # to denote a change in colour
\end_layout

\begin_layout Plain Layout

    partition = vcat(
\end_layout

\begin_layout Plain Layout

        [begin
\end_layout

\begin_layout Plain Layout

            z[end]=0; z
\end_layout

\begin_layout Plain Layout

        end for z in ones.(Cint,size.(colorsarray))]...
\end_layout

\begin_layout Plain Layout

    )
\end_layout

\begin_layout Plain Layout

    return (labels,partition)
\end_layout

\begin_layout Plain Layout

end
\end_layout

\begin_layout Plain Layout


\backslash
end{minted}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Implementation in Julia of labelling and partition for use of nauty on coloured
 graphs
\begin_inset CommandInset label
LatexCommand label
name "alg:nauty-labelpart"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Itemize
The ability to call nauty from Julia, shown in algorithm 
\begin_inset CommandInset ref
LatexCommand ref
reference "alg:nauty-interface"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float algorithm
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{minted}[linenos=false,mathescape=true]{julia}
\end_layout

\begin_layout Plain Layout

struct nautyreturn
\end_layout

\begin_layout Plain Layout

    canong::NautyGraph
\end_layout

\begin_layout Plain Layout

    partition::Array{Cint}
\end_layout

\begin_layout Plain Layout

end
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

# Get the canonical form of a graph from nauty given the colouring
\end_layout

\begin_layout Plain Layout

function baked_canonical_form_color(g, labelling, partition)
\end_layout

\begin_layout Plain Layout

    g = lg_to_nauty(g)
\end_layout

\begin_layout Plain Layout

    (num_vertices, num_setwords) = size(g, 1, 2)
\end_layout

\begin_layout Plain Layout

    stats = statsblk()
\end_layout

\begin_layout Plain Layout

    outgraph = zero(g)
\end_layout

\begin_layout Plain Layout

    orbits = zero(labelling)
\end_layout

\begin_layout Plain Layout

    ccall((:baked_options_color, LIB_FILE), Cvoid,
\end_layout

\begin_layout Plain Layout

           (NautyGraphC, Ptr{Cint},
\end_layout

\begin_layout Plain Layout

            Ptr{Cint}, Ptr{Cint},
\end_layout

\begin_layout Plain Layout

            Ref{statsblk}, Cint,
\end_layout

\begin_layout Plain Layout

            Cint, NautyGraphC),
\end_layout

\begin_layout Plain Layout

        g, labelling, partition,
\end_layout

\begin_layout Plain Layout

        orbits, stats, num_setwords,
\end_layout

\begin_layout Plain Layout

        num_vertices, outgraph
\end_layout

\begin_layout Plain Layout

    )
\end_layout

\begin_layout Plain Layout

    return nautyreturn(outgraph, partition)
\end_layout

\begin_layout Plain Layout

end
\end_layout

\begin_layout Plain Layout

                
\end_layout

\begin_layout Plain Layout

function nauty(g; egonet=false, ego=0 )
\end_layout

\begin_layout Plain Layout

    labels, partition = nautylabelspartition(
\end_layout

\begin_layout Plain Layout

        g;model_t=egonet,ego=ego
\end_layout

\begin_layout Plain Layout

    )
\end_layout

\begin_layout Plain Layout

    nautyrtn = baked_canonical_form_color(g,labels,partition)
\end_layout

\begin_layout Plain Layout

    return nautyrtn
\end_layout

\begin_layout Plain Layout

end
\end_layout

\begin_layout Plain Layout


\backslash
end{minted}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Interface to nauty from Julia
\begin_inset CommandInset label
LatexCommand label
name "alg:nauty-interface"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Kavosh adaptation
\end_layout

\begin_layout Standard
The key changes made to the algorithm to support colour and work in Julia
 are as follows:
\end_layout

\begin_layout Itemize
Hashing and equality operators for Julia on the nauty return values to enable
 us to use them as keys for a dictionary, as shown in algorithm 
\begin_inset CommandInset ref
LatexCommand ref
reference "alg:nauty-hash"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float algorithm
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{minted}[linenos=false,mathescape=true]{julia}
\end_layout

\begin_layout Plain Layout

function Base.hash(m::nautyreturn)
\end_layout

\begin_layout Plain Layout

    return hash((m.canong,m.partition))
\end_layout

\begin_layout Plain Layout

end
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

function Base.:(==)(a::nautyreturn, b::nautyreturn)
\end_layout

\begin_layout Plain Layout

    # two canonical graphs are the same if they have the same layout
\end_layout

\begin_layout Plain Layout

    # and the same colourings
\end_layout

\begin_layout Plain Layout

    return a.canong == b.canong && a.partition == b.partition
\end_layout

\begin_layout Plain Layout

end
\end_layout

\begin_layout Plain Layout


\backslash
end{minted}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Implementation of equality and hashing operators on nauty return values
 in Julia
\begin_inset CommandInset label
LatexCommand label
name "alg:nauty-hash"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Itemize
As Kavosh iterates through each unique (per node labels) induced subgraph,
 we increment the counter corresponding to the key corresponding to the
 nauty return value for that graph.
 This routine is not shown here for brevity; it may be found online from
 links in chapter 
\begin_inset Quotes xld
\end_inset


\begin_inset CommandInset ref
LatexCommand nameref
reference "chap:Availability-of-data"
plural "false"
caps "false"
noprefix "false"

\end_inset


\begin_inset Quotes xrd
\end_inset

 
\begin_inset CommandInset ref
LatexCommand vpageref
reference "chap:Availability-of-data"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Subsection
Limitations
\end_layout

\begin_layout Standard
Our implementation of Kavosh only supports graphlets of at least size two
 and graphs 
\begin_inset Formula $\mathcal{G}=(\mathcal{V},\mathcal{E})$
\end_inset

 with a maximum of two colours, i.e.
\begin_inset Foot
status open

\begin_layout Plain Layout
here we use the notation 
\begin_inset Formula $\{i:i\in S\}\equiv\{i\mid i\in S\}$
\end_inset

 to prevent ambiguity with the cardinality of a set
\end_layout

\end_inset

 
\begin_inset Formula $\lvert\{\pi(i):i\in\mathcal{V}\}\rvert=2$
\end_inset

, where 
\begin_inset Formula $\pi(i)$
\end_inset

 is the colour of the node 
\begin_inset Formula $i$
\end_inset

.
 One of these colours must have a maximum of one node, that is, if we let
 
\begin_inset Formula $\Pi=\{\pi(i):i\in\mathcal{V}\}$
\end_inset

, then
\begin_inset Formula 
\[
\text{min}(\{\lvert\{i:\pi(i)=a\}\rvert:a\in\Pi\})\leq1\text{.}
\]

\end_inset

This limitation is because nauty has no memory of the entire graph and so
 it cannot tell the difference, for example, between a red node connected
 to a green node and a yellow node connected to a blue node on separate
 calls.
\end_layout

\begin_layout Standard
Support for all coloured graphs could be added by adding a new disconnected
 node of each colour to every subgraph as it is sent to nauty acting as
 a key.
 This was not necessary to distinguish between the single ego 
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego
\end_layout

\end_inset

 and alters
\begin_inset Index idx
status open

\begin_layout Plain Layout
alters
\end_layout

\end_inset

 in ego-networks.
\end_layout

\begin_layout Section
Node characterisation from graphlet profiles of ego-networks
\begin_inset CommandInset label
LatexCommand label
name "chap:Node-characterisation-graphlet"

\end_inset


\end_layout

\begin_layout Standard
Graphlets, or induced motifs
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!induced motifs
\end_layout

\end_inset


\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!induced motifs|seealso
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\end_layout

\end_inset

metrics!graphlets
\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset

, have long been used to find important medium-scale structures in directed
 networks 
\begin_inset CommandInset citation
LatexCommand citep
key "milo_network_2002"
literal "false"

\end_inset

.
 The works introduced in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Graphlets"
plural "false"
caps "false"
noprefix "false"

\end_inset

 such as 
\begin_inset CommandInset citation
LatexCommand cite
key "Cunningham2013"
literal "false"

\end_inset

 have a clear area for improvement: they do not retain information about
 the location of the node of interest within its local topology.
 Our coloured graphlet profiles offer a clear way to retain this information
 from the employment of coloured 
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego-network
\end_layout

\end_inset

ego-networks, where the ego
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego
\end_layout

\end_inset

 is given a distinct colour from the alters
\begin_inset Index idx
status open

\begin_layout Plain Layout
alters
\end_layout

\end_inset

, as the location of the node of interest (the ego) is denoted by its colour.
 Here we present a method using the composition of coloured graphlets in
 ego-networks to characterise nodes:
\end_layout

\begin_layout Enumerate
For each node, extract its 
\begin_inset Formula $n$
\end_inset

-ego-network
\end_layout

\begin_layout Enumerate
Compose coloured graphlet profiles for the ego-network of each node, where
 the ego is one colour and the alters are another, for each size of graphlet
 considered
\end_layout

\begin_layout Enumerate
For each node, combine the graphlet profiles into a single vector to characteris
e that node; these vectors can then be used in further analysis such as
 multi-linear regression as described in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:mullin"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Subsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego-network
\end_layout

\end_inset

Ego-networks and colour
\begin_inset Index idx
status open

\begin_layout Plain Layout
graph colour
\end_layout

\end_inset

ed graphlets
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlets
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:Ego-networks-and-coloured"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "colouredgraphlets-1"

\end_inset


\end_layout

\begin_layout Standard
The ego 
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego
\end_layout

\end_inset

 in each ego-network is given a colour distinct from all of the other nodes,
 used in the coloured subgraph matching.
 This allows us to understand the importance of an ego within a graphlet,
 if it is there at all.
\end_layout

\begin_layout Standard
Each graphlet is given a label consisting of a decimal followed by an alphabetic
 character.
 These labels follow the convention that the number before the decimal place
 is the number of nodes in the graphlet, the number after the decimal place
 labels the monochromatic automorphism group in ascending order of the Shannon
 entropy
\begin_inset Index idx
status open

\begin_layout Plain Layout
Shannon entropy
\end_layout

\end_inset

 of their degree distributions (so less-uniform distributions have a higher
 label 
\begin_inset CommandInset citation
LatexCommand citep
after "pp. 191-192"
key "Costa2007"
literal "false"

\end_inset

) and the character signifies colour differences between graphlets within
 the same monochromatic automorphism group, as shown in figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:motifs"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/2.1a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/2.1b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset VSpace smallskip
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/3.1a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/3.1b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/3.2a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/3.2b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/3.2c.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset VSpace smallskip
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.1a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.1b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.2a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.2b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.3a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset VSpace smallskip
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.3b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.3c.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.4a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.4b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.4c.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset VSpace smallskip
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.5a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.5b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.5c.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.6a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.6b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset VSpace smallskip
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.6c.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.6d.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
All possible undirected graphlets with at most a single differently-coloured
 node (i.e, the possible graphlets within ego-networks where the ego is one
 colour and the alters are another colour) up to size 
\begin_inset Formula $4$
\end_inset


\begin_inset Argument 1
status open

\begin_layout Plain Layout
All undirected graphlets with up to a single distinguishable node up to
 size 4
\end_layout

\end_inset

.
 The smaller black node corresponds to the ego, if it exists, and the coloured
 nodes are the alters.
 The graphlets are labelled as described in Sect.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Ego-networks-and-coloured"
plural "false"
caps "false"
noprefix "false"

\end_inset

; the colours are chosen such that similar graphlets have similar colours
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:motifs"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
From graphlet profiles to node characterisation
\begin_inset CommandInset label
LatexCommand label
name "sec:From-graphlet-profiles"

\end_inset


\end_layout

\begin_layout Standard
Similar to previous work 
\begin_inset CommandInset citation
LatexCommand citep
key "Cunningham2013"
literal "false"

\end_inset

 as described in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Graphlet-profile"
plural "false"
caps "false"
noprefix "false"

\end_inset

, we normalise each graphlet profile
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlet profile
\end_layout

\end_inset

 of graphlets with 
\begin_inset Formula $n$
\end_inset

 nodes, 
\begin_inset Formula $\mathbf{g_{n}}$
\end_inset

, by the total number of induced subgraphs of size 
\begin_inset Formula $n$
\end_inset

 in the ego-network, 
\begin_inset Formula 
\begin{equation}
%\label{eqn:norm}
\hat{g}_{ni}=\frac{g_{ni}}{\sum_{i}{g_{ni}}}\;,
\end{equation}

\end_inset

where each 
\begin_inset Formula $ni$
\end_inset

 corresponds to a specific automorphism group 
\begin_inset Formula $i$
\end_inset

 of size 
\begin_inset Formula $n$
\end_inset

.
 We normalise predominantly to allow for meaningful comparison between networks
 of different sizes.
\end_layout

\begin_layout Standard
This is sufficient compared to the much more computationally expensive significa
nce profiles approach of comparing the counts to the the counts in ensemble
 of configuration models, calculating a so-called z-score 
\begin_inset CommandInset citation
LatexCommand citep
key "milo_network_2002,kashani_kavosh:_2009,meira_acc-motif:_2014"
literal "false"

\end_inset

 as we are later comparing these profiles with many other profiles: we are
 not concerned with how they compare to configuration models.
 We want to know how they differ from other networks in same domain.
\end_layout

\begin_layout Standard
After normalisation, we combine the graphlet profile
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlet profile
\end_layout

\end_inset

s for each graphlet size considered in a single vector, 
\begin_inset Formula $\mathbf{g}$
\end_inset

, where each dimension corresponds to a specific graphlet such that profiles
 between nodes are directly comparable.
 
\end_layout

\begin_layout Section
Conclusion
\end_layout

\begin_layout Standard
We have described a technique for characterising nodes and networks from
 coloured graphlet profiles, extending previous monochromatic treatments.
 We will employ it in part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Matrix-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

 as part of a data matrix to characterise whole networks.
 In the next chapter, we demonstrate its usefulness compared to previous
 monochromatic treatments and the standard node centralities in one example
 application on a real-world piecewise static network.
\end_layout

\begin_layout Standard
\begin_inset Newpage newpage
\end_inset


\end_layout

\begin_layout Section*
Bibliography
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "monorefs"
options "authordate4"

\end_inset


\end_layout

\begin_layout Chapter
Career length in academic collaboration networks
\begin_inset CommandInset label
LatexCommand label
name "chap:Career-length-in"

\end_inset


\end_layout

\begin_layout Standard

\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
epigraph{``Time is a great teacher, it is said; the misfortune is that he
 is a brutal teacher who kills all his students"}{--- 
\backslash
textup{Hector Berlioz}}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
MyLettrine{O}{ur technique}
\end_layout

\end_inset

 introduced in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Node-characterisation-graphlet"
plural "false"
caps "false"
noprefix "false"

\end_inset

 is used here to predict the numbers of years researchers are active from
 their temporal collaboration networks.
 We compare our success with simpler metrics; we find that the use of coloured
 graphlets improves predictive performance compared to colour-blind graphlets;
 that 4-star graphlets centred on an author early in their career are predictors
 of an eventual long career; and that this effect is not degenerate to centralit
ies, i.e.
\begin_inset space \thinspace{}
\end_inset

graphlets have a contribution to career length which is not explained by
 other centralities.
 
\end_layout

\begin_layout Standard
We compare our coloured graphlet profiles to other methods of characterisation
 of nodes, namely the standard centralities such as node degree
\begin_inset Index idx
status open

\begin_layout Plain Layout
centralities!degree
\end_layout

\end_inset

, betweenness
\begin_inset Index idx
status open

\begin_layout Plain Layout
centralities!betweenness
\end_layout

\end_inset

 centrality and the 
\begin_inset Index idx
status open

\begin_layout Plain Layout
centralities!local clustering coefficient
\end_layout

\end_inset

local clustering coefficient.
 We apply this node characterisation technique in a specific application:
 predicting the 
\begin_inset Quotes xld
\end_inset

lifetime
\begin_inset Quotes xrd
\end_inset

 of researchers from the shape of their collaborations.
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "sec:1"

\end_inset


\end_layout

\begin_layout Standard
The research lifetime of researchers, that is, the number of years they
 are active publishing research, follows a power-law, so the overwhelming
 majority of researchers have short research lifetimes, as shown in figure
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:lifedist"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and found in other research 
\begin_inset CommandInset citation
LatexCommand citep
key "Milojevic2018"
literal "false"

\end_inset

.
 It has been suggested that this distribution is due to the voluntary career
 changes of PhD students 
\begin_inset CommandInset citation
LatexCommand citep
key "Biryukov2010"
literal "false"

\end_inset

, but this would imply a bi-modal distribution which we do not see in figure
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:lifedist"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 However, other research has found that there is evidence that lifetime
 distributions are often 
\begin_inset Quotes xld
\end_inset

U
\begin_inset Quotes xrd
\end_inset

-shaped, which are clearly bi-modal 
\begin_inset CommandInset citation
LatexCommand citep
key "fedotov2017anomalous"
literal "false"

\end_inset

.
 This could be consistent with our data as our distributions are truncated
 as detailed in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Experimental-design-and"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
liningnums{53}
\end_layout

\end_inset

% of all UK workers do not wish to change career 
\begin_inset CommandInset citation
LatexCommand citep
key "LSBF2015"
literal "false"

\end_inset

, suggesting that a large proportion of career changes away from academia
 are involuntary.
 Our motivation is to find information that could help early career researchers
 extend their careers in academia, as many of them clearly wish to do so.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% this comes up as tech.
 rep.
 - is that right? what is it? Is it actually peer-reviewed? Try to find
 the actual underlying research!
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

The shape of a collaboration network is relatively easy to change as one
 chooses who to collaborate with.
 Our work aims to uncover whether particular shapes of research collaboration
 predict career longevity and to explore the importance of colour
\begin_inset Index idx
status open

\begin_layout Plain Layout
graph colour
\end_layout

\end_inset

, or nodes which fall into discrete categories, in ego-networks.
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego-network
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/lifedist.pdf
	width 48text%

\end_inset

 
\begin_inset Caption Standard

\begin_layout Plain Layout
A log-lin histogram of showing the distribution of researcher lifetimes
\begin_inset Argument 1
status open

\begin_layout Plain Layout
A log-lin histogram of showing the distribution of researcher lifetimes
\end_layout

\end_inset

 in the subsample of the Arnetminer co-authorship dataset 
\begin_inset CommandInset citation
LatexCommand cite
key "Zhuang2013"
literal "false"

\end_inset

 used throughout this chapter.
 The lowest career length is 5 years and the highest career length is 23
 years.
 Both of these artefacts are due to choices we have made to ensure we do
 not have an incomplete picture of any careers 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% should also change it to read % of sample so the numbers aren't confusing
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% should change this to be a scatter plot so that it looks better in print
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% probably should fit power law and include exponent
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:lifedist"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In previous work, uncoloured graphlets in co-authorship networks have been
 used to predict success in terms of citation count 
\begin_inset CommandInset citation
LatexCommand citep
key "Krumov2011"
literal "false"

\end_inset

.
 It was found that 
\begin_inset Quotes xld
\end_inset

box motifs
\begin_inset Quotes xrd
\end_inset

, i.e.
 4-cycles, were beneficial.
 Whether this effect could be better explained by other centralities was
 not explored.
 More information on academic collaboration networks was given in section
 
\begin_inset CommandInset ref
LatexCommand vref
reference "par:Academic-coauthorship-networks"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
The rest of this chapter is organised as follows: section
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sec:methods"
plural "false"
caps "false"
noprefix "false"

\end_inset

 explains our methodological contributions; section
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sec:results"
plural "false"
caps "false"
noprefix "false"

\end_inset

 demonstrates the use of our technique with a real-world application; and
 section
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sec:conclusion"
plural "false"
caps "false"
noprefix "false"

\end_inset

 summarises our results and possible directions for future work.
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "sec:background"

\end_inset


\end_layout

\begin_layout Standard
In our application, the ego
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego
\end_layout

\end_inset

 is the author whose lifetime we predict and the alters
\begin_inset Index idx
status collapsed

\begin_layout Plain Layout
alters
\end_layout

\end_inset

 are their co-authors, their co-authors' co-authors and so on.
\end_layout

\begin_layout Section
Methods
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "sec:methods"

\end_inset


\end_layout

\begin_layout Standard
We predict the lifetimes of nodes in an evolving complex network using a
 simple linear regression model that combines all of our features.
 This allows us to disregard degenerate features.
 Our features consist of graphlet profile
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlet profile
\end_layout

\end_inset

s and the standard node centralities.
 These methods are explored in full in this subsection.
\end_layout

\begin_layout Subsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego-network
\end_layout

\end_inset

Ego-networks and colour
\begin_inset Index idx
status open

\begin_layout Plain Layout
graph colour
\end_layout

\end_inset

ed graphlets
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlets
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "colouredgraphlets"

\end_inset


\end_layout

\begin_layout Standard
We follow the technique introduced in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Ego-networks-and-coloured"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 We chose to consider the 3-ego-networks of each node, where 
\begin_inset Formula $3$
\end_inset

 corresponds to the maximum distance from the node of interest.
 Three was chosen as it was found that any more than three did not improve
 predictions and led to somewhat increased computational complexity, considerabl
y so in the worst cases; fewer than three would have meant that some 4-graphlets
 were underrepresented.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%reword?
\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Graphlet profiles
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "sec:profiles"

\end_inset


\end_layout

\begin_layout Standard
The graphlet profiles are calculated as in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:From-graphlet-profiles"
plural "false"
caps "false"
noprefix "false"

\end_inset

 with the following caveat: we produce coloured and uncoloured graphlet
 profiles and compare them; we refer to them as the coloured and uncoloured
 or monochromatic treatments.
 Monochromatic graphlets can appear in both types of profile but we refer
 to them with a trailing 
\begin_inset Quotes xld
\end_inset

a
\begin_inset Quotes xrd
\end_inset

 in the coloured treatment to signify that they represent the complete absence
 of the ego.
 Since we use the graphlet profile as part of multilinear regression, we
 in effect compare the graphlets from each ego-network with every other
 ego-network to determine what is out of the ordinary and what is normal,
 further suggesting that our choice of normalisation in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:From-graphlet-profiles"
plural "false"
caps "false"
noprefix "false"

\end_inset

 is sound.
\end_layout

\begin_layout Standard
Graphlet profiles for graphlets of size 
\begin_inset Formula $n=2,3,4$
\end_inset

 were calculated.
 Larger graphlets are not used due to their increased computational complexity
 and our experimental findings that they did not improve predictions, in
 agreement with previous studies such as 
\begin_inset CommandInset citation
LatexCommand cite
key "Cunningham2013"
literal "false"

\end_inset

.
 
\end_layout

\begin_layout Subsection
Other centralities considered
\end_layout

\begin_layout Standard
Table
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "tab:allfeats"
plural "false"
caps "false"
noprefix "false"

\end_inset

 summarises all of the features we consider in this work, many of which
 were introduced in sections 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Node-characterisation"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Network-characterisation"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 We refer to all of the features that are not graphlet profiles as the 
\begin_inset Quotes xld
\end_inset

standard centralities
\begin_inset Quotes xrd
\end_inset

, but not all of the features are centralities in the traditional sense
 as they characterise graphs rather than nodes.
 However, because we are using them to describe the ego-network of specific
 nodes, they can in some sense be considered to characterise those nodes,
 so we take the liberty of referring to them as centralities.
\end_layout

\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Features we use to characterise egos in ego-networks
\end_layout

\end_inset

Features used to characterise egos in ego-networks.
 See section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Node-characterisation"
plural "false"
caps "false"
noprefix "false"

\end_inset

 for more detail
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "tab:allfeats"

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Add citations.
 I think de Weck had a nice table.
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
centering
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset Tabular
<lyxtabular version="3" rows="9" columns="3">
<features tabularvalignment="middle">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<row>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 Feature
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Description
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Range
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 Betweenness
\begin_inset Index idx
status open

\begin_layout Plain Layout
centralities!betweenness
\end_layout

\end_inset


\begin_inset Formula $^{a}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Prop.
 of shortest paths passing through ego
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $[0,1]$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Index idx
status open

\begin_layout Plain Layout
centralities!degree
\end_layout

\end_inset

Degree
\begin_inset Formula $^{a}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Number of edges attached to ego
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $[0,\infty)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Index idx
status open

\begin_layout Plain Layout
centralities!closeness
\end_layout

\end_inset

Closeness
\begin_inset Formula $^{a}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Reciprocal of distance of ego to alters
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $(0,1]$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!global clustering coefficient
\end_layout

\end_inset

Clustering
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Ratio of triangles present to possible triangles
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $[0,1]$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Index idx
status open

\begin_layout Plain Layout
centralities!local clustering coefficient
\end_layout

\end_inset

Ego-clustering
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Clustering of 1-ego-network of ego
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $[0,1]$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Edges
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Number of edges in ego-network
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $[0,\infty)$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Graphlet profile
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Normalised graphlet counts as described
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $[0,1]^{d}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Formula $^{a}$
\end_inset

 Commonly known with the suffix 
\begin_inset Quotes xld
\end_inset

centrality
\begin_inset Quotes xrd
\end_inset

, omitted here for brevity.
 NB: 
\begin_inset Formula $^{d}$
\end_inset

 is the number of unique graphlets and is not a footnote; it is equal to
 
\begin_inset Formula $24$
\end_inset

 in the coloured treatment and 
\begin_inset Formula $9$
\end_inset

 in the monochromatic treatment
\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Estimation of confidence intervals
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% this should be updated to explain that it just gives errors on the correlation
 plots, probably.
 also why aren't we bonferronying?
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

We estimate errors via a 
\begin_inset Quotes xld
\end_inset

bootstrap method
\begin_inset Quotes xrd
\end_inset

: the standard error is calculated for the correlation with lifetime for
 ten subsamples of the graphlet profiles.
 This error is then scaled by the square root of the number of subsamples.
 The error bars plotted correspond to twice the standard error and are therefore
 approximations of the 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
liningnums{95}
\end_layout

\end_inset

% confidence interval, assuming a normal distribution.
\end_layout

\begin_layout Subsection
Multiple linear regression
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "sec:mullin"

\end_inset


\end_layout

\begin_layout Standard
As introduced in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Multiple-linear-regression"
plural "false"
caps "false"
noprefix "false"

\end_inset

, multiple linear regression
\begin_inset Index idx
status open

\begin_layout Plain Layout
multiple linear regression
\end_layout

\end_inset

 is a simple method for investigating the relationship between many continuous
 so-called explanatory variables 
\begin_inset Formula $x_{i}$
\end_inset

 and a single dependent variable, 
\begin_inset Formula $y$
\end_inset

, 
\begin_inset CommandInset citation
LatexCommand citep
key "montgomery2012"
literal "false"

\end_inset

 with
\begin_inset Formula 
\[
y=\Sigma_{i}\beta_{i}x_{i}\text{.}
\]

\end_inset


\end_layout

\begin_layout Standard
The explanatory variables are the features enumerated in table
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "tab:allfeats"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and the dependent variable in our application is the research lifetime
 of the authors.
 The 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "beta"
symbol "$\\beta_{\\text{reduced}}$"
description "a coefficient of a variable in a linear regression multiplied by the standard deviation of that variable"
literal "true"

\end_inset


\begin_inset Formula $\beta_{\text{reduced}}$
\end_inset

 coefficients, 
\begin_inset Formula 
\[
\beta_{i_{\text{reduced}}}=\frac{\beta_{i}}{\text{std}(x_{i})}\text{,}
\]

\end_inset

where std is the standard deviation of the variable 
\begin_inset Formula $x_{i}$
\end_inset

, are particularly important for our application as the variables have very
 different ranges, as shown in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:allfeats"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
We apply a Bonferroni correction
\begin_inset Index idx
status open

\begin_layout Plain Layout
Bonferroni correction
\end_layout

\end_inset

 
\begin_inset CommandInset citation
LatexCommand citep
key "Bonferroni1950"
literal "false"

\end_inset

 as explained in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Bonferroni-correction"
plural "false"
caps "false"
noprefix "false"

\end_inset

 to ensure that the large number of statistical tests we perform are valid.
 Since we have 30 features—each of the items in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:allfeats"
plural "false"
caps "false"
noprefix "false"

\end_inset

, our corrected significance level
\begin_inset Index idx
status open

\begin_layout Plain Layout
p-value
\end_layout

\end_inset

 corresponding to a 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
liningnums{95}
\end_layout

\end_inset

% confidence level is approximately 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none
0.0017
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\xout default
\uuline default
\uwave default
\noun default
\color inherit
.
 The software package GLM.jl 
\begin_inset CommandInset citation
LatexCommand citep
key "bates_2019"
literal "false"

\end_inset

 was used for prediction.
\end_layout

\begin_layout Subsection
Experimental design and data
\begin_inset CommandInset label
LatexCommand label
name "subsec:Experimental-design-and"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% correct title?
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The dataset chosen was Arnetminer's disambiguated temporal
\begin_inset Index idx
status open

\begin_layout Plain Layout
temporal networks
\end_layout

\end_inset

 collaboration network 
\begin_inset CommandInset citation
LatexCommand citep
key "Zhuang2013"
literal "false"

\end_inset

.
 The window size on this dataset is 3 years: each collaboration is instantaneous
 in the original datasets, but Arnetminer expands these collaborations to
 exist for three years past this.
 The resolution is a single year, with data for every year from 1986 up
 to and including 2012.
 The growth of a particularly successful researcher's collaboration network
 is shown in figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:sevenego"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
center
\end_layout

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/seven_lifecycle.pdf
	width 95col%

\end_inset

 
\begin_inset Caption Standard

\begin_layout Plain Layout
The 3-ego-networks, anti-clockwise from top-left, of one researcher's first,
 fifth and fifteenth year
\begin_inset Argument 1
status open

\begin_layout Plain Layout
3-ego-networks of one researcher's first, fifth and fifteenth year
\end_layout

\end_inset

 in their career as a researcher.
 The ego is shown in red; the alters are shown in purple
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:sevenego"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The data were cleaned in order to consider researchers for whom we have
 a full picture of their careers.
 Any researchers who were active in the first year of the dataset were removed
 to exclude researchers who started their careers before the data began
 and any researchers still active at the end of the dataset, with the assumption
 that these researchers may not have finished their careers yet, were also
 removed.
 This eliminated approximately 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
liningnums{50}
\end_layout

\end_inset

% of the dataset.
 10,000 authors were then sampled to reduce the computational complexity.
 Finally, any authors that did not have careers of at least 5 years were
 discarded—effectively discarding all PhD students—as this was found to
 make prediction easier with stronger correlations, leaving us with about
 2,000 authors.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% really need to justify this / reword / something is weird here
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

The lifetime distribution is shown in figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:lifedist"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% which fields are included? how is it mined?
\end_layout

\end_inset


\end_layout

\begin_layout Standard
We postulate the following: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% wording - also perhaps this should come first?
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Ask Eva if this is OK
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Need to be made more precise / answerable
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{hyp}
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "h1"

\end_inset

 Some components of graphlet profiles of researchers' co-authorship ego-networks
 at an early stage of their careers correlate with their career lengths,
 after controlling for potentially confounding variables.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{hyp}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{hyp}
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "h2"

\end_inset

 Graphlet profiles coloured to distinguish egos from the alters correlate
 more strongly with career length.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{hyp}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Finally, we conclude this section with a summary of the experimental design
 explained in this section:
\end_layout

\begin_layout Enumerate
clean data—ensure that we have a complete life cycle for all nodes considered,
 
\end_layout

\begin_layout Enumerate
choose a subsample of these nodes to limit computation time, 
\end_layout

\begin_layout Enumerate
calculate lifetimes of the chosen nodes, 
\end_layout

\begin_layout Enumerate
extract the 3-ego-networks of each node at a specific early stage in their
 life cycle, 
\end_layout

\begin_layout Enumerate
extract features: calculate centralities and graphlet profiles on these
 ego-networks, 
\end_layout

\begin_layout Enumerate
calculate the linear correlation of features with lifetime, 
\end_layout

\begin_layout Enumerate
check the pairwise correlation between features to check for possible degeneracy
, 
\end_layout

\begin_layout Enumerate
and finally, perform multiple linear regression to determine whether correlation
s are better explained by other features.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 
\backslash
item Check hypotheses
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Section
Results
\begin_inset CommandInset label
LatexCommand label
name "sec:results"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/2.1a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/2.1b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset VSpace smallskip
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/3.1a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/3.1b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/3.2a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/3.2b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/3.2c.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset VSpace smallskip
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.1a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.1b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.2a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.2b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.3a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset VSpace smallskip
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.3b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.3c.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.4a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.4b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.4c.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset VSpace smallskip
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.5a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.5b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.5c.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.6a.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.6b.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset VSpace smallskip
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.6c.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.6d.pdf
	display false
	width 19text%
	groupId graphlets

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
All possible undirected graphlets with at most a single differently-coloured
 node (i.e, the possible graphlets within ego-networks where the ego is one
 colour and the alters are another colour) up to size 
\begin_inset Formula $4$
\end_inset


\begin_inset Argument 1
status open

\begin_layout Plain Layout
All undirected graphlets with up to a single distinguishable node up to
 size 4
\end_layout

\end_inset

.
 The smaller black node corresponds to the ego, if it exists, and the coloured
 nodes are the alters.
 The graphlets are labelled as described in Sect.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Ego-networks-and-coloured"
plural "false"
caps "false"
noprefix "false"

\end_inset

; the colours are chosen such that similar graphlets have similar colours.
 This figure is identical to figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:motifs"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and is included again here for the benefit of the reader
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:motifs-1"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Correlation of features with lifetime
\end_layout

\begin_layout Standard
The correlation of each graphlet with career longevity is shown in figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:importantgraphlets"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 There are no strong correlations, but we can see that 4.5c, 3.2c, 4.3b and
 4.6b have weak positive correlations.
 If one cross-references with figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:motifs-1"
plural "false"
caps "false"
noprefix "false"

\end_inset

, it can be seen that these are the graphlets where the ego is central and
 of higher degree than its neighbours.
 Contrary to what one might expect, 4.2b and 3.1b, the all-to-all graphlets
 which in some sense correspond to 
\begin_inset Quotes xld
\end_inset

maximum collaboration
\begin_inset Quotes xrd
\end_inset

, have a very weak negative correlation with lifetime.
 
\end_layout

\begin_layout Standard
On the whole, the uncoloured treatment has weaker correlations with lifetime
 and misses out on some subtlety: where monochrome graphlets have equivalent
 coloured graphlets that are both negatively and positively correlated with
 lifetime, the net effect is to have an overall correlation which is closer
 to zero, which could be misleading.
 This is especially evident with graphlet 4.3: 4.3b is one of the strongest
 positive predictors of lifetime, but 4.3a is negatively correlated in the
 coloured treatment, albeit not significantly so.
 This leads to 4.3 having no overall correlation with lifetime in the uncoloured
 treatment, which could explain why previous work found this graphlet to
 be unremarkable (
\begin_inset CommandInset citation
LatexCommand citealp
key "Krumov2011"
literal "false"

\end_inset

, where our 4.3 graphlet corresponds to their 
\begin_inset Quotes xld
\end_inset

Motif 3
\begin_inset Quotes xrd
\end_inset

).
\end_layout

\begin_layout Standard
The higher correlation with lifetime for 3.2c and 4.3b could be because they
 are proxies for the number of papers published; 4.5c could be a proxy for
 the number of papers published by your collaborators.
 The fact that the correlation is stronger than that of degree centrality
 could be because degree centrality is a poor measure of the number of papers
 published in a co-authorship network; the negative correlation of the clusterin
g coefficients is further evidence in favour of this as a higher number
 of authors per paper will result in a higher clustering coefficient.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/7pcacolour_lifetimes_motcor.pdf
	width 48text%

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/7pcamono_lifetimes_motcor.pdf
	width 48text%

\end_inset

 
\begin_inset Caption Standard

\begin_layout Plain Layout
The correlation of each graphlet frequency with lifetime
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Correlation of each graphlet frequency with lifetime
\end_layout

\end_inset

 for graphlets in the coloured treatment (left) and uncoloured treatment
 (right).
 The right-hand figure also includes the standard node centralities
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Make scales the same.
 meh.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:importantgraphlets"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
To investigate whether graphlet profiles add any new insight compared to
 the standard node centralities, pairwise correlation analysis on the graphlet
 profile components and the centralities was performed, as shown in figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:correlations"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 It can be seen that the monochromatic graphlets 4.5a, 4.3a, 4.6a, 3.2a and
 2.1a are closely related, which is as expected as they are all subgraphs
 of 4.6a.
 Of particular interest is the that graphlets which correlate with lifetime
 also correlate with the various centralities: 4.3b, 4.5c, 4.6b and 3.2c all
 correlate strongly with betweenness centrality.
 Generally, this correlation plot was unsurprising and confirmed the need
 for further analysis to rule out degeneracies.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphletbccorr.pdf
	width 90text%

\end_inset

 
\begin_inset Caption Standard

\begin_layout Plain Layout
A pairwise correlation matrix of the various features on the subsample of
 co-authorship 3-ego-networks
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Correlation matrix of features of co-authorship 3-ego-networks
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Finish caption.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Might be worth segregating by 2,3,4 size as these can overlap
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% bottom line is overlapping - probably best to reduce the font size
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% need to include all centralities here
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:correlations"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Prediction of researcher lifetimes
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% This figure doesn't exist! Need to add it to importantgraphlets - also,
 we reference it twice now, which is probably not good.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Our results are shown in Fig~
\backslash
ref{fig:importantgraphlets}: various correlation measures are given for
 the metrics compared to researcher lifetime.
 While there are no strong correlations, betweenness centrality, degree,
 clustering and the graphlet frequencies are the least weak.
 The negative correlation with clustering is somewhat surprising and could
 warrant further investigation---it suggests that the efficiency of the
 number of papers per co-author could be important.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% these results should come after the correlation plots - don't want to
 give the game away too early.
 Narrative and all that.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% probably need to define features and mention clearly that the centralities
 refer to the centralities on the ego-networks
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% as to why we did this...
 dunno? seemed like a good idea at the time? centrality on whole network
 would be computationally a bit nuts
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Due to the high pairwise correlations between some of the centralities and
 the graphlets that were most highly correlated with lifetime, we tested
 for degeneracy amongst the features by performing linear regression analysis.
 The results can be seen in table
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "tab:regmono"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and table
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "tab:regpoly"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% def need more in methods on this
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Graphlet profile components with particularly insignificant results have
 been omitted for brevity.
 Many of the most strongly correlated graphlets with lifetime, 4.5c, 3.2c
 and 4.6b, were found to be degenerate to betweenness centrality.
 It can be seen that many of the centralities are much more significant
 than the graphlet profile
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlet profile
\end_layout

\end_inset

 components.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% include box motif that other paper talks about for comparison 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% do I need to explain degenerate? is it degenerate to or with?
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Comparing table
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "tab:regmono"
plural "false"
caps "false"
noprefix "false"

\end_inset

 with table
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "tab:regpoly"
plural "false"
caps "false"
noprefix "false"

\end_inset

, we can accept hypothesis 
\begin_inset CommandInset ref
LatexCommand ref
reference "h2"
plural "false"
caps "false"
noprefix "false"

\end_inset

: on the whole, the coloured graphlet profiles have more significant and
 stronger relationships with lifetime than in the monochrome treatment.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% poorly substantiated
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

We can also accept hypothesis 
\begin_inset CommandInset ref
LatexCommand ref
reference "h1"
plural "false"
caps "false"
noprefix "false"

\end_inset

: graphlet 4.3b is strongly and significantly linked with career lifetime,
 even after accounting for degree and betweenness centrality, with which
 it is correlated.
 It is displayed in figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:goodbadgraphlets"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 The large coefficient suggests that the graphlet's presence could explain
 a large amount of research lifetime, which could warrant further investigation.
 There are no other graphlets which are significantly linked to lifetime.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Standardise the coefficients by dividing by std !
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Some of the centralities are highly significant; betweenness centrality
 in particular has a very significant link, although it should be noted
 that it can only explain 4 years of lifetime.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% need to check whether this is actually new work.
 It seems highly unlikely.
 // Expand / reword
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

The significance of betweenness centrality contradicts earlier research,
 although the significant but small influence of degree on lifetime is in
 agreement 
\begin_inset CommandInset citation
LatexCommand citep
key "Li2013"
literal "false"

\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Read paper carefully - they seem to have similar hypotheses to us.
 Surely we should mention in background?
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Criticisms: have not controlled for field - could be that e.g, bc is higher
 in other fields (is that possible?)
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Regression analysis for graphlets in the monochromatic treatment versus
 research lifetime
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "tab:regmono"

\end_inset

 
\begin_inset FormulaMacro
\newcommand{\qrtr}{0.25\textwidth}
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Todo: centre table, stretch, check for rogue > signs
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% would be nice if significant values were bolded
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% give better feature names
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Is one supposed to discard the insignificant predictors and re-run?
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Have I done the right thing by commenting out the insignificant results?
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Check on e-5 etc presentation
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 
\backslash
begin{tabular}{p{
\backslash
qrtr}}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
centering
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset Tabular
<lyxtabular version="3" rows="9" columns="5">
<features tabularvalignment="middle">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<row>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 Feature
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\beta$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Std.Error
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
t value
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Pr(
\begin_inset Formula $>|t|$
\end_inset

)
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% preferred 
\backslash
svhline
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% -
\backslash
?
\backslash
d*
\backslash
.
\backslash
d* - regex to make dollars - need to check that e's aren't left out
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 3.1a
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.4$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.45$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.9$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.3688$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 3.2a & $-4.1$ & $2.8$ & $-1.5$ & $0.1413$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.1a & $-13.0$ & $50.0$ & $-0.26$ & $0.7920$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.2a & $-0.097$ & $0.41$ & $-0.24$ & $0.8106$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

4.3a
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $1.8$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.78$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $2.3$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.0227$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.4a & $2.5$ & $1.2$ & $2.2$ & $0.0308$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

4.5a
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $2.9$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.92$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $3.1$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.0018$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.6a & $1.9$ & $0.91$ & $2.1$ & $0.0323$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Betweenness
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $3.9$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.5$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $7.7$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $<1e-13$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Closeness & $-0.26$ & $0.81$ & $-0.32$ & $0.7503$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Clustering
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-0.69$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.22$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-3.2$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.0016$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Degree
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.1$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.024$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $4.4$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $<1e-4$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Edges
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-0.0026$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.00086$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-3.0$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.0027$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Ego-clustering & $-3.2$ & $2.5$ & $-1.3$ & $0.1924$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%   "footnotes" go here
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%   $^a$ The principal components as calculated in Sect.~
\backslash
ref{sec:methods}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

The features are described in detail in table
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "tab:allfeats"
plural "false"
caps "false"
noprefix "false"

\end_inset

; for the meanings of the other columns see section
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sec:mullin"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%   $^b$ Number of edges in ego-network
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Regression analysis for coloured graphlets versus research lifetime
\begin_inset CommandInset label
LatexCommand label
name "tab:colouredgraphlets_author_lifetime"

\end_inset


\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "tab:regpoly"

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Todo: probably need to include mean / s.d.
 for each so we get an idea of the explanatory power
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
centering
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset Tabular
<lyxtabular version="3" rows="10" columns="6">
<features tabularvalignment="middle">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<row>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% l = left align.
 c = centre
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 Feature
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\beta$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\beta_{\text{reduced}}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Std.Error
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
t value
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Pr(
\begin_inset Formula $>|t|$
\end_inset

)
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% preferred 
\backslash
svhline
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% -
\backslash
?
\backslash
d*
\backslash
.
\backslash
d* - regex to make dollars
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 2.1a & $4.8$ & $4.1e6$ & $1.2e-6$ & $1.0000$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 2.1b & $15.0$ & $4.1e6$ & $3.7e-6$ & $1.0000$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 3.1a
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $1.8$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $10.0$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $6.9$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $1.5$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.1385$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3.1b
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.95$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $4.0$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $1.6$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $2.6$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.0095$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 3.2a & $15.0$ & $7.4$ & $2.1$ & $0.0360$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 3.2b & $5.9$ & $4.2$ & $1.4$ & $0.1619$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 3.2c & $0.14$ & $3.7$ & $0.037$ & $0.9709$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.1a & $-5.6$ & $60.0$ & $-0.093$ & $0.9256$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.1b & $-85.0$ & $100.0$ & $-0.82$ & $0.4118$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.2a & $0.19$ & $3.9$ & $0.049$ & $0.9612$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.2b & $0.15$ & $1.0$ & $0.15$ & $0.8826$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

4.3a
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-4.1$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-0.95$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $3.4$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-1.2$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.2280$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.3b
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $10.0$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.34$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $2.7$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $3.8$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.0001$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.3c
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $1.3$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.077$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $1.8$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.69$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.4882$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.4a & $-4.9$ & $3.9$ & $-1.2$ & $0.2115$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.4b & $1.2$ & $1.6$ & $0.76$ & $0.4469$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.4c & $3.2$ & $2.1$ & $1.5$ & $0.1247$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.5a & $-0.74$ & $3.1$ & $-0.24$ & $0.8080$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.5b & $-3.5$ & $2.5$ & $-1.4$ & $0.1628$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.5c & $0.88$ & $1.4$ & $0.62$ & $0.5349$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.6a & $-4.0$ & $3.2$ & $-1.3$ & $0.2053$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.6b & $1.4$ & $1.6$ & $0.93$ & $0.3514$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.6c & $1.6$ & $1.6$ & $1.0$ & $0.2980$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 4.6d & $-0.64$ & $1.7$ & $-0.37$ & $0.7144$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Betweenness
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $4.2$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.74$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.75$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $5.7$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $<1e-7$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Closeness & $-2.3$ & $1.4$ & $-1.6$ & $0.1013$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Clustering
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-0.82$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-0.33$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.23$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $-3.5$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.0005$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Degree
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.12$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.42$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $0.028$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $4.3$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $<1e-4$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Edges & $-0.0024$ & $0.00093$ & $-2.5$ & $0.0110$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Ego-clustering & $2.1$ & $3.4$ & $0.63$ & $0.5290$
\backslash

\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
noalign
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset VSpace smallskip
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="none" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%   "footnotes" go here
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Dislike this - not very clear that it is a footnote
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

The features are described in detail in table
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "tab:allfeats"
plural "false"
caps "false"
noprefix "false"

\end_inset

; for the meanings of the other columns see section
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "sec:mullin"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
centering
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% % % % 
\backslash
% includegraphics[width=0.1% 
\backslash
% textwidth]{{{figures/graphlet_menagerie/3.2b}}}% 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.3b.pdf
	width 20text%

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Graphlet 4.3b
\end_layout

\end_inset

Graphlet 4.3b from figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:importantgraphlets"
plural "false"
caps "false"
noprefix "false"

\end_inset

, the only graphlet found to have a significant link with career lifetime,
 after controlling for many standard node centralities
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% not any more.
 reword.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "fig:goodbadgraphlets"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
Curiously, 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% not allowed
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

graphlets 4.1a and 4.1b do not appear to have any link with career length,
 despite their previously being found to be important for success as measured
 by citation count 
\begin_inset CommandInset citation
LatexCommand citep
key "Krumov2011"
literal "false"

\end_inset

.
 This distinction could be because that previous work looked at motifs (i.e,
 all subgraphs, not just induced ones), but one would expect to see the
 4.2 and 4.4 graphlets, of which 4.1 is a subgraph, correlating positively
 with lifetime with some certainty, which is not the case.
 4.2 seems to be negatively correlated with lifetime, if at all.
\end_layout

\begin_layout Standard
Finally, it is pertinent to note that none of the features tested have a
 very large impact on career-length, especially when considering the reduced
 coefficients, suggesting that there is a large element of random chance,
 or that some important factor has been left out of our analysis, which
 is consistent with other work 
\begin_inset CommandInset citation
LatexCommand citep
key "Milojevic2018"
literal "false"

\end_inset

.
\end_layout

\begin_layout Section
Conclusions
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "sec:conclusion"

\end_inset

 We have demonstrated that coloured graphlet profiles can be predictors
 for a real-world characteristic of nodes and this predictive power is not
 explained by the other standard centralities; particularly, degree centrality
 and betweenness centrality.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% In our application, our technique is as strongly correlated with the item
 we are trying to predict as the best commonly used metrics, to a confidence
 level of 95
\backslash
%.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
In our application, the 4.3b graphlet, the 4-star centred on the author of
 interest, was found to be the strongest predictor of career length in researche
rs in the fifth year of their career.
 Therefore, the best advice we can give to young researchers is that they
 should try to collaborate broadly with people who do not collaborate with
 each other.
 However, the correlations we have found do come with the usual caveat:
 there could be some other confounding variables that are at play—for example,
 it could be that the kind of people who collaborate with lots of people
 are interdisciplinarians who are particularly intelligent and therefore
 were more likely to be successful anyway, or that the success of the 4.3b
 graphlet compared to degree centrality could be because 4.3b requires more
 papers to be published than degree centrality; indeed, other work has found
 that the number of papers published is one of the main factors in determining
 career-length 
\begin_inset CommandInset citation
LatexCommand citep
key "Milojevic2018"
literal "false"

\end_inset

.
 This could be solved by a more thorough study with more information about
 each author.
 It could also be beneficial to use a dataset spanning more time as our
 current approach limits the maximum career length to 23 years—we are forced
 disregard any researcher with a career longer than that in order to ensure
 that we have a complete picture of all the careers of the researchers we
 consider.
\end_layout

\begin_layout Standard
Finally, it would be interesting to see if our technique has similar or
 greater success in other applications.
 Analysis of any network where there are interesting properties on the nodes
 which are linked to the medium scale structure of the network could plausibly
 benefit; for example, it could be used to detect groups of users on a social
 network spreading information in an inauthentic fashion.
 
\end_layout

\begin_layout Standard
Our main conclusion is that when characterising a node through its ego-network,
 it can be useful to retain the information as to which node is the ego
 and which are the alters.
 This information can be used by existing graph packages by colouring the
 ego one colour and the alters another.
 
\end_layout

\begin_layout Standard
\begin_inset Newpage newpage
\end_inset


\end_layout

\begin_layout Section*
Bibliography
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "monorefs"
options "authordate4"

\end_inset


\end_layout

\begin_layout Part
Network characterisation through matrix decompositions 
\begin_inset CommandInset label
LatexCommand label
name "part:Matrix-decomposition"

\end_inset


\end_layout

\begin_layout Chapter
Construction of data matrices and matrix decomposition
\begin_inset CommandInset label
LatexCommand label
name "chap:Construction-of-data"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
epigraph{``Things will explain themselves to you if you just know how to
 listen"}{--- 
\backslash
textup{Dad}}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
MyLettrine{T}{he dimensionality}
\end_layout

\end_inset

, or the number of degrees of freedom, in the topology of complex networks
 naively appears to be very high—of the order of 
\begin_inset Formula $N^{2}$
\end_inset

, where 
\begin_inset Formula $N$
\end_inset

 is the number of nodes in a network.
 In reality, many of those dimensions are redundant.
 Evidence for this was found in the previous part in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:correlations"
plural "false"
caps "false"
noprefix "false"

\end_inset

 where it was shown that 
\begin_inset Quotes xld
\end_inset

features
\begin_inset Quotes xrd
\end_inset

 of networks are highly correlated, in line with previous work 
\begin_inset CommandInset citation
LatexCommand citep
key "bounova_overview_2012,Ronqui2015"
literal "false"

\end_inset

.
 It stands to reason that, if one could combine the dimensions in such a
 way that only the 
\begin_inset Quotes xld
\end_inset

true
\begin_inset Quotes xrd
\end_inset

 degrees of freedom for complex networks were dimensions in some reduced
 space, that space could make characterising complex networks substantially
 easier.
 A more general form of this problem is well-recognised in many other fields;
 techniques for addressing it are termed 
\begin_inset Quotes xld
\end_inset

dimension reductions
\begin_inset Quotes xrd
\end_inset

, as was explored in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Matrix-decompositions"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
In this part, we build on work introduced in sections 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Network-characterisation"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Matrix-decompositions"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 There is some prior work applying dimension reductions to the task of character
ising whole networks; similar techniques were employed by 
\begin_inset CommandInset citation
LatexCommand citet
after "pp. 221-234"
key "Costa2007"
literal "false"

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand citet
key "bounova_overview_2012"
literal "false"

\end_inset

 and 
\begin_inset CommandInset citation
LatexCommand citet
key "Ronqui2015"
literal "false"

\end_inset

.
 The graphlet profiles explored in the previous part are also explored here.
\end_layout

\begin_layout Standard
The first task in finding a low-dimensional representation of a network
 is to find a high-dimensional representation of a network to which dimension
 reductions may be applied.
 That is the focus of this chapter, which is split into two sections: section
 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Construction-of-a"
plural "false"
caps "false"
noprefix "false"

\end_inset

 introduces the process by which we construct data matrices and section
 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Dimension-reductions"
plural "false"
caps "false"
noprefix "false"

\end_inset

 briefly discusses how we apply dimension reductions to these.
\end_layout

\begin_layout Section
Construction of a data matrix for network characterisation
\begin_inset CommandInset label
LatexCommand label
name "sec:Construction-of-a"

\end_inset


\end_layout

\begin_layout Standard
The standard form for high-dimensional representations of observations is
 the data matrix.
 
\begin_inset Quotes xld
\end_inset

Data matrix
\begin_inset Quotes xrd
\end_inset

 is a term from the field of machine learning which we adopt; each column
 is termed a 
\begin_inset Quotes xld
\end_inset

feature
\begin_inset Quotes xrd
\end_inset

 and each row is called an 
\begin_inset Quotes xld
\end_inset

example
\begin_inset Quotes xrd
\end_inset

.
 In our application, each example corresponds to a network.
 Each feature is an observable property of a network which is directly comparabl
e across all networks considered; for example, one feature might be 
\begin_inset Quotes xld
\end_inset

first moment of the degree distribution
\begin_inset Quotes xrd
\end_inset

 and another might be 
\begin_inset Quotes xld
\end_inset

network diameter
\begin_inset Quotes xrd
\end_inset

.
 The creation of data matrices allows us to easily apply standard techniques
 from other fields to characterise networks, as was explored by previous
 work 
\begin_inset CommandInset citation
LatexCommand citep
key "Costa2007,bounova_overview_2012,Ronqui2015"
literal "false"

\end_inset

.
 However, as was discussed in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Matrix-decompositions"
plural "false"
caps "false"
noprefix "false"

\end_inset

, these previous works used existing metrics as features which we believe
 is somewhat against the spirit of machine learning; one should apply it
 to the fullest representation of the data possible and allow the technique
 to discover which features are the most important.
 In this chapter, we explore how we can provide high-dimensional descriptions
 of networks to these data matrices.
\end_layout

\begin_layout Subsection
Requirements for data matrix
\end_layout

\begin_layout Standard
Each feature in the data matrix must be directly comparable between examples;
 if two examples have the same entry for a feature, it must mean that this
 observable property is in some way the same.
 An additional constraint is that all of the features must have physical
 dimensions consistent with the transformation applied by the machine learning
 techniques used; for example, a linear transformation demands that the
 features must have the same physical dimensions, because the addition of
 features must be a valid operation.
 We normalise our features to make them dimensionless.
 It is important to note that there is often no penalty to including features
 that do not provide discriminatory power, as the machine learning procedures
 will just discard them.
 Clearly, some features must have some discriminatory power for the procedure
 to provide anything useful.
 In this and the next chapter, we consider the features separately such
 that we can understand their usefulness separately.
 In a real application one would concatenate all of the relevant feature
 vectors into a single feature vector, which could increase the richness
 of the results.
\end_layout

\begin_layout Subsection
Features considered
\end_layout

\begin_layout Standard
In this section, we present a series of metrics and network representations
 and discuss whether they meet our criteria for features.
 In addition to the absolute requirements mentioned above, a good feature
 should also have the following properties:
\end_layout

\begin_layout Enumerate
similar networks should have small Euclidean distances between one another,
\end_layout

\begin_layout Enumerate
dissimilar networks, such as configuration models and the original networks,
 should have a fairly large distance between one another,
\end_layout

\begin_layout Enumerate
invariance with respect to the number of nodes in the graph.
\end_layout

\begin_layout Standard
In our view, as was discussed in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Matrix-decompositions"
plural "false"
caps "false"
noprefix "false"

\end_inset

, features should also be simple—each feature should have as little 
\begin_inset Quotes xld
\end_inset

human design
\begin_inset Quotes xrd
\end_inset

 in it as possible to allow the machine learning technique to make its own
 decisions.
\end_layout

\begin_layout Standard
All matrices are converted into feature vectors by simply flattening them
 such that
\begin_inset Formula 
\[
x_{i+(j-1)n}=a_{ij}\text{,}
\]

\end_inset

where 
\begin_inset Formula $\boldsymbol{x}$
\end_inset

 is the feature vector corresponding to the 
\begin_inset Formula $n\times m$
\end_inset

 matrix 
\begin_inset Formula $A$
\end_inset

.
\end_layout

\begin_layout Subsubsection
Adjacency matrices
\end_layout

\begin_layout Standard
Adjacency matrices
\begin_inset Index idx
status open

\begin_layout Plain Layout
network representations!adjacency matrix
\end_layout

\end_inset

 do not initially appear to satisfy the requirements on features: there
 is no guarantee that node labels (i.e.
 
\begin_inset Formula $\{i|v_{i}\in\mathcal{V}\}$
\end_inset

 where 
\begin_inset Formula $i\in\mathbb{N\setminus}\{0\}$
\end_inset

) between different graphs have the same meaning.
 We therefore relabel the nodes such that the node with the lowest degree
 has the lowest label and the node with the highest degree has the highest
 label; any nodes with the same degree are labelled randomly.
 This does mean that only networks with the same number of nodes can be
 meaningfully considered, even with padding—the decomposition would otherwise
 spot the hard cut-off as small networks had empty entries for any node
 labels higher than their number of nodes.
 It is unclear whether similar networks will have small distances between
 each other, nor whether dissimilar networks will have large distances between
 each other.
 This is something we will determine in testing in the next chapter.
 The feature is, however, very simple—it is the most common representation
 of networks.
\end_layout

\begin_layout Standard
We note with interest that node labels in synthetic graphs often hold a
 great deal of meaning.
 In many implementations of Watts-Strogatz small-world networks, nodes with
 adjacent or near-adjacent labels are neighbours and so have a far greater
 probability of being linked than nodes with labels which are very far apart.
 Similarly, Barabási-Albert models are usually constructed such that nodes
 with lower labels have higher degrees than nodes with higher labels as
 the first nodes added to the network are most likely to have links added
 to them.
 It is therefore very easy to be misled by the quality of these matrices;
 one receives very good separation between separate models due to the wealth
 of information contained in their labels.
 This information is clearly not 
\begin_inset Quotes xld
\end_inset

real
\begin_inset Quotes xrd
\end_inset

 in the sense that one is generally unable to construct a real network with
 labels which are as useful as that, so we take great care to discard it.
\end_layout

\begin_layout Subsubsection
\begin_inset Index idx
status open

\begin_layout Plain Layout
network representations!Laplacian matrix
\end_layout

\end_inset

Laplacian matrices
\end_layout

\begin_layout Standard
Laplacian matrices, similar to adjacency matrices, do not appear to satisfy
 the requirements on features as node labels are not directly comparable.
 Relabelling the nodes based on their degree mitigates this.
 Since we only consider networks with no self-loops, the Laplacian as a
 feature vector is clearly equivalent to including the degree sequence at
 the end of the adjacency matrix.
 As such, the Laplacian matrix adheres to our criteria in the same way as
 the adjacency matrices.
\end_layout

\begin_layout Subsubsection
Degree correlation matrices
\end_layout

\begin_layout Standard
The degree mixing matrix or degree correlation matrix
\begin_inset Index idx
status open

\begin_layout Plain Layout
degree correlation matrix
\end_layout

\end_inset

, 
\begin_inset Formula $D$
\end_inset

, is defined such that each entry
\begin_inset Formula 
\[
d_{i\leftarrow j}=P(\text{edge chosen at random connects a node of degree }j\text{ to a node of degree }i)\text{.}
\]

\end_inset

As it is a joint probability distribution, it is already normalised and
 suitable for usage with networks of any number of nodes.
 The degree mixing matrices are quick to calculate and take relatively little
 memory as, for most real-world networks, the maximum degree in a network
 grows very slowly with the size of the network.
 Networks with the same degree correlations will appear closely together;
 whether this is a good measure of similarity is an open question.
 For example, two Watts-Strogatz models with parameters of two and four
 neighbours could appear totally unrelated.
 The correlation matrix is simple, however; it can be seen as an adjacency
 matrix tallied up by degree and normalised.
\end_layout

\begin_layout Subsubsection
Graphlet profile
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlet profile
\end_layout

\end_inset

s
\end_layout

\begin_layout Standard
We generate graphlet profiles using the routines introduced in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Coloured-graphlet-profiles"
plural "false"
caps "false"
noprefix "false"

\end_inset

 but without any consideration of colour.
 We apply the same normalisation presented in sections 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Graphlet-profile"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:From-graphlet-profiles"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:profiles"
plural "false"
caps "false"
noprefix "false"

\end_inset

, again reasoning that we don't need to compare our networks to configuration
 models when we're comparing them to many other networks.
 With regards to our criteria, similar graphs will clearly have little differenc
e between them as they will have similar graphlet profiles.
 However, graphlet profiles do not consider the global topology of networks—for
 example, a network with a single connected component could potentially
 appear the same as a network with many disconnected components—so dissimilar
 networks might not appear far apart.
 Whether this is a problem in practice remains to be seen.
 Graphlet profiles are somewhat arbitrarily constructed as they have a couple
 of parameters: the size of the graphlets and the way that they are normalised.
 They are clearly simpler than some other network metrics such as betweenness
 centrality, however.
\end_layout

\begin_layout Section
Dimension reductions
\begin_inset CommandInset label
LatexCommand label
name "sec:Dimension-reductions"

\end_inset


\end_layout

\begin_layout Standard
As was mentioned as the start of this chapter, these data matrices are very
 high-dimensional characterisations of the networks.
 To make these characterisations interpretable, we must compress the number
 of dimensions to a more manageable number, such as two.
 We do this via matrix decomposition.
\end_layout

\begin_layout Standard
To decompose our data matrices, we use 
\begin_inset Index idx
status open

\begin_layout Plain Layout
principal component analysis
\end_layout

\end_inset

principal component analysis (PCA, 
\begin_inset CommandInset citation
LatexCommand citealp
key "wold_principal_1987"
literal "false"

\end_inset

) as introduced in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Matrix-decomposition-theory"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Many other techniques could be applied—related work applies multidimensional
 scaling or canonical variable analysis 
\begin_inset CommandInset citation
LatexCommand citep
key "Costa2007,schieber2017quantification"
literal "false"

\end_inset

.
 PCA was chosen because of its widespread usage and computational simplicity.
 We discuss this further and show our results in the following chapter.
 
\end_layout

\begin_layout Standard
\begin_inset Newpage newpage
\end_inset


\end_layout

\begin_layout Section*
Bibliography
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "monorefs"
options "authordate4"

\end_inset


\end_layout

\begin_layout Chapter
Application of matrix decomposition to characterisation of networks
\begin_inset CommandInset label
LatexCommand label
name "chap:Application-of-PCA"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
epigraph{``[Experimentalists must] seek and examine, as if they were indifferent
 divine beings, what is and not what pleases."}{--- 
\backslash
textup{Johann Wolfgang von Goethe,} Experiment as mediator between subject
 and object, 1792}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
MyLettrine{T}{he data}
\end_layout

\end_inset

 matrices introduced in the previous chapter are used here to characterise
 networks through the application of 
\begin_inset Index idx
status open

\begin_layout Plain Layout
principal component analysis
\end_layout

\end_inset

principal component analysis (PCA), a kind of matrix decomposition.
 We explore each feature described in the previous chapter in turn.
 We demonstrate the usage of our technique in distinguishing synthetic networks,
 distinguishing between networks and their configuration models, characterising
 piecewise static networks and assessing network damage.
\end_layout

\begin_layout Standard
In this section we consider only unweighted, undirected and uncoloured networks,
 with no multi-edges or self-loops.
 All networks we consider are converted to this form before any further
 processing is carried out; directed edges are replaced with undirected
 ones.
\end_layout

\begin_layout Standard
The features considered can clearly all be applied to directed and undirected
 networks simultaneously, but we believe this is unsound.
 In an directed network, 
\begin_inset Formula $a_{i\leftarrow j}=0$
\end_inset

 does not have the same implications as in an undirected network where 
\begin_inset Formula $a_{i\leftarrow j}=0\iff a_{j\leftarrow i}=0$
\end_inset

; the absence of an entry in an adjacency or Laplacian matrix means a very
 different thing.
 In practice, we found that PCA immediately separated the directed and undirecte
d networks and lost all nuance within directed and undirected networks,
 which we do not show for brevity.
 To compare the two kinds of networks meaningfully while preserving directedness
 would require substantially more research and thought than we have had
 time for.
 It is a matter of regret that we thereby contribute to the general neglect
 directed graphs face in network science 
\begin_inset CommandInset citation
LatexCommand citep
key "johnson2019digraphs"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard
This chapter comprises five sections: first, in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Types-of-network"
plural "false"
caps "false"
noprefix "false"

\end_inset

 we discuss the synthetic and real-world networks we attempt to characterise
 in this chapter; section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Characterisation-of-piecewise"
plural "false"
caps "false"
noprefix "false"

\end_inset

 characterises static networks; section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Characterisation-of-configuratio"
plural "false"
caps "false"
noprefix "false"

\end_inset

 shows how we can distinguish between networks and configuration models
 thereof for some of our data matrices; section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Characterising-temporal-networks"
plural "false"
caps "false"
noprefix "false"

\end_inset

 explores how we can track the evolution of real and synthetic piecewise
 static networks; and finally section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Conclusions-&-further"
plural "false"
caps "false"
noprefix "false"

\end_inset

 summarises our findings from this chapter and provides an outline of some
 possible directions for further research.
\end_layout

\begin_layout Section
Types of network considered
\begin_inset CommandInset label
LatexCommand label
name "sec:Types-of-network"

\end_inset


\end_layout

\begin_layout Standard
Here, we will briefly mention the different networks we attempt to characterise
 throughout this chapter.
 See sections 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Network-models"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Real-world-complex"
plural "false"
caps "false"
noprefix "false"

\end_inset

 for more detail.
\end_layout

\begin_layout Subsection
Synthetic networks
\end_layout

\begin_layout Standard
We use four types of synthetic networks, introduced in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Network-models"
plural "false"
caps "false"
noprefix "false"

\end_inset

, with various parameters, holding the number of nodes and density constant
 where features are sensitive to them.
 These networks are Erdős-Rényi 
\begin_inset CommandInset citation
LatexCommand citep
key "Erdos1959"
literal "false"

\end_inset

, Barabási-Albert 
\begin_inset CommandInset citation
LatexCommand citep
key "barabasi1999emergence"
literal "false"

\end_inset

, Watts-Strogatz 
\begin_inset CommandInset citation
LatexCommand citep
key "watts_collective_1998"
literal "false"

\end_inset

 and configuration models 
\begin_inset CommandInset citation
LatexCommand citep
key "newman_finding_2004"
literal "false"

\end_inset

.
 The specific implementations of these models are from the LightGraphs.jl
 package for Julia 
\begin_inset CommandInset citation
LatexCommand citep
key "seth_bromberger_2017"
literal "false"

\end_inset

.
 The Watts-Strogatz models are shown with various rewiring probabilities.
 The exact parameters are detailed in each figure and table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Models-considered-and"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Tabular
<lyxtabular version="3" rows="7" columns="3">
<features booktabs="true" tabularvalignment="middle" tabularwidth="98text%">
<column alignment="center" valignment="top" width="38text%">
<column alignment="center" valignment="top" width="10text%">
<column alignment="center" valignment="top" width="45text%">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Graph/model name
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Abbreviation
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Parameters/description
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!Erdős-Rényi
\end_layout

\end_inset

Erdős-Rényi
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
ER
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
liningnums{0.8}
\end_layout

\end_inset

% edge selection probability
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!Barabási-Albert
\end_layout

\end_inset

Barabási-Albert
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
BA
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
liningnums{2}
\end_layout

\end_inset

 edges per additional node
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!Watts-Strogatz
\end_layout

\end_inset

Watts-Strogatz
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
WS
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
liningnums{4}
\end_layout

\end_inset

 neighbours per node, various rewiring probabilities
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Configuration model
\begin_inset Index idx
status open

\begin_layout Plain Layout
network models!configuration model
\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Cfg model
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
The degree sequence of a graph 
\begin_inset Formula $\mathcal{G}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Food webs
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Eco
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Real-world ecological food webs with around a hundred nodes and hundreds
 of links each 
\begin_inset CommandInset citation
LatexCommand citep
key "Thompson"
literal "false"

\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Suffragette social tie network
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Suf
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
A network representing social interactions between members of the Suffragette
 suffragist movement with 112 nodes and 2690 edges 
\begin_inset CommandInset citation
LatexCommand citep
key "edwards_measures_2009"
literal "false"

\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Models considered and their parameters
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Models considered for dimension reductions and their parameters
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "tab:Models-considered-and"

\end_inset

; all models had 500 nodes and 1000 edges
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Real-world networks
\begin_inset Index idx
status open

\begin_layout Plain Layout
real-world networks
\end_layout

\end_inset


\end_layout

\begin_layout Standard
We only consider two types of real-world network as the figures became too
 hard to interpret when there were too many types of network displayed at
 once.
 This problem could be solved with creating interactive plots where metadata
 could be displayed about the networks as one hovered over them.
 Additionally, due to the fact that the adjacency and Laplacian networks
 can only compare networks with the same number of nodes, the real-world
 networks are left out for these as they are not all the same size.
\end_layout

\begin_layout Standard
The two types of network we do consider are a series of ecological
\begin_inset Index idx
status open

\begin_layout Plain Layout
real-world networks!ecological network
\end_layout

\end_inset

 food web networks 
\begin_inset CommandInset citation
LatexCommand citep
key "Thompson"
literal "false"

\end_inset

 and a 112-person clandestine network corresponding to social ties within
 the social network of the Suffragette
\begin_inset Index idx
status open

\begin_layout Plain Layout
real-world networks!Suffragette
\end_layout

\end_inset

, Emmeline Pankhurst 
\begin_inset CommandInset citation
LatexCommand citep
key "edwards_measures_2009"
literal "false"

\end_inset

.
 More detail on these real-world networks is given in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Real-world-complex"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Section
Characterisation of static networks
\begin_inset CommandInset label
LatexCommand label
name "sec:Characterisation-of-piecewise"

\end_inset


\end_layout

\begin_layout Standard
We first demonstrate the general use of all of the features considered on
 static networks.
 Later in the chapter, we drop the graphlet sizes which were found to be
 too small to be useful or too large to be practical from a computational
 perspective.
\end_layout

\begin_layout Subsection
Adjacency and Laplacian matrices
\begin_inset CommandInset label
LatexCommand label
name "subsec:Adjacency-and-Laplacian"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/pca_with_components/thesis_adj_pca_500n_1000e.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
Adjacency matrices of various networks whose nodes are labelled according
 to degree
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA applied to the adjacency matrices of various networks
\end_layout

\end_inset

, as detailed in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Construction-of-data"
plural "false"
caps "false"
noprefix "false"

\end_inset

, projected onto their first two principal components.
 Each point corresponds to a single network
\begin_inset CommandInset label
LatexCommand label
name "fig:Adjacency-matrix"

\end_inset

 and the abbreviations used are explained in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Models-considered-and"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/pca_with_components/thesis_lap_pca_500n_1000e.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
Laplacian matrices of various networks whose nodes are labelled according
 to degree
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA applied to the Laplacian matrices of various networks
\end_layout

\end_inset

, as detailed in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Construction-of-data"
plural "false"
caps "false"
noprefix "false"

\end_inset

, projected onto their first two principal components.
 Each point corresponds to a single network
\begin_inset CommandInset label
LatexCommand label
name "fig:Laplacian-matrix"

\end_inset

 and the abbreviations used are explained in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Models-considered-and"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/pca_with_components/thesis_adj_pca_500n_1000e_c1.pdf.jpg
	display false
	width 50text%
	groupId sidebyside

\end_inset


\begin_inset Graphics
	filename figures/pca_with_components/thesis_adj_pca_500n_1000e_c2.pdf.jpg
	display false
	width 50text%
	groupId sidebyside

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
The first two principal components of adjacency matrices
\end_layout

\end_inset

The first (left) and second (right) principal components of the adjacency
 matrices corresponding to those in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Adjacency-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

, reconstructed as matrices
\begin_inset CommandInset label
LatexCommand label
name "fig:adj-components"

\end_inset

.
 Contrast has been enhanced as described in 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Adjacency-and-Laplacian"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/pca_with_components/thesis_lap_pca_500n_1000e_c1.pdf.jpg
	display false
	width 50text%
	groupId sidebyside

\end_inset


\begin_inset Graphics
	filename figures/pca_with_components/thesis_lap_pca_500n_1000e_c2.pdf.jpg
	display false
	width 50text%
	groupId sidebyside

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
The first two principal components of Laplacian matrices
\end_layout

\end_inset

The first (left) and second (right) principal components of the Laplacian
 matrices corresponding to those in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Laplacian-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

, reconstructed as matrices
\begin_inset CommandInset label
LatexCommand label
name "fig:lap-components"

\end_inset

.
 Contrast has been enhanced as described in 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Adjacency-and-Laplacian"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 The diagonals are shown separately in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Laplacian-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/thesis_lap_diag_c1.pdf
	display false
	width 50text%
	groupId sidebyside

\end_inset


\begin_inset Graphics
	filename figures/thesis_lap_diag_c2.pdf
	display false
	width 50text%
	groupId sidebyside

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
The diagonals of the first two principal components of Laplacian matrices
\end_layout

\end_inset

The diagonals of the first (left) and second (right) principal components
 of the Laplacian matrices corresponding to those in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Laplacian-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

, reconstructed as matrices
\begin_inset CommandInset label
LatexCommand label
name "fig:lap-components-diags"

\end_inset

.
 These diagonals correspond to the degree
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
The dimension reductions of the Laplacian
\begin_inset Index idx
status open

\begin_layout Plain Layout
network representations!Laplacian matrix
\end_layout

\end_inset

 and adjacency matrices
\begin_inset Index idx
status open

\begin_layout Plain Layout
network representations!adjacency matrix
\end_layout

\end_inset

, figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Laplacian-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Adjacency-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

 respectively, show similar results.
 The Laplacian matrices show a greater separation than the adjacency matrices.
 Both figures show that there is a clear separation between the scale-free
 graphs on one side and the small-world and random graphs on the other.
 Strikingly in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Laplacian-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and less clearly in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Adjacency-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

, one can observe that as Watts-Strogatz models have their wiring probability
 increased, they become less like the Erdős-Rényi random graphs, which is
 quite counter-intuitive.
 One would expect as more randomness was added to a nearest-neighbour graph
 that it would become more like a random graph.
 This warrants further investigation, so we will now examine the principal
 components in detail.
\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: this may be questioned in viva, especially for adjacency
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The first two principal components are shown reconstructed as matrices in
 figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:adj-components"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:lap-components"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 We have enhanced the contrast in these plots (and figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:mix-components"
plural "false"
caps "false"
noprefix "false"

\end_inset

) by mapping the value of each cell, 
\begin_inset Formula $x$
\end_inset

, such that
\begin_inset Formula 
\[
x\mapsto\text{sgn}(x)\sqrt{\lvert x\rvert}\text{,}
\]

\end_inset

where 
\begin_inset Formula $\text{sgn}$
\end_inset

 is the signum or sign function.
 This transformation improves legibility around zero, aiding interpretation.
 The contrast was then further enhanced after plotting by editing the 
\begin_inset Quotes xld
\end_inset

curves
\begin_inset Quotes xrd
\end_inset

 in an image editor.
\end_layout

\begin_layout Standard
The first component of the adjacency matrix, shown in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:adj-components"
plural "false"
caps "false"
noprefix "false"

\end_inset

, has positive elements for nodes of higher degree linking to nodes of all
 degrees, and negative elements for nodes of lower degree linking to nodes
 of lower degree; i.e.
\begin_inset space \space{}
\end_inset

it looks like the adjacency matrix of a scale-free network.
 This explains why the scale-free network is on the right hand side of figure
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Adjacency-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 In figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Adjacency-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

, there is no discernible pattern along the second principal component.
 It is therefore unsurprising that when one inspects the second component
 in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:adj-components"
plural "false"
caps "false"
noprefix "false"

\end_inset

, it appears to be uniformly random.
\end_layout

\begin_layout Standard
The components relating to the Laplacian matrices, shown in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:lap-components"
plural "false"
caps "false"
noprefix "false"

\end_inset

, seem to be composed from the components given for the adjacency matrices
 with the addition of a diagonal, as one might expect given the definition
 of the Laplacian matrix.
 What is unexpected is that the first component seems to have had its adjacency
 matrix negated but without any effect on where the scale-free networks
 lie in the projected space.
 The diagonals are interesting and are understood more readily from figure
 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:lap-components-diags"
plural "false"
caps "false"
noprefix "false"

\end_inset

: the first component's diagonal suggests that the first moment of the degree
 of the network is tightly centred on the highest degree nodes; i.e.
\begin_inset space \space{}
\end_inset

they are scale-free networks, with a smooth negative region for the rest
 of the nodes.
 The diagonal of the second component is quite pronounced: it has discrete
 steps where degree strength falls at around 50, 200 and 350 nodes, before
 it peaks sharply in the positive direction around node 500.
 If we take the negative direction of this, it represents a degree distribution
 where most nodes have a degree of one of four levels.
\end_layout

\begin_layout Standard
From this, the positions of the networks can be explained: Erdős-Rényi graphs
 have a negative first component, corresponding to a smooth, fairly even
 degree distribution, and a positive second component which cancels out
 the spike at the end of the first component.
 Watts-Strogatz graphs have varying levels of a negative first component
 and the second component can become positive, which vary in tandem according
 to the rewiring probability.
 More rewiring probability leads to more distance from the Erdős-Rényi graphs.
 To understand this, one must go back to the algorithm in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Network-models"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Only a single link per node can be rewired, so a hundred-percent rewiring
 probability only corresponds to (number of nodes)
\begin_inset Formula $\cdot$
\end_inset

(number of neighbours)
\begin_inset Formula $^{-1}$
\end_inset

 edges being rewired, and these rewired edges will have half of their ends
 evenly distributed throughout the nodes.
 We therefore must have a degree distribution which is quite tightly centred
 on the original distribution (such as the one shown in figure 
\begin_inset CommandInset ref
LatexCommand vref
reference "fig:ws-deg-dist-mix"
plural "false"
caps "false"
noprefix "false"

\end_inset

), hence the large discrete steps in the second component, which the degree
 distributions of the Erdős-Rényi models (such as the one shown in figure
 
\begin_inset CommandInset ref
LatexCommand vref
reference "fig:er-deg-dist-mix"
plural "false"
caps "false"
noprefix "false"

\end_inset

) do not possess.
 The Watts-Strogatz graphs with a low rewiring probability will have tight
 degree distributions and so the diagonal of the Laplacian matrix will be
 a constant, which is closer to the first principal component than it is
 to the second.
 These facts all explain why the increased rewiring probability moves Watts-Stro
gatz further from Erdős-Rényi for the Laplacian matrices, which is the opposite
 to intuition.
\end_layout

\begin_layout Standard
It is still unclear why the plots of the adjacency matrix suggest that Erdős-Rén
yi graphs are closer to Watts-Strogatz with low rewiring probability.
\end_layout

\begin_layout Subsection
Degree correlation
\begin_inset Index idx
status open

\begin_layout Plain Layout
degree correlation matrix
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The decomposed degree correlation matrices (figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Degree-correlation-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

) show good separation between different models.
 It illustrates clearly how, with rewiring, a Watts-Strogatz model becomes
 more like an Erdős-Rényi random graph as the rewiring probability increases
 from the bottom-right to the middle-top.
 It would be pertinent to note that this is exactly the opposite behaviour
 to that seen in figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Adjacency-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Laplacian-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

, which we explore in this section.
\end_layout

\begin_layout Standard
The figure also shows the similarities between the ecological networks,
 the Suffragette network and the Barabási-Albert network which are all clustered
 together in the top left.
\end_layout

\begin_layout Standard
Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:mix-components"
plural "false"
caps "false"
noprefix "false"

\end_inset

 shows how the projection manages such good separation between the scale-free
 and small-world networks.
 The first principal component has negative elements for low-degree nodes
 linking to high-degree nodes—i.e.
\begin_inset space \space{}
\end_inset

it is negative for the types of degree mixing matrices one would expect
 from a scale-free graph, which is why the scale-free models appear on the
 left in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Degree-correlation-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 It has positive elements for nodes of degree 4 linking to nodes of degree
 4 or 3, and some strength for nodes of degree 5 linking to nodes of degree
 5.
 This matches the parameters in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Models-considered-and"
plural "false"
caps "false"
noprefix "false"

\end_inset

 so the positive direction along this component therefore represents nearest-nei
ghbour and potentially small-world networks.
\end_layout

\begin_layout Standard
The second principal component, also shown in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:mix-components"
plural "false"
caps "false"
noprefix "false"

\end_inset

, has a strong positive element for nodes of degree 4 linking to nodes of
 degree 4, surrounded by a 
\begin_inset Quotes xld
\end_inset

smear
\begin_inset Quotes xrd
\end_inset

 of negative elements for, e.g.
\begin_inset space \space{}
\end_inset

nodes of degree 5 linking to nodes of degree 5.
 It has positive elements for nodes of low degree linking to nodes of high
 degree, with a characteristic striping going out to very high degree, signifyin
g the small numbers of nodes with high degree in scale-free networks.
 This principal component therefore does double duty: its positive end represent
s nearest-neighbour networks with no shortcuts and scale-free networks;
 its negative end represents Poisonnian degree distributions such as those
 found in Erdős-Rényi graphs or small-world graphs with a very high number
 of shortcuts.
\end_layout

\begin_layout Standard
There are some clear limitations of this technique, however.
 It is very sensitive to the density of the small-world and Erdős-Rényi
 networks: any increase in density would move the 
\begin_inset Quotes xld
\end_inset

blobs
\begin_inset Quotes xrd
\end_inset

 seen in the bottom left of the reconstructed components diagonally upwards,
 such that the projection would portray the networks as totally unrelated.
 It is less sensitive to the density of scale-free networks, however, as
 the smooth fall-off in the likelihood of higher-degree nodes linking to
 higher-degree nodes means that networks with higher average degree will
 just appear as more 
\begin_inset Quotes xld
\end_inset

extreme
\begin_inset Quotes xrd
\end_inset

 versions of the less dense scale-free networks.
 Additionally, unlike the adjacency and Laplacian matrices, there is no
 requirement that the networks are the same size.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/pca_with_components/thesis_mix_pca_500n_1000e.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
Degree correlation matrices of various networks
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA applied to the degree mixing matrices of various networks
\end_layout

\end_inset

, as detailed in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Construction-of-data"
plural "false"
caps "false"
noprefix "false"

\end_inset

, projected onto their first two principal components.
 Each point corresponds to a single network
\begin_inset CommandInset label
LatexCommand label
name "fig:Degree-correlation-matrix"

\end_inset

 and the abbreviations used are explained in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Models-considered-and"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/pca_with_components/thesis_mix_pca_500n_1000e_c1.pdf.jpg
	display false
	width 50text%
	groupId sidebyside

\end_inset


\begin_inset Graphics
	filename figures/pca_with_components/thesis_mix_pca_500n_1000e_c2.pdf.jpg
	display false
	width 50text%
	groupId sidebyside

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
The first two principal components of degree mixing matrices
\end_layout

\end_inset

The first (left) and second (right) principal components of the degree correlati
on matrices corresponding to those in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Degree-correlation-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

, reconstructed as matrices
\begin_inset CommandInset label
LatexCommand label
name "fig:mix-components"

\end_inset

.
 Contrast has been enhanced as described in 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Adjacency-and-Laplacian"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Graphlets
\end_layout

\begin_layout Standard
It is important to use the smallest graphlet possible: the complexity of
 the graphlet enumeration algorithm we use scales with the number of unique
 subgraphs of the size of the graphlet in the graph.
 For random graphs of constant density, the number of unique subgraphs of
 size 
\begin_inset Formula $n$
\end_inset

 is approximately 
\begin_inset Formula ${E \choose n}\propto E^{n}$
\end_inset

, where 
\begin_inset Formula $E$
\end_inset

 is the number of edges in the graph.
 We therefore tested a few sizes of graphlet and chose the one which struck
 the best balance between computation time and discriminatory power.
\end_layout

\begin_layout Subsubsection
3-graphlets
\end_layout

\begin_layout Standard
The figure for the three-graphlets, figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:PCA-applied-to-1"
plural "false"
caps "false"
noprefix "false"

\end_inset

, is not particularly informative; it does not show much separation between
 the different graphs and seems to be totally linear with no second principal
 component.
 Watt-Strogatz graphs with a rewiring probability of 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
liningnums{10}
\end_layout

\end_inset

% can be distinguished from 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
liningnums{30}
\end_layout

\end_inset

%.
 All of the others are hard to distinguish with many networks totally obscured
 by others.
 This is because an uncoloured 3-graphlet profile is a proxy for the global
 clustering co-efficient: it simply measures the percentages of paths of
 length 3 which are closed and open.
 It is therefore unsurprising that it is one-dimensional.
 It is interesting to note that the Suffragette network appears to have
 a very different clustering coefficient to all of the other graphs.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/thesis_mot_pca_500n_1000e_3motifs.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA applied to a data matrix formed from the graphlet profile
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlet profile
\end_layout

\end_inset

s of 3-graphlets 
\end_layout

\end_inset

A data matrix formed from the graphlet profiles of 3-graphlets of various
 networks of various types.
 The networks are shown projected onto their first two principal components;
 each point is a single network
\begin_inset CommandInset label
LatexCommand label
name "fig:PCA-applied-to-1"

\end_inset

 and the abbreviations used are explained in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Models-considered-and"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
4-graphlets
\end_layout

\begin_layout Standard
The figure for the four-graphlets, figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:PCA-applied-to-2"
plural "false"
caps "false"
noprefix "false"

\end_inset

, is much more informative.
 There is clear separation between many of the models and the similarities
 between the Barabási-Albert model and the ecological networks are clear.
 It is interesting that the Suffragette network is so much of an outlier
 in this plot compared to the degree correlation figure (figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Degree-correlation-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

); it is presumably due to the high clustering coefficient as was seen in
 figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:PCA-applied-to-1"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 The transformation of the Watts-Strogatz model into something more like
 the Erdős-Rényi model is once again clear from the middle-left to the bottom-le
ft.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/thesis_mot_pca_500n_1000e_4motifs.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA applied to a data matrix formed from the graphlet profiles of 4-graphlets
\end_layout

\end_inset

A data matrix formed from the graphlet profile
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlet profile
\end_layout

\end_inset

s of 4-graphlets of various networks of various types.
 The networks are shown projected onto their first two principal components;
 each point is a single network
\begin_inset CommandInset label
LatexCommand label
name "fig:PCA-applied-to-2"

\end_inset

 and the abbreviations used are explained in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Models-considered-and"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
5-graphlets
\end_layout

\begin_layout Standard
The figure for the five-graphlets, figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:PCA-applied-to-3"
plural "false"
caps "false"
noprefix "false"

\end_inset

, also shows separation between the different graphs.
 Curiously, compared to figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:PCA-applied-to-2"
plural "false"
caps "false"
noprefix "false"

\end_inset

, there is slightly more separation between the Barabási-Albert model and
 the ecological networks.
 This could be because the number of 
\begin_inset Index idx
status open

\begin_layout Plain Layout
trophic level
\end_layout

\end_inset

trophic levels in ecological networks is often no higher than five 
\begin_inset CommandInset citation
LatexCommand citep
key "lindeman1942trophic"
literal "false"

\end_inset

, whereas long chains are quite possible in Barabási-Albert models.
\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: Harri says we should expand on this a bit
\end_layout

\begin_layout Plain Layout
I'm reluctant because my external examiner is a world expert on ecological
 networks...
\end_layout

\end_inset

 There is not a great deal of difference between this and the case with
 the four-graphlets, but this takes far longer to run (approximately ten
 minutes rather than ten seconds), so from now on we only consider four-graphlet
s.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/thesis_mot_pca_500n_1000e_5motifs.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA applied to a data matrix formed from the graphlet profiles of 5-graphlets
\end_layout

\end_inset

A data matrix formed from the graphlet profiles of 5-graphlets of various
 networks of various types.
 The networks are shown projected onto their first two principal components;
 each point is a single network
\begin_inset CommandInset label
LatexCommand label
name "fig:PCA-applied-to-3"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Characterisation of configuration models
\begin_inset CommandInset label
LatexCommand label
name "sec:Characterisation-of-configuratio"

\end_inset


\end_layout

\begin_layout Standard
A configuration model
\begin_inset Index idx
status open

\begin_layout Plain Layout
configuration model
\end_layout

\end_inset

, as described in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Configuration-model"
plural "false"
caps "false"
noprefix "false"

\end_inset

, generates graphs similar to an existing graph.
 Here, the configuration model ensures that the generated graphs have the
 same degree sequence, 
\begin_inset Formula $\{k_{i}\vert v_{i}\in\mathcal{V}\}$
\end_inset

, as the original graph.
 Whether our techniques can distinguish between 
\begin_inset Quotes xld
\end_inset

real graphs
\begin_inset Quotes xrd
\end_inset

 and their configuration models is a good test of how well they characterise
 the networks; configuration models are only superficially similar to the
 original networks 
\begin_inset CommandInset citation
LatexCommand citep
key "newman_finding_2004"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard
Figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Adjacency-matrices-cfg"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Laplacian-matrices-cfg"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Degree-mixing-matrices-cfg"
plural "false"
caps "false"
noprefix "false"

\end_inset

, corresponding to adjacency, Laplacian and degree correlation matrices
 respectively, show no difference between the configuration models and the
 original networks.
 Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:4-graphlet-profiles-cfg"
plural "false"
caps "false"
noprefix "false"

\end_inset

 from the four-graphlets shows a clear difference, which is unsurprising
 as configuration models are well-known for reducing the clustering coefficient,
 which graphlet profiles can capture.
\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: expand if time
\end_layout

\end_inset

 It is a surprise that the degree correlation matrix performed so badly,
 as our configuration models only impose the constraint that the degree
 sequence is retained; all edges are otherwise assigned at random, so one
 would expect the correlations to be lost.
 Presumably the homogeneity of the Watts-Strogatz model has ensured that
 the degree correlations are preserved.
 It could be fruitful to run this on more models, particularly more heterogeneou
s ones, and their configuration models and extract the principal components.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/cfg_model/GFPCA_adj_motifs.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA applied to adjacency matrices of configuration models
\end_layout

\end_inset

Adjacency matrices, as in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Adjacency-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

, projected onto their first two principal components of Watts-Strogatz
 synthetic networks and configuration models of those networks
\begin_inset CommandInset label
LatexCommand label
name "fig:Adjacency-matrices-cfg"

\end_inset

.
 The abbreviations used are explained in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Models-considered-and"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/cfg_model/GFPCA_laplacian_motifs.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA applied to Laplacian matrices of configuration models
\end_layout

\end_inset

Laplacian matrices, as in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Laplacian-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

, projected onto their first two principal components of Watts-Strogatz
 synthetic networks and configuration models of those networks
\begin_inset CommandInset label
LatexCommand label
name "fig:Laplacian-matrices-cfg"

\end_inset

.
 The abbreviations used are explained in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Models-considered-and"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/cfg_model/GFPCA_degree_cf.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA applied to degree mixing matrices of configuration models
\end_layout

\end_inset

Degree mixing matrices, as in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Degree-correlation-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

, projected onto their first two principal components of Watts-Strogatz
 synthetic networks and configuration models of those networks
\begin_inset CommandInset label
LatexCommand label
name "fig:Degree-mixing-matrices-cfg"

\end_inset

.
 The abbreviations used are explained in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Models-considered-and"
plural "false"
caps "false"
noprefix "false"

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: this figure is real but we can tell the difference between BA configuratio
n and original which is interesting for viva
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/cfg_model/GFPCA_mf_cf.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA applied to 4-graphlet profiles of configuration models
\end_layout

\end_inset

4-graphlet profiles, as in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:PCA-applied-to-2"
plural "false"
caps "false"
noprefix "false"

\end_inset

, projected onto their first two principal components of Watts-Strogatz
 synthetic networks and configuration models of those networks
\begin_inset CommandInset label
LatexCommand label
name "fig:4-graphlet-profiles-cfg"

\end_inset

.
 The abbreviations used are explained in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Models-considered-and"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Characterising 
\begin_inset Index idx
status open

\begin_layout Plain Layout
temporal networks
\end_layout

\end_inset

temporal networks
\begin_inset CommandInset label
LatexCommand label
name "sec:Characterising-temporal-networks"

\end_inset


\end_layout

\begin_layout Standard
Our ultimate vision with this section is to eventually end up with something
 analogous to the famous Hertzsprung-Russell diagrams 
\begin_inset CommandInset citation
LatexCommand citep
key "Russell1914"
literal "false"

\end_inset

 from stellar astronomy where we can plot the life cycle of networks and
 categorise new networks based on where they appear within our projected
 diagrams.
 This clearly doesn't make much sense when comparing networks across domains—it
 strikes us as unlikely that a food web is likely to turn into a road network,
 for example—so in the second part of this section we return to the collaboratio
n network investigated in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Career-length-in"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Our first section considers damage to synthetic networks.
\end_layout

\begin_layout Subsection
Synthetic network damage
\begin_inset CommandInset label
LatexCommand label
name "subsec:network-damage"

\end_inset


\end_layout

\begin_layout Standard
Figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Degree-mixing-matrices"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Degree-mixing-matrices-trained"
plural "false"
caps "false"
noprefix "false"

\end_inset

 show the same mix of synthetic networks characterised by their projections
 onto their first two principal components according to their 
\begin_inset Index idx
status open

\begin_layout Plain Layout
degree correlation matrix
\end_layout

\end_inset

degree correlation matrices shown in previous sections.
 We have added a new network: a Watts-Strogatz network which has had a single
 link, chosen uniformly at random, severed at each time-step.
 The journey of the network from something indistinguishable to a small-world
 network to something more like a scale-free network can clearly be seen
 as it travels from right to left.
 It is interesting to note that the space it travels through is a kind of
 
\begin_inset Quotes xld
\end_inset

no-man's land
\begin_inset Quotes xrd
\end_inset

 where no undamaged network is projected.
\end_layout

\begin_layout Standard
The only difference between the two figures is that figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Degree-mixing-matrices"
plural "false"
caps "false"
noprefix "false"

\end_inset

 is only 
\begin_inset Quotes xld
\end_inset

trained
\begin_inset Quotes xrd
\end_inset

 on the undamaged networks, a so-called 
\begin_inset Quotes xld
\end_inset

online
\begin_inset Quotes xrd
\end_inset

 method, whereas 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Degree-mixing-matrices-trained"
plural "false"
caps "false"
noprefix "false"

\end_inset

 is 
\begin_inset Quotes xld
\end_inset

trained
\begin_inset Quotes xrd
\end_inset

 on all of the networks, an 
\begin_inset Quotes xld
\end_inset

offline
\begin_inset Quotes xrd
\end_inset

 approach; i.e.
\begin_inset space ~
\end_inset

in the online approach, the transformation matrix 
\begin_inset Formula $W$
\end_inset

 is calculated from a data matrix 
\begin_inset Formula $X$
\end_inset

 which excludes the undamaged network.
 This could be important for real-world applications where data are streaming
 in in real-time, as a very large ensemble of reference networks could be
 used to find the projection matrices once and new data could be projected
 as it was added.
 It can be seen that there are limited differences between the two figures;
 predominantly that the end-point of the damaged small-world network is
 not exactly within the scale-free cluster in the 
\begin_inset Quotes xld
\end_inset

trained on all networks
\begin_inset Quotes xrd
\end_inset

 version, i.e.
\begin_inset space \space{}
\end_inset

figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Degree-mixing-matrices-trained"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/ChoppedJustProjectedNotTrainedSW01.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Degree-mixing-matrices"

\end_inset


\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA applied to degree mixing matrices showing network damage with an 
\begin_inset Quotes xld
\end_inset

online
\begin_inset Quotes xrd
\end_inset

 method
\end_layout

\end_inset

Degree mixing matrices projected onto their first two principal components,
 as in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Degree-correlation-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

, for 100-node networks via an 
\begin_inset Quotes xld
\end_inset

online
\begin_inset Quotes xrd
\end_inset

 method.
 Particularly of note are the 
\begin_inset Quotes xld
\end_inset

Chopped SW
\begin_inset Quotes xrd
\end_inset

 networks which show a network created according to a Watts-Strogatz model
 with a 10% rewiring probability where a random link has been removed at
 each time step which can be see as the network moves from right to left.
 The abbreviations used are explained in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Models-considered-and"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/ChoppedSW01.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig:Degree-mixing-matrices-trained"

\end_inset


\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA applied to degree mixing matrices showing network damage with an 
\begin_inset Quotes xld
\end_inset

offline
\begin_inset Quotes xrd
\end_inset

 method
\end_layout

\end_inset

Degree mixing matrices projected onto their first two principal components,
 as in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Degree-correlation-matrix"
plural "false"
caps "false"
noprefix "false"

\end_inset

, for 100-node networks via an 
\begin_inset Quotes xld
\end_inset

offline
\begin_inset Quotes xrd
\end_inset

 method.
 Particularly of note are the 
\begin_inset Quotes xld
\end_inset

Chopped SW
\begin_inset Quotes xrd
\end_inset

 networks which show a network created according to a Watts-Strogatz model
 with a 10% rewiring probability where a random link has been removed at
 each time step which can be see as the network moves from right to left.
 The abbreviations used are explained in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Models-considered-and"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Characterisation of ego-networks
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego-network
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sec:Characterisation-of-ego-networks"

\end_inset


\end_layout

\begin_layout Standard
The techniques described in this part can be applied to the problem of describin
g and comparing individual nodes within a graph by considering the neighbourhood
 of each node as a separate graph.
 This work expands on that of chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Career-length-in"
plural "false"
caps "false"
noprefix "false"

\end_inset

: we apply matrix decompositions to the coloured and monochromatic graphlet
 profiles of authors in an academic collaboration network
\begin_inset Index idx
status open

\begin_layout Plain Layout
real-world networks!co-authorship network
\end_layout

\end_inset

.
 
\begin_inset CommandInset citation
LatexCommand citet
key "Cunningham2013"
literal "false"

\end_inset

 propose a similar technique for characterising nodes by the decomposition
 of data matrices formed from their graphlet profiles but do not use colour
 to denote the position of the ego or examine the effect of colour.
 Here, in contrast to chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Career-length-in"
plural "false"
caps "false"
noprefix "false"

\end_inset

, we try to understand how the patterns of collaboration change over a researche
r's career rather than trying to determine which graphlets make a long career
 more likely.
\end_layout

\begin_layout Standard
As in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Career-length-in"
plural "false"
caps "false"
noprefix "false"

\end_inset

, we use disambiguated data from Arnetminer 
\begin_inset CommandInset citation
LatexCommand citep
key "Tang2008"
literal "false"

\end_inset

, covering the years 1986–2012.
 The window size was set to three years.
\end_layout

\begin_layout Standard
We picked one author as an example to investigate, known only as 
\begin_inset Quotes xld
\end_inset

number 7
\begin_inset Quotes xrd
\end_inset

, as they seemed to have a relatively long career.
 Most authors do not: they fizzle out after a few years, as was shown in
 figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:lifedist"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Motifs"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Motifs-(coloured)"
plural "false"
caps "false"
noprefix "false"

\end_inset

 show the changes in the graphlet profiles of the 3-ego-networks of our
 author of interest.
 These diagrams are quite hard to interpret; one must cross-reference with
 figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:motifs"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 With some effort, it can be seen from figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Motifs"
plural "false"
caps "false"
noprefix "false"

\end_inset

 that the diversity of graphlets and the number of more complex graphlets
 (graphlets which are less likely to appear by random chance) increases
 as time goes on.
 These more complex graphlets include long chains and loops.
\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: Harri asks what we conclude from this.
 I think the answer is probably 
\begin_inset Quotes xld
\end_inset

not much
\begin_inset Quotes xrd
\end_inset

.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/researcherlifecycle_colour/figures/sevenmotifs_old.png
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
The proportion of monochrome graphlets
\begin_inset Argument 1
status collapsed

\begin_layout Plain Layout
Proportion of monochrome graphlets in one authors' ego-network
\end_layout

\end_inset

 of each type found in the 3-ego-networks of the author known as 
\begin_inset Quotes xld
\end_inset

7
\begin_inset Quotes xrd
\end_inset

 in the Arnetminer dataset.
 The graphlet labels correspond to those in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Motifs-(coloured)"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\begin_inset CommandInset label
LatexCommand label
name "fig:Motifs"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
This similarly can be seen in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Motifs-(coloured)"
plural "false"
caps "false"
noprefix "false"

\end_inset

, with the added complication that as time goes on, the ego
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego
\end_layout

\end_inset

 can be initially be found in more of the graphlets, but this subsides after
 the first decade—i.e.
 the 
\begin_inset Quotes xld
\end_inset

a
\begin_inset Quotes xrd
\end_inset

 variants of graphlets become more common—and could potentially reflect
 the author becoming more of a bridge for knowledge rather than an active
 researcher themselves.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/researcherlifecycle_colour/figures/sevenmotifs.png
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Proportion of coloured graphlets in one authors' ego-network
\end_layout

\end_inset

The proportion of coloured graphlets of each type found in the 3-ego-networks
 of the author known as 
\begin_inset Quotes xld
\end_inset

7
\begin_inset Quotes xrd
\end_inset

 in the Arnetminer dataset.
 The graphlet labels correspond to those in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:motifs"
plural "false"
caps "false"
noprefix "false"

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:Motifs-(coloured)"

\end_inset

.
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/researcherlifecycle_colour/figures/7pcamono_lifetimes.png
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA of a data matrix of the monochromatic graphlet profiles of author 
\begin_inset Quotes xld
\end_inset

7
\begin_inset Quotes xrd
\end_inset


\end_layout

\end_inset

A data matrix formed from the monochromatic graphlet profile
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlet profile
\end_layout

\end_inset

s of 4-graphlets of the ego-networks of author 
\begin_inset Quotes xld
\end_inset

7
\begin_inset Quotes xrd
\end_inset

 from each year of their career the Arnetminer dataset and a random sample
 of other authors.
 The networks are shown projected onto their first two principal components;
 each point is a single network.
 The colours of the other authors corresponds to their age in years.
 Author 
\begin_inset Quotes xld
\end_inset

7
\begin_inset Quotes xrd
\end_inset

's life cycle is shown with yellow circles for each year and green arrows
 showing the direction from one year to the next 
\begin_inset CommandInset label
LatexCommand label
name "fig:7's-motif-lifecycle"

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: oh ffs, what have we done to year 1? why does it have a bar now?
\end_layout

\begin_layout Plain Layout
lol, is it even possible to not be in a four graphlet? it must be, right?
 right?
\end_layout

\begin_layout Plain Layout
need to push the legend up just a tiny bit (nothing is actually cut off,
 though)
\end_layout

\begin_layout Plain Layout
what do the other nodes represent? which years in their lifetimes? is it
 all years, is it random, is it the last year? are we including all years
 of 7 in this plot or the same as the others, i.e.
 +5 years?
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/researcherlifecycle_colour/figures/7pcacolour_lifetimes.png
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA of a data matrix of the coloured graphlet profiles of author 
\begin_inset Quotes xld
\end_inset

7
\begin_inset Quotes xrd
\end_inset


\end_layout

\end_inset

A data matrix formed from the coloured graphlet profile
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlet profile
\end_layout

\end_inset

s of 4-graphlets of the ego-networks of author 
\begin_inset Quotes xld
\end_inset

7
\begin_inset Quotes xrd
\end_inset

 from each year of their career the Arnetminer dataset and a random sample
 of other authors.
 The networks are shown projected onto their first two principal components;
 each point is a single network.
 The colours of the other authors corresponds to their age in years.
 Author 
\begin_inset Quotes xld
\end_inset

7
\begin_inset Quotes xrd
\end_inset

's life cycle is shown with yellow circles for each year and green arrows
 showing the direction from one year to the next 
\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva / corrections: todo: swap to pdf, needs more pixels and bigger arrows
 - Researcher life cycle disambiguated tidied.ipynb
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:7's-graphlet-lifecycle-colour"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
In some sense, the same information can be seen in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:7's-graphlet-lifecycle-colour"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 The author starts life in an area sparsely populated by other authors,
 briefly goes towards the mass of authors residing around 
\begin_inset Formula $(0.8,-0.25)$
\end_inset

, and then spends most of their life in regions that are sparsely populated.
 We can therefore tell that this author is unusual, at least with respect
 to how they co-author papers.
\end_layout

\begin_layout Standard
There are clear limitations of this matrix decomposition approach on display
 here.
 Figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:7's-motif-lifecycle"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:7's-graphlet-lifecycle-colour"
plural "false"
caps "false"
noprefix "false"

\end_inset

, where the nodes have been coloured according to their ultimate career-length,
 show no clear relationship between position and lifetime.
 The more traditional analysis in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Career-length-in"
plural "false"
caps "false"
noprefix "false"

\end_inset

 found a statistically significant positive link between a particular graphlet
 (4.3b) and career-length.
 Unlisted in that chapter but mentioned here for completeness, we also statistic
ally tested for any link between the principal components found here and
 career-length and found none.
 As was shown in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Career-length-in"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and particularly figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:importantgraphlets"
plural "false"
caps "false"
noprefix "false"

\end_inset

, when it comes to academic collaboration networks, a large portion of the
 variation in topology has nothing to do with research career-length.
 Given that PCA is predominantly concerned with preserving variation, it
 is unsurprising that it is of little use for investigating a particular
 hidden variable that is not associated with a great deal of variance.
\end_layout

\begin_layout Section
Conclusions & further work
\begin_inset CommandInset label
LatexCommand label
name "sec:Conclusions-&-further"

\end_inset


\end_layout

\begin_layout Standard
We have shown that matrix decompositions of data matrices formed from network
 representations and graphlet profiles can be used to discriminate between
 different types of synthetic networks and real-world networks.
 Furthermore, its use in displaying the damage done to a network was demonstrate
d.
 This work is all novel; the closest previous work had only considered data
 matrices formed from centrality distributions or metrics 
\begin_inset CommandInset citation
LatexCommand citep
key "Costa2007,bounova_overview_2012,Ronqui2015"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard
There is an obvious criticism of much of this chapter: there is no quantitative
 comparison of our technique with existing metrics.
 In the absence of a more thorough study, we can compare our plots with
 the plots made by 
\begin_inset CommandInset citation
LatexCommand citet
key "bounova_overview_2012"
literal "false"

\end_inset

.
 Our technique seems to show a similar level of separation between networks
 as the best metrics and decompositions in their work.
\end_layout

\begin_layout Standard
However, comparison with other attempts at characterising networks in the
 literature is only a small part of the story.
 Complex networks are only of interest because of the systems they represent.
 Therefore the true test of any metric that describes how different networks
 are should consider how different the dynamics those networks support is
 and a measure of the amount of damage done to a network (as discussed in
 section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:network-damage"
plural "false"
caps "false"
noprefix "false"

\end_inset

) should be a measure of how much damage is done to the underlying node
 dynamics; this would be a fascinating avenue to explore for further work.
\end_layout

\begin_layout Standard
A further limitation of this technique is that because it relies so heavily
 on the variance in the data, it can be rendered useless by outliers in
 the data.
 This was shown by the fact that when we compared undirected and directed
 networks, that difference was found at the expense of all others.
 It was also shown in other testing where we neglected to first remove self-loop
s from our real world networks and, once again, that became the only difference
 that was noticeable.
 A related risk is that if one is interested in a characteristic that isn't
 associated with a great deal of variance in the topology, the technique
 is of limited use, as was shown in the stark contrast between the usefulness
 and interpretability of the results in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Characterisation-of-ego-networks"
plural "false"
caps "false"
noprefix "false"

\end_inset

, where all we could say is that 
\begin_inset Quotes xld
\end_inset

there's some variation
\begin_inset Quotes xrd
\end_inset

 and 
\begin_inset Quotes xld
\end_inset

this particular author's graphlet profile changes a lot over time
\begin_inset Quotes xrd
\end_inset

, compared to section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:results"
plural "false"
caps "false"
noprefix "false"

\end_inset

, where we showed that the 4.3b graphlet is correlated with career length
 after controlling for centralities, in the previous chapter.
 It would also be desirable to thoroughly test our techniques on directed
 networks, especially the data matrices formed from graphlet profiles, as
 direction greatly increases the total number of possible graphlets.
 Recent work has found that food webs fall neatly into categories when clustered
 based on directed graphlet profile
\begin_inset Index idx
status open

\begin_layout Plain Layout
metrics!graphlet profile
\end_layout

\end_inset

s 
\begin_inset CommandInset citation
LatexCommand citep
key "klaise2017origin"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard
Our selection of graph models and real-world networks could also be improved.
 Recent work has suggested that many real-world networks are not as close
 to scale-free networks as once thought 
\begin_inset CommandInset citation
LatexCommand citep
key "house2015testing,broido2019scale"
literal "false"

\end_inset

.
 Our technique could potentially inform this debate given enough real-world
 networks and competing models with approximately scale-free nature; the
 real-world networks should simply appear closest to their closest models
 in the projected space.
\end_layout

\begin_layout Standard
\begin_inset CommandInset citation
LatexCommand citet
key "schieber2017quantification"
literal "false"

\end_inset

 published some related work after this work was conducted.
 Their approach is similar to ours; they form high dimensional descriptions
 of networks from characterising nodes and then apply a dimension reduction
 (specifically, multi-dimensional scaling) to see how networks are similar
 or dissimilar.
 The authors test their metric on automorphism groups, ensuring that their
 metric returns zero for distances within the group, which is a sound way
 of testing network characterisation.
 Our reduced Laplacian and adjacency network data matrices should return
 near zero for this test, since they cannot tell the difference between
 models and their configuration models, but we cannot be certain without
 testing.
 The degree mixing matrices and graphlet profiles, however, are clearly
 invariant within automorphism groups so we can be certain that these data
 matrices would pass this test.
\end_layout

\begin_layout Standard
It would also be of interest to apply other dimension reductions and machine
 learning techniques to our data matrices.
 
\begin_inset CommandInset citation
LatexCommand citet
key "mcinnes2018umap"
literal "false"

\end_inset

 in particular created a novel dimension reduction, UMAP, which is particularly
 computationally efficient, making it possible to analyse larger adjacency
 and Laplacian matrices than was possible at the time this work was conducted.
 Additionally, these 
\begin_inset Quotes xld
\end_inset

manifold-preserving
\begin_inset Quotes xrd
\end_inset

 dimension reductions are far less susceptible to outliers than variance-based
 dimension reductions such as PCA, so we could see better distinction between
 networks which are similar.
 
\begin_inset CommandInset citation
LatexCommand citet
key "torres2019non"
literal "false"

\end_inset

 have applied UMAP to networks through a data matrix constructed from a
 novel graph representation.
 It should be noted that many of these competing dimension reductions are
 not invertible transformations; i.e.
\begin_inset space ~
\end_inset

one cannot map back to the original space from the reduced space.
 This greatly harms their interpretability; figures such as figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:mix-components"
plural "false"
caps "false"
noprefix "false"

\end_inset

 would not be possible.
\end_layout

\begin_layout Standard
\begin_inset CommandInset citation
LatexCommand citet
key "Rossi2018"
literal "false"

\end_inset

 have conducted similar work on the creation of data matrices, which they
 call feature matrices.
 The authors use graphlets as features by which they characterise networks
 for the purpose of inter-network comparison; and graphlets of ego-networks
 to characterise nodes.
\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: It is implied that the authors consider the node colour in their graphlet
 enumeration but the enumeration routine they use does not appear to support
 this.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
In the next part, we apply a related technique—tensor decomposition—to discover
 and characterise communities in networks.
 There, each tensor represents a single network whereas in this chapter
 each matrix represented an ensemble of networks; this change in approach
 (previously pursued by other authors) is what allows us to characterise
 communities rather than whole networks.
\end_layout

\begin_layout Standard
\begin_inset Newpage newpage
\end_inset


\end_layout

\begin_layout Section*
Bibliography
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "monorefs"
options "authordate4"

\end_inset


\end_layout

\begin_layout Part
Temporal community characterisation
\begin_inset CommandInset label
LatexCommand label
name "part:Tensor-decomposition"

\end_inset


\end_layout

\begin_layout Chapter
Finding an optimum number of communities
\begin_inset Index idx
status open

\begin_layout Plain Layout
tensor decomposition
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
epigraph{``Life is mostly grief and labour 
\backslash

\backslash
 Two things get you through 
\backslash

\backslash
 Chortling when it hits a neighbour 
\backslash

\backslash
 Whingeing when it hits you"}{--- 
\backslash
textup{Kingsley Amis on local optima}}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
MyLettrine{T}{his part}
\end_layout

\end_inset

 explores temporal community characterisation through the use of tensor
 decompositions.
 As was explored in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Community-characterisation"
plural "false"
caps "false"
noprefix "false"

\end_inset

, tensor decompositions are a good method of characterising temporal communities
 as they provide a detailed view into their evolution.
 They are the sole method of community characterisation we directly consider
 in this part.
\end_layout

\begin_layout Standard
Tensor decompositions are mathematically closely related to concepts introduced
 in part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Matrix-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Here, we focus on using the decomposition techniques to detect communities
 in temporal networks and then calculate membership and relative activity
 of those communities over time.
 We decompose adjacency tensors, as introduced in section 
\begin_inset CommandInset ref
LatexCommand vref
reference "subsec:Adjacency-tensor"
plural "false"
caps "false"
noprefix "false"

\end_inset

, relating to single networks
\begin_inset Index idx
status open

\begin_layout Plain Layout
network representations!adjacency tensor
\end_layout

\end_inset

 rather than data matrices relating to multiple networks.
 In this way we find similarities within rather than across networks.
 This approach gives us more detail about the behaviour within each network—comp
are this with the previous part where we distilled networks down to a single
 point—and is often easier to interpret.
 A clear example of this is in our novel application of the technique to
 software dependency networks in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Software"
plural "false"
caps "false"
noprefix "false"

\end_inset

, where the difference between CRAN, the R language's repository, and other
 software ecosystems can be seen as predominantly due to CRAN's rolling-release
 cycle 
\begin_inset CommandInset citation
LatexCommand citep
key "BlanthornAPNS2019"
literal "false"

\end_inset

; if we had performed the same analysis as in part 
\begin_inset CommandInset ref
LatexCommand ref
reference "part:Matrix-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

, we may well have seen the point relating to CRAN moving around more than
 the other ecosystems, but we would not have known why.
 
\end_layout

\begin_layout Standard
Tensor decompositions also make dealing with multilayer network
\begin_inset Index idx
status open

\begin_layout Plain Layout
multilayer network
\end_layout

\end_inset

s possible.
 The decompositions can be viewed as a kind of soft clustering
\begin_inset Index idx
status open

\begin_layout Plain Layout
clustering
\end_layout

\end_inset

, making them natural for use in networks where community membership is
 loosely defined.
\end_layout

\begin_layout Standard
In this part, a new technique for determining a locally optimal number of
 components into which a tensor representing a temporal complex network
 should be decomposed is introduced in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Locally-optimal-numbers"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Its use is demonstrated in chapters 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Software"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Linguistic-evolution"
plural "false"
caps "false"
noprefix "false"

\end_inset

, where it is used to analyse two types of directed temporal networks: software
 collaboration networks, where we have shown the impact that the rolling-release
 cycle in R 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "R"
symbol "R"
description "the R programming language"
literal "true"

\end_inset

has on its software ecosystem.
 This chapter is simply split into two sections: the first presents our
 technique for finding locally optimal numbers of communities and the second
 provide a short summary of the chapter and technique.
\end_layout

\begin_layout Section
Locally optimal numbers of communities for decomposition
\begin_inset CommandInset label
LatexCommand label
name "chap:Locally-optimal-numbers"

\end_inset


\end_layout

\begin_layout Standard
One of the major downsides to tensor decompositions is that one must choose
 the number of communities or components, 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "R"
symbol "$R$"
description "the number of components in a tensor decomposition"
literal "true"

\end_inset


\begin_inset Formula $R$
\end_inset

, the current approaches to which are given in 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Finding-locally-optimal"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 In the rest of this chapter, we present our approach to finding locally
 optimal numbers of communities.
 Due to the local nature of the technique, where we compare decompositions
 of certain numbers of communities with higher and lower numbers of communities,
 we cannot make any statement about whether the number of communities is
 globally optimal.
\end_layout

\begin_layout Standard
As will be seen in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Number-of-communities"
plural "false"
caps "false"
noprefix "false"

\end_inset

, unlike previous works, in practice we found that the NRSS
\begin_inset Index idx
status open

\begin_layout Plain Layout
NRSS|see
\begin_inset ERT
status open

\begin_layout Plain Layout

{
\end_layout

\end_inset

residual sum of squares
\begin_inset ERT
status open

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset Index idx
status open

\begin_layout Plain Layout
residual sum of squares
\end_layout

\end_inset

, the normalised residual sum of squares, was not useful for determining
 locally optimal numbers of components because no 
\begin_inset Quotes xld
\end_inset

elbows
\begin_inset Quotes xrd
\end_inset

 were visible 
\begin_inset CommandInset citation
LatexCommand citep
key "BlanthornAPNS2019"
literal "false"

\end_inset

.
 We also found that CORCONDIA 
\begin_inset CommandInset citation
LatexCommand citep
key "Bro2003b"
literal "false"

\end_inset

 gave no pronounced 
\begin_inset Quotes xld
\end_inset

elbows
\begin_inset Quotes xrd
\end_inset

 either, but, as we were using our own implementation, we felt that we could
 not be certain that it was not due to an implementation error and therefore
 excluded it from our results in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Number-of-communities"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Previous works have had very clearly defined 
\begin_inset Quotes xld
\end_inset

real
\begin_inset Quotes xrd
\end_inset

 communities—such as pupils separated into classes at school 
\begin_inset CommandInset citation
LatexCommand citep
key "Gauvin2014b"
literal "false"

\end_inset

—or have not been concerned with picking an optimal number of components
 
\begin_inset CommandInset citation
LatexCommand citep
key "kang2012gigatensor"
literal "false"

\end_inset

, but our applications (chapters 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Software"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Linguistic-evolution"
plural "false"
caps "false"
noprefix "false"

\end_inset

) have no ground truth.
 This could account for the differences in utility of NRSS.
 As is explored in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:AMIforDecomps"
plural "false"
caps "false"
noprefix "false"

\end_inset

, we reason that if multiple tensor decompositions arrive at similar clusterings
—i.e.
 partitions of 
\begin_inset Formula $\mathcal{V}$
\end_inset

 which are approximately the same—despite the stochastic nature of the decomposi
tion algorithms, those clusterings are likely to be the right ones.
 We therefore turned to another metric for evaluating clustering methods:
 adjusted mutual information, a measure of the level of agreement between
 clusterings.
 
\end_layout

\begin_layout Subsection
Measuring the level of agreement between clusterings
\end_layout

\begin_layout Standard
Adjusted mutual information is a measure of the level of agreement between
 two sets of clusterings, independent of cluster labels.
 If the two clusterings split the elements into the same subsets, it is
 maximal; if the two clusterings have the same level of overlap between
 their subsets as if they had been partitioned randomly, it is 
\begin_inset Formula $0$
\end_inset

.
\end_layout

\begin_layout Standard
More rigorously, the adjusted mutual information (AMI) between two partitions
 
\begin_inset Formula $U$
\end_inset

 and 
\begin_inset Formula $V$
\end_inset

, where 
\begin_inset Formula $U=\{U_{1},U_{2},\ldots\}$
\end_inset

 is a finite-length partition of some set of length 
\begin_inset Formula $N$
\end_inset

 and 
\begin_inset Formula $V$
\end_inset

 is another finite partition of that set, is
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\text{AMI}(U,V)=\frac{I(U;V)-E[I(U;V)]}{\text{max}(S(U),S(V))-E[I(U;V)]}\text{,}
\]

\end_inset


\end_layout

\begin_layout Standard
\noindent
where the entropy 
\begin_inset Formula $S$
\end_inset

 is
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
S(U)=-\sum_{i}\frac{|U_{i}|}{N}\text{log}\frac{|U_{i}|}{N}\text{,}
\]

\end_inset


\end_layout

\begin_layout Standard
\noindent
with 
\begin_inset Formula $0\text{\ensuremath{\cdot}log}(0)=0$
\end_inset

, the mutual information between 
\begin_inset Formula $U$
\end_inset

 and 
\begin_inset Formula $V$
\end_inset

 is
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
I(U;V)=\sum_{i=1}^{|U|}\sum_{j=1}^{|V|}\frac{|U_{i}\cap V_{j}|}{N}\log\frac{N|U_{i}\cap V_{j}|}{|U_{i}||V_{j}|}\text{,}
\]

\end_inset


\end_layout

\begin_layout Standard
\noindent
and 
\begin_inset Formula $E[W]$
\end_inset

 is the expectation of a random variable 
\begin_inset Formula $W$
\end_inset

.
 Here, it can be calculated using a hypergeometric model of all possible
 random clusterings 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{Vinh:2009}
\end_layout

\end_inset

 which we omit for brevity.
 AMI measures the amount of agreement between two clusterings.
 It has an upper bound of 
\begin_inset Formula $1$
\end_inset

.
 An AMI score of 
\begin_inset Formula $0$
\end_inset

 corresponds to the amount of agreement one would expect if one of the clusterin
gs was totally random and 
\begin_inset Formula $1$
\end_inset

 corresponds to perfect agreement between the two sets of clusters.
 The AMI score can be negative if there is more disagreement between the
 clusters than one would expect from random chance.
 It is hard to interpret the meaning of scores between 
\begin_inset Formula $0$
\end_inset

 and 
\begin_inset Formula $1$
\end_inset

, but it is important to bear in mind that anything greater than 
\begin_inset Formula $0$
\end_inset

 is a good score in the sense that it is better than random guessing.
 AMI is quite sensitive to the number of clusters and awards higher scores
 to pairs of clusterings that agree on the same number of assignments but
 have a larger number of clusters, to the limit where if each element is
 given its own cluster the AMI must be 
\begin_inset Formula $1$
\end_inset

.
\end_layout

\begin_layout Subsection
Application of AMI to choosing the number of components in tensor decompositions
\begin_inset CommandInset label
LatexCommand label
name "subsec:AMIforDecomps"

\end_inset


\end_layout

\begin_layout Standard
The first task is to consider which values of 
\begin_inset Formula $R$
\end_inset

 to test.
 Generally in this work, we don't have any knowledge as to the likely number
 of communities.
 We therefore start with two, since 
\begin_inset Formula $R=1$
\end_inset

 will always give a maximal AMI.
 We run the decomposition multiple times for each likely 
\begin_inset Formula $R$
\end_inset

 to determine whether the decomposition is stable, that is, if multiple
 runs produce the same results.
 As with many experiments, the exact number of runs required will depend
 on the exact application: as the number of runs increases the standard
 error of the mean pairwise AMI will reduce.
 This makes it easier to determine with more certainty where there is a
 peak in AMI.
 In our experiments, we found that thirty repeats were sufficient.
 We reason that a good choice of the number of communities to decompose
 a tensor into should result in the tensor decomposition algorithms choosing
 roughly the same nodes to assign to each community as this implies that
 true communities are not being artificially split up.
 That would result in nodes being assigned fairly randomly to each of the
 communities a true community was split into and so would not result in
 what we term a stable decomposition.
\end_layout

\begin_layout Standard
We look to see if nodes appear in the same communities as each other by
 considering the mean pairwise adjusted mutual information between repeated
 runs for a single 
\begin_inset Formula $R$
\end_inset

.
 We use hard clustering for this process by assigning nodes to communities
 for each non-temporal factor from the maximum strength, i.e.
\begin_inset space ~
\end_inset

for a single-layer temporal network, 
\begin_inset Formula $\text{Community}(i,\boldsymbol{v})=\text{argmax}(v_{ir})$
\end_inset

, for each factor matrix in equation 
\begin_inset CommandInset ref
LatexCommand vref
reference "eq:tensordecomp"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 For many applications it is clear which factor to prioritise; for a single-laye
red temporal network with a scale-free out-degree distribution but a normal
 in-degree distribution, such as the software dependency networks considered
 in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Software"
plural "false"
caps "false"
noprefix "false"

\end_inset

, it is probably best to consider communities based on the factor corresponding
 to the in-degree distribution, as the out-degree factor will be very sparse.
\end_layout

\begin_layout Standard
\noindent
Since we are using AMI to compare clusterings with each other and have far
 fewer clusters (or communities) than nodes, we believe the interpretability
 of the absolute value of the AMI is not particularly important.
 We are concerned primarily with the relative changes in AMI as we move
 from one number of communities to another; a peak in AMI with respect to
 the number of communities means that that number of communities is more
 stable and is therefore more likely to be an optimal number of communities.
 We therefore plot AMI against 
\begin_inset Formula $R$
\end_inset

 and look for the point with a negative second derivative—the second local
 maxima—to guard against the initial advantage that very low numbers of
 communities have with AMI.
 Rather than an an 
\begin_inset Quotes xld
\end_inset

elbow
\begin_inset Quotes xrd
\end_inset

 in the plot of AMI versus the number of communities, we therefore look
 for the second maxima in a graph that looks like a 
\begin_inset Quotes xld
\end_inset

U-bend
\begin_inset Quotes xrd
\end_inset

.
\end_layout

\begin_layout Subsection
Subjective checks
\begin_inset CommandInset label
LatexCommand label
name "sec:Eyeballing-it"

\end_inset


\end_layout

\begin_layout Standard
One can check visually to see if the temporal activity of communities looks
 similar across multiple decompositions into a single number of communities.
 If the decomposition is unstable, i.e.
\begin_inset space \space{}
\end_inset

if multiple runs produce wildly different results, it suggests that ground-truth
 communities are being merged or split up.
 Another check that can be performed is to look at the metadata associated
 with nodes from each community, making a qualitative judgement as to whether
 nodes in a single community are similar.
 For example, if a numerical computing software package and a front-end
 web development software package are in the same community in a software
 dependency network, the number of communities is probably too small.
 A final check is whether adding an extra component adds interesting new
 behaviour to the temporal community activity, or if it instead seems to
 break an existing community into noisy subcommunities.
 All of these approaches are quite subjective, so there is quite a large
 uncertainty in exactly how many communities is ideal for any application.
 This is not a major issue as we are not particularly interested in the
 exact number of communities our applications have; indeed, we do not believe
 this is a meaningful question as there are no 
\begin_inset Quotes xld
\end_inset

ground truth
\begin_inset Quotes xrd
\end_inset

 communities, i.e.
\begin_inset space ~
\end_inset

communities which have some real basis, for example, the classes children
 are assigned to at school.
 We are interested in the activity over time of communities and the membership
 of these communities—it does not matter a great deal if two communities
 are seen as erroneously one in our analysis.
\end_layout

\begin_layout Section
Summary
\end_layout

\begin_layout Standard
In this chapter we have explored a method for determining a locally optimal
 number of communities, 
\begin_inset Formula $R$
\end_inset

, to decompose a tensor into.
 In the next chapter, we apply this technique to analyse temporal collaboration
 networks: software dependency networks corresponding to several languages.
\end_layout

\begin_layout Section*
\begin_inset Newpage newpage
\end_inset

Bibliography
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "monorefs"
options "authordate4"

\end_inset


\end_layout

\begin_layout Chapter
Community detection in temporal software dependency networks
\begin_inset CommandInset label
LatexCommand label
name "chap:Software"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
epigraph{``Computers are useless.
 They can only give you answers."}{--- 
\backslash
textup{Pablo Picasso}}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Index idx
status open

\begin_layout Plain Layout
real-world networks!software dependency network
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
MyLettrine{F}{actorisation of}
\end_layout

\end_inset

 tensors
\begin_inset Index idx
status open

\begin_layout Plain Layout
tensor decomposition
\end_layout

\end_inset

, and our own work on determining the optimal number of communities from
 chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Locally-optimal-numbers"
plural "false"
caps "false"
noprefix "false"

\end_inset

, is used in this chapter to explore the dynamics of communities of software
 and then compare these dynamics between languages on a dataset of approximately
 1 million software projects.
 We establish that adjusted mutual information is a valid metric by which
 to assess the number of communities in a tensor decomposition and find
 that there are striking differences between the communities found across
 different software ecosystems and that communities do experience large
 and interpretable changes in activity over time.
 The differences between the elm and R software ecosystems, which see some
 communities decline over time and the more conventional software ecosystems
 of Python, Java and JavaScript, which do not see many declining communities,
 are particularly marked.
 A paper describing some of this work has been accepted and will shortly
 be published 
\begin_inset CommandInset citation
LatexCommand citep
key "BlanthornAPNS2019"
literal "false"

\end_inset

.
\end_layout

\begin_layout Section
Importance of communities in software dependency networks
\end_layout

\begin_layout Standard
Contemporary software authors routinely depend on and re-use the software
 packages of authors with whom they have no contact.
 This unco-ordinated process creates what have recently been called 
\begin_inset Quotes xld
\end_inset

software ecosystems
\begin_inset Quotes xrd
\end_inset

 
\begin_inset CommandInset citation
LatexCommand citep
key "Decan2018h"
literal "false"

\end_inset

: extensive networks of interdependent software components that are used
 and maintained by large communities of contributors all over the world.
 These ecosystems are complex networks whose nodes and edges both evolve
 over time.
 Throughout this chapter, terminology such as 
\begin_inset Quotes xld
\end_inset

package
\begin_inset Quotes xrd
\end_inset

, 
\begin_inset Quotes xld
\end_inset

package manager
\begin_inset Quotes xrd
\end_inset

, 
\begin_inset Quotes xld
\end_inset

dependency
\begin_inset Quotes xrd
\end_inset

 follow the usual software engineering convention, as documented by 
\begin_inset CommandInset citation
LatexCommand citep
key "Decan2018h"
literal "false"

\end_inset

.
 With this work we would like to contribute to the debate on software dependenci
es, which recently re-emerged due to the hijacking of the 
\family typewriter
event-stream
\family default
 package after previous incidents 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{Baldwin2018,Schlueter2016,Durumeric2014c}
\end_layout

\end_inset

, by giving a clearer picture of the structure of software dependency networks.
 For this, we propose a novel framework to model and analyse the formation,
 long-term behaviour and change with time of communities of software packages
 and compare these behaviours across several programming languages.
\end_layout

\begin_layout Standard
The evolution of software and package dependency networks has been extensively
 studied by using network science techniques over the past 15 years.
 An early such work is 
\begin_inset CommandInset citation
LatexCommand cite
key "Myers2003b"
literal "false"

\end_inset

, which has been followed by others 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{Pan2011b,Xu2005b,Zheng2008d}
\end_layout

\end_inset

.
 A recent survey on this topic is given by 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citet{Savic2019}
\end_layout

\end_inset

.
 Methods and tools used for social networks have also been applied to analyse
 software evolution 
\begin_inset CommandInset citation
LatexCommand citep
key "Chatzigeorgiou2012b"
literal "false"

\end_inset

, reinforcing the importance of the social component in evolving software
 networks.
 There has been recent success in applying standard network metrics to analyse
 how software ecosystems have evolved with time 
\begin_inset CommandInset citation
LatexCommand citep
key "Decan2018h,Decan2018a"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard
Collaborative software networks—mainly in free and open-source software
 (FOSS) environments—can be considered as adaptive, evolving or temporal
 networks.
 Most of the published works highlight the fact that software networks exhibit
 scale-free network properties with a power-law-type node degree distribution
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{Cai2009b,LianWen2009,Louridas2008b}
\end_layout

\end_inset

 and a clear preferential attachment in the network growth process 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{Ai2013a,Chaikalis2015b}
\end_layout

\end_inset

, confirming the heterogeneity and hierarchical characteristics of networks
 of software.
 Other key properties in software systems like maintainability and reliability
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{Chong2015b}
\end_layout

\end_inset

, robustness 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{Gao2014b}
\end_layout

\end_inset

 and modularity 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{Zanetti2012b}
\end_layout

\end_inset

 have been also analysed using the complex network paradigm.
\end_layout

\begin_layout Standard
The identification of communities in software dependency networks that evolve
 over time is one of the main motivations of this chapter.
 General background on characterisation of communities in temporal networks
 was given in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Community-characterisation"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 There has been much less work on finding clusters or communities in software
 dependency networks: some representative works are 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citet{Dietrich2008,Paymal2011b,Concas2013b}
\end_layout

\end_inset

 and 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citet{Savic2012b}
\end_layout

\end_inset

.
 However, it is still a challenge to give satisfactory solutions for the
 dynamic treatment of these clusters and inter-language comparison.
 
\begin_inset CommandInset citation
LatexCommand citep
key "Savic2012b"
literal "false"

\end_inset

, for example, give some dynamic treatment of communities within the 
\begin_inset Quotes xld
\end_inset

class collaboration
\begin_inset Quotes xrd
\end_inset

 network of Apache Ant.
 This is a single piece of software in which the classes are more witting
 participants in the software than in general software dependency networks
 where the authors of a package may have no idea where their work is being
 used.
 Additionally, the community detection methods used are static, so it is
 the metrics on the types of communities found at each version of Apache
 Ant which are compared rather than the qualities of any individual communities
 being tracked across time.
\end_layout

\begin_layout Standard
The novelty of our work is the detection of communities in temporal software
 dependency networks, the use of tensor decompositions on software ecosystems
 and the use of adjusted mutual information (AMI, introduced in chapter
 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Locally-optimal-numbers"
plural "false"
caps "false"
noprefix "false"

\end_inset

) to assist in choosing the number of communities.
 Additionally, some of the networks we study are amongst the largest temporal
 networks to which tensor factorisation has been applied, although there
 have been studies considering much larger static networks 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{kang2012gigatensor}
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
This chapter addresses three research questions corresponding to unsolved
 problems in collaborative large-scale software development and evolution
 over time:
\end_layout

\begin_layout Itemize
\begin_inset Formula $RQ_{1}$
\end_inset

 What are the differences between different software ecosystems? 
\end_layout

\begin_layout Itemize
\begin_inset Formula $RQ_{2}$
\end_inset

 What do communities of commonly-used-together software packages look like?
\end_layout

\begin_layout Itemize
\begin_inset Formula $RQ_{3}$
\end_inset

 How do these communities change with time? 
\end_layout

\begin_layout Standard
By answering these questions, we identify communities of packages in the
 ecosystems of several languages.
 Namely: Elm; JavaScript; Rust; Python; R; and Java.
 These languages were chosen because data on their package ecosystems are
 readily available and they represent a variety of uses and ages: low-level
 systems languages; scientific computing; and web development; with histories
 between 3 and 20 years long.
\end_layout

\begin_layout Standard
The rest of this chapter is structured as follows.
 Firstly, in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Methods"
plural "false"
caps "false"
noprefix "false"

\end_inset

, we detail the data, mathematical tools, software and hardware used to
 conduct this study.
 Next, section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Results-and-discussion"
plural "false"
caps "false"
noprefix "false"

\end_inset

 presents our results and provides some discussion of why they might have
 arisen.
 The final section, section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Conclusions"
plural "false"
caps "false"
noprefix "false"

\end_inset

, compares our results with prior work and provides a summary of this works'
 limitations while highlighting further avenues of research.
\end_layout

\begin_layout Section
Methods
\begin_inset CommandInset label
LatexCommand label
name "sec:Methods"

\end_inset


\end_layout

\begin_layout Subsection
What is a community?
\end_layout

\begin_layout Standard
Software packages associated with a single programming language and package
 manager form an 
\emph on
ecosystem
\emph default
 and a 
\emph on
community
\emph default
 is a collection of packages which are more likely to depend on one another
 than depend on packages from outside that community.
 Communities may be related by some theme and by co-occurrence of activity
 over time.
 They might be collections of numerical computing packages, or a community
 of packages that use a particular library, or a community of older packages
 that became obsolete.
 Packages may be in more than one community, which is modelled by vectors
 of continuous strengths of membership to each community as detailed in
 the 
\begin_inset Quotes xld
\end_inset

Tensor decompositions
\begin_inset Quotes xrd
\end_inset

 section below.
 Communities may wax and wane over time.
\end_layout

\begin_layout Standard
Further detail on how we calculate communities and membership strengths
 thereof is given in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Tensor-decompositions"
plural "false"
caps "false"
noprefix "false"

\end_inset

, chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Locally-optimal-numbers"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Choosing-the-number"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Subsection
Shape of the data
\end_layout

\begin_layout Standard
We considered the following package managers (corresponding to specific
 languages): elm-get (Elm), npm (JavaScript), crates.io (Rust), PyPI (Python),
 CRAN (R) and Maven (Java).
 All publicly registered packages for these package managers were included
 in our analysis.
 To simplify our analysis, we do not consider specific versions of dependent
 packages—each package depends on some other packages at each time, each
 of which is assumed to be the most up-to-date package available at that
 time.
 We only consider packages registered with these package managers and do
 not consider cross-platform (inter-language) dependencies.
 The data cover 1 million projects listed in package managers.
 This is sourced from Libraries.io 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{jeremy_katz_2018_2536573}
\end_layout

\end_inset

.
 The network of Elm dependencies at the final time-step is shown in figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:elmnet"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Preview

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
includegraphics[width=0.55
\backslash
paperwidth]{../ans-ml-cfp/paper/src/figures/elm_network/elm_network.pdf}
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Elm network
\end_layout

\end_inset

Elm network visualised using 
\family typewriter
Gephi
\family default
.
 Node and label size corresponds to out-degree.
 Each node 
\begin_inset Formula $i$
\end_inset

 is coloured according to the strongest strength of community membership
 given by 
\begin_inset Formula $\boldsymbol{\kappa}_{i}$
\end_inset

 as described in the section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Methods"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Legend: light-green corresponds to the community we labelled as 
\family typewriter
elm-lang/core
\family default
, purple to 
\family typewriter
elm-lang/http
\family default
, blue to 
\family typewriter
elm-lang/html
\family default
 and orange to 
\family typewriter
evancz/elm-html
\family default
, dark-green to unassigned nodes, i.e.
\begin_inset space \space{}
\end_inset

those without any dependencies
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:elmnet"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
Brief introduction to each ecosystem
\end_layout

\begin_layout Standard
Here, we will briefly introduce each language and ecosystem considered.
\end_layout

\begin_layout Paragraph
Elm
\end_layout

\begin_layout Standard
(
\begin_inset CommandInset href
LatexCommand href
name "elm-lang.org"
target "https://elm-lang.org/"
literal "false"

\end_inset

) is a relatively small new language created in 2012 as a functional language
 for developing web applications.
 It is unusually focused on a single domain and unusually restrictive in
 what can be expressed in listed packages.
 The language authors explicitly discourage proliferation of packages in
 favour of a unified and carefully designed standard package for each task.
 It aims to eliminate run-time errors and many classes of logical errors
 through static analysis and careful language design.
\end_layout

\begin_layout Paragraph
Python
\end_layout

\begin_layout Standard
is a very popular general-purpose interpreted programming language.
 It was first released in 1991.
 Its package manager, PyPI (
\begin_inset CommandInset href
LatexCommand href
name "https://pypi.org/"
target "pypi.org"
literal "false"

\end_inset

), was released in 2002.
 Python is notable for having a large standard library; many features such
 as HTTP and GUI support that would be external packages in other languages
 are included by default.
 We therefore suspect that Python's package ecosystem will look quite different
 to many of the other languages we consider.
 It is perhaps pertinent to note that some of the packages listed in PyPI
 are designed to be used by end-users, not developers.
\end_layout

\begin_layout Paragraph
JavaScript
\end_layout

\begin_layout Standard
is another popular general-purpose interpreted programming language.
 It was first released in 1995 as part of the Netscape browser and was initially
 used for client-side scripts on web pages, but now has several server-side
 implementations, most notably Node.js.
 JavaScript has a very small standard library, with the exception of the
 DOM interface (Document Object Model—a representation of a web page), which
 is included.
 We therefore expect most JavaScript packages to have to depend on many
 other packages.
 JavaScript's most common package manager is npm (
\begin_inset CommandInset href
LatexCommand href
target "npmjs.com"
literal "false"

\end_inset

), the Node.js package manager, which was released in 2010.
 The JavaScript ecosystem we describe in this paper is the public npm registry,
 which is the 
\shape italic
de facto
\shape default
 canonical list of JavaScript packages.
 Despite its name, the registry is used for all types of JavaScript, not
 just server-side applications.
 Like PyPI, some of the packages listed in npm are designed to be used by
 end-users, not developers.
\end_layout

\begin_layout Paragraph
Rust
\end_layout

\begin_layout Standard
is another relatively new systems programming language.
 Unlike many of the other ecosystems we have considered, it is statically
 compiled.
 One of its main aims is safe concurrent programming.
 We suspect that because of this conservative focus, packages written in
 Rust will have relatively few dependencies.
 Rust's package manager is Cargo, which was released in 2014 and its registry
 is 
\begin_inset CommandInset href
LatexCommand href
name "crates.io"
target "https://crates.io/"
literal "false"

\end_inset

.
 Like PyPI, some of the packages listed on crates.io are designed to be used
 by end-users, not developers.
\end_layout

\begin_layout Paragraph
R
\end_layout

\begin_layout Standard
is an interpreted programming language for statistical computing.
 It was first released in 1993.
 Its standard library for statistical computing is large.
 Its package manager is the Comprehensive R Archive Network (CRAN 
\begin_inset CommandInset href
LatexCommand href
target "cran.r-project.org"
literal "false"

\end_inset

).
\end_layout

\begin_layout Paragraph
Java
\end_layout

\begin_layout Standard
is a general-purpose typed compiled programming language, first released
 in 1995.
 Its standard library is large.
 Java has no official package manager, but the most widely used is Apache
 Maven (
\begin_inset CommandInset href
LatexCommand href
name "maven.apache.org"
target "https://maven.apache.org/"
literal "false"

\end_inset

), which was released in 2004.
\end_layout

\begin_layout Subsection
Representing the data
\end_layout

\begin_layout Standard
We represent each language's network as a tensor
\begin_inset Index idx
status open

\begin_layout Plain Layout
network representations!adjacency tensor
\end_layout

\end_inset

, 
\begin_inset Formula $\boldsymbol{A}_{\text{lang}}$
\end_inset

, where each 
\begin_inset Formula 
\[
a_{ijt}\in\{0,1\}
\]

\end_inset


\end_layout

\begin_layout Standard
\noindent
denotes whether the package 
\begin_inset Formula $i$
\end_inset

 depends on the package 
\begin_inset Formula $j$
\end_inset

 at time 
\begin_inset Formula $t$
\end_inset

.
 We consider links as going from 
\begin_inset Formula $j$
\end_inset

 to 
\begin_inset Formula $i$
\end_inset

, so the direction of each link reflects the flow of code.
 Software packages have unweighted directed links to packages that they
 depend on at time 
\begin_inset Formula $t$
\end_inset

 according to the most recently available version of the software package
 at that time.
 The time is quantised into time-steps.
 We chose a resolution of one month on the basis that packages are long-lived
 and change dependencies rarely; we looked at other resolutions but they
 didn't qualitatively change our results.
\end_layout

\begin_layout Subsection
Tensor decompositions
\begin_inset CommandInset label
LatexCommand label
name "subsec:Tensor-decompositions"

\end_inset


\end_layout

\begin_layout Standard
As was detailed in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:bg-Tensor-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

, a rank-
\begin_inset Formula $N$
\end_inset

 tensor 
\begin_inset Formula $X$
\end_inset

 can be approximated as 
\begin_inset Formula 
\[
X\approx\sum_{r=1}^{R}\text{reduce}_{n=1}^{N}(\otimes,\boldsymbol{a}_{nr})\text{,}
\]

\end_inset


\end_layout

\begin_layout Standard
\noindent
where 
\begin_inset Formula $\text{reduce}_{n=1}^{N}(\odot,b_{n})$
\end_inset

 returns the reduction of 
\begin_inset Formula $\{b_{n}|n\in1..N\}$
\end_inset

 by applying an arbitrary associative binary operator 
\begin_inset Formula $\odot$
\end_inset

 repeatedly, i.e.
 
\begin_inset Formula 
\[
\text{reduce}_{n=1}^{N}(\odot,b_{n})=b_{1}\odot b_{2}\odot b_{3}\odot\ldots\odot b_{N}\text{;}
\]

\end_inset


\end_layout

\begin_layout Standard
\noindent
\begin_inset Formula $\otimes$
\end_inset

 is the tensor product
\begin_inset Index idx
status open

\begin_layout Plain Layout
tensor product
\end_layout

\end_inset

; and each 
\begin_inset Formula $\boldsymbol{a}_{nr}\in\mathbb{R}^{d_{n}+}$
\end_inset

 where 
\begin_inset Formula $d_{n}$
\end_inset

 is the dimension of the 
\begin_inset Formula $n^{\text{th}}$
\end_inset

-mode of the tensor 
\begin_inset Formula $X$
\end_inset

.
\end_layout

\begin_layout Standard
Each 
\begin_inset Formula $\boldsymbol{a}_{nr}$
\end_inset

 for a specific 
\begin_inset Formula $n$
\end_inset

 is called a factor and maps its indices (in our case, software packages
 or time) to strength of membership to the community 
\begin_inset Formula $r$
\end_inset

.
 The total number of communities or components, 
\begin_inset Formula $R$
\end_inset

, is a fixed chosen parameter.
 Each 
\begin_inset Formula $\boldsymbol{a}_{nr}$
\end_inset

 for a specific 
\begin_inset Formula $r$
\end_inset

 is called a community or component.
 The concept of a temporal index having strengths of community membership
 may seem odd; it is perhaps more easily understood as the amount of community
 activity at a certain time 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
citep{Gauvin2014b}
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
There are many algorithms for performing tensor decompositions of this kind,
 as we discussed in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:bg-Tensor-decomposition"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 We selected a non-negative method with a public implementation that would
 allow us to later extend our analysis to large rank-
\begin_inset Formula $N$
\end_inset

 tensors: Alternating Proximal Gradient (APG-TF
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Implementation: 
\begin_inset CommandInset href
LatexCommand href
name "https://www.caam.rice.edu/\\textasciitilde{}optimization/bcu/ncp/"
target "https://www.caam.rice.edu/~optimization/bcu/ncp/"
literal "true"

\end_inset


\end_layout

\end_inset

, 
\begin_inset CommandInset citation
LatexCommand citealp
key "Xu2013b"
literal "false"

\end_inset

), an efficient non-negative CANDECOMP
\begin_inset Index idx
status open

\begin_layout Plain Layout
CANDECOMP
\end_layout

\end_inset

/PARAFAC
\begin_inset Index idx
status open

\begin_layout Plain Layout
PARAFAC
\end_layout

\end_inset

 (CP) method.
 We think the use of tensor decomposition 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{TensorReview}
\end_layout

\end_inset

 is better suited to community detection in temporal networks than other
 static methods, such as the Louvain algorithm 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{Aynaud2010b}
\end_layout

\end_inset

, because tensor decompositions explicitly integrate evolution over time
 and can scale very efficiently for large graphs.
\end_layout

\begin_layout Standard
For our specific application, we approximate our temporal adjacency matrix
 as 
\begin_inset Formula 
\begin{equation}
a_{i\leftarrow j,t}\approx\sum_{r=1}^{R}\kappa_{ir}h_{jr}\tau_{tr}\text{,}\label{eqn:decomp}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
\noindent
where 
\begin_inset Formula $\kappa_{ri}$
\end_inset

 are the elements of a matrix 
\begin_inset Formula $K\in(\mathbb{R}^{+})^{N_{p}\times R}$
\end_inset

 that describe the strengths of community membership of each node 
\begin_inset Formula $i$
\end_inset

 based on outgoing links, 
\begin_inset Formula $h_{rj}$
\end_inset

 are the elements of a matrix 
\begin_inset Formula $H\in(\mathbb{R}^{+})^{N_{p}\times R}$
\end_inset

 which represent the strengths of community membership of each node 
\begin_inset Formula $j$
\end_inset

 based on incoming links, 
\begin_inset Formula $\tau_{rt}$
\end_inset

 are the elements of a matrix 
\begin_inset Formula $T\in(\mathbb{R}^{+})^{N_{p}\times R}$
\end_inset

 that describe the activity levels of each community 
\begin_inset Formula $r$
\end_inset

 at each time step 
\begin_inset Formula $t$
\end_inset

 and 
\begin_inset Formula $N_{p}$
\end_inset

 and 
\begin_inset Formula $N_{t}$
\end_inset

 are the number of packages considered and the number of time steps chosen,
 respectively.
\end_layout

\begin_layout Standard
We will now describe 
\begin_inset Formula $H$
\end_inset

 and 
\begin_inset Formula $K$
\end_inset

 in more detail.
 We will use two facts about software dependency networks 
\begin_inset CommandInset citation
LatexCommand citep
key "Decan2018h"
literal "false"

\end_inset

:
\end_layout

\begin_layout Itemize
the number of outgoing links from each package (number of packages that
 depend on a package) follows a power-law; 
\end_layout

\begin_layout Itemize
the number of incoming links to each package (number of packages that a
 package depends on) is fairly evenly distributed.
 
\end_layout

\begin_layout Standard
\noindent
Since 
\begin_inset Formula $H$
\end_inset

 represents groups of nodes which are linked to by similar nodes, it is
 very sparse with only a few hub nodes having any community membership due
 to high strengths in 
\begin_inset Formula $H$
\end_inset

.
 
\begin_inset Formula $K$
\end_inset

 is much more evenly distributed as it represents groups of nodes which
 link to similar nodes; most packages are strongly associated with at least
 one community.
\end_layout

\begin_layout Standard
If a community has significantly declined from its peak according to a plot
 of 
\begin_inset Formula $\boldsymbol{\tau}_{r}$
\end_inset

, we will define its lifetime as the full width at half maximum.
 A summary of the relevant symbols and terms is provided in table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:Nomenclature-for-decomposed"
plural "false"
caps "false"
noprefix "false"

\end_inset

 for the reader's convenience.
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Tabular
<lyxtabular version="3" rows="11" columns="2">
<features booktabs="true" tabularvalignment="middle" tabularwidth="98text%">
<column alignment="center" valignment="middle" width="20text%">
<column alignment="center" valignment="middle" width="75text%">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Term
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Description
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
dependency
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
a package whose code is used by another package (the dependent)
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
dependent package
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
a package which uses another package's code (the dependency)
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $A=[a_{i\leftarrow j,t}]$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
the adjacency tensor of the software dependency matrix where the direction
 of links denotes the flow of code from dependencies to dependent packages
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $i$
\end_inset

, 
\begin_inset Formula $\text{max}(i)=N_{p}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
the index corresponding to packages with incoming links, i.e.
\begin_inset space ~
\end_inset

dependencies
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $j$
\end_inset

, 
\begin_inset Formula $\text{max}(j)=N_{p}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
the index corresponding to packages with outgoing links, i.e.
\begin_inset space ~
\end_inset

dependent packages
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $t$
\end_inset

, 
\begin_inset Formula $\text{max}(t)=N_{t}$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
an index corresponding to time measured in months elapsed since records
 began
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $r$
\end_inset

, 
\begin_inset Formula $\text{max}(r)=R$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
an index corresponding to communities
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $K=[\kappa_{ir}]$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
a matrix which comprises strengths of membership of package 
\begin_inset Formula $i$
\end_inset

 to community 
\begin_inset Formula $r$
\end_inset

 based on their dependencies for all 
\begin_inset Formula $i,r$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $H=[h_{jr}]$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
a matrix which comprises strengths of membership of package 
\begin_inset Formula $j$
\end_inset

 to community 
\begin_inset Formula $r$
\end_inset

 based on dependent packages for all 
\begin_inset Formula $j,r$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $T=[\tau_{tr}]$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
a matrix which comprises temporal activities of community 
\begin_inset Formula $r$
\end_inset

 at time 
\begin_inset Formula $t$
\end_inset

 for all 
\begin_inset Formula $r,t$
\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Nomenclature for decomposed adjacency tensors of software dependency networks
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Nomenclature for decomposed software dependency networks
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "tab:Nomenclature-for-decomposed"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Choosing the number of communities, 
\begin_inset Formula $R$
\end_inset


\begin_inset CommandInset label
LatexCommand label
name "subsec:Choosing-the-number"

\end_inset


\end_layout

\begin_layout Standard
One of the major downsides to tensor decompositions is that one must choose
 the number of communities, 
\begin_inset Formula $R$
\end_inset

, as explored in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Locally-optimal-numbers"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 For this application, we employ all techniques mentioned: NRSS, AMI and
 the more subjective measures mentioned in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Eyeballing-it"
plural "false"
caps "false"
noprefix "false"

\end_inset

, adapted to the application of software communities by looking at the metadata
 associated with a sample of software projects (e.g.
\begin_inset space ~
\end_inset

the names of the packages, their authors, the descriptions of the packages)
 from each community and making a qualitative judgement as to whether they
 are similar.
 For example, if a numerical computing package and a front-end web development
 package are in the same community, the number of communities is probably
 too small.
\end_layout

\begin_layout Subsection
Labelling communities
\begin_inset CommandInset label
LatexCommand label
name "subsec:Labelling-soft-communities"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $H$
\end_inset

 is a sparse matrix that denotes the main packages that are depended on
 in each community.
 The package name associated with a community 
\begin_inset Formula $r$
\end_inset

 is the name of the package 
\begin_inset Formula $p_{r}$
\end_inset

 that is maximal in 
\begin_inset Formula $\boldsymbol{h}_{r}$
\end_inset

, such that 
\begin_inset Formula $p_{r}$
\end_inset

 does not have a higher 
\begin_inset Formula $\boldsymbol{h}_{r'}$
\end_inset

 value in any other community 
\begin_inset Formula $r'$
\end_inset

.
 That is, 
\begin_inset Formula 
\[
p_{r}=\text{indmax}([\text{mask}(H)]_{r})\text{,}
\]

\end_inset

where 
\begin_inset Formula $\text{indmax}(\boldsymbol{x})$
\end_inset

 of any vector 
\begin_inset Formula $\boldsymbol{x}$
\end_inset

 returns the index of its maximum value and 
\begin_inset Formula 
\begin{equation}
[\text{mask}(M)]_{ij}=\begin{cases}
m_{ij} & \text{if }m_{ij}=\text{max}(\boldsymbol{m}_{i})\\
0 & \text{otherwise,}
\end{cases}\label{eqn:labelling}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
\noindent
where 
\begin_inset Formula $m_{ij}$
\end_inset

 and 
\begin_inset Formula $\boldsymbol{m}_{i}$
\end_inset

 are the elements and column vectors of an arbitrary matrix 
\begin_inset Formula $M$
\end_inset

.
\end_layout

\begin_layout Subsection
Hardware and software used
\end_layout

\begin_layout Standard
Julia 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{bezanson2012julia}
\end_layout

\end_inset

, LightGraphs.jl 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{seth_bromberger_2017}
\end_layout

\end_inset

, Plots.jl and Jupyter 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{kluyver2016jupyter}
\end_layout

\end_inset

 with IJulia were used for exploratory data analysis, data pre-processing,
 figures and coordinating external software.
 The actual tensor decompositions were performed using MATLAB.jl, MATLAB
 and TensorToolbox 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{TTB_Sparse}
\end_layout

\end_inset

.
 Adjusted mutual information was calculated using ScikitLearn.jl and scikit-learn
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{Pedregosa2011}
\end_layout

\end_inset

.
 The decompositions were performed on 20-core Xeon servers with 250GB of
 RAM provided by MARC1, part of the High Performance Computing and Leeds
 Institute for Data Analytics (LIDA) facilities at the University of Leeds,
 UK.
\end_layout

\begin_layout Section
Results and discussion
\begin_inset CommandInset label
LatexCommand label
name "sec:Results-and-discussion"

\end_inset


\end_layout

\begin_layout Standard
We examine the pairwise average adjusted mutual information (AMI) score
 and 
\begin_inset Formula $\text{NRSS}_{R}(X)$
\end_inset

 as defined in equation
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand vref
reference "NRSS-2"
plural "false"
caps "false"
noprefix "false"

\end_inset

 to identify a suitable number of communities for each ecosystem.
 A high AMI would indicate that the communities created by the decomposition
 are stable across repeated runs of the algorithm.
 Troughs between peaks in our AMI plots would indicate that communities
 are being split inappropriately.
 We will interpret the first peak in AMI as the minimum valid number of
 communities for an ecosystem.
 Each community will then be labelled according to equation
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eqn:labelling"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
Table
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "tbl:results"
plural "false"
caps "false"
noprefix "false"

\end_inset

 shows that there are clear differences between the language ecosystems
 in all the statistics we calculate.Python packages have a mean number of
 
\begin_inset Formula $2.4\pm2.9$
\end_inset

 dependencies or dependents (i.e.
\begin_inset space ~
\end_inset

their mean degree is 
\begin_inset Formula $2.4\pm2.9$
\end_inset

) where JavaScript packages have 
\begin_inset Formula $9.9\pm22$
\end_inset

.
 Figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:elmdist"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:pypidist"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:npmdist"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:cargodist"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:crandist"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:mavendist"
plural "false"
caps "false"
noprefix "false"

\end_inset

 show that the distributions of in- and out-degree look fairly similar in
 shape across all software ecosystems considered; broadly, all out-degree
 distributions follow a power-law and the in-degree distributions follow
 more Poisonnian distributions.
 Respectively, this means that a few packages are depended on by most packages,
 and that packages have similar numbers of dependencies within ecosystems.
\end_layout

\begin_layout Standard
Npm, the JavaScript ecosystem, in an outlier here.
 Its in-degree distribution is bimodal: there are more packages with the
 highest numbers of dependencies than there are with middling numbers of
 dependencies.
 Upon further investigation, it became apparent that the distribution is
 skewed by a few hundred joke/malicious packages
\begin_inset Foot
status open

\begin_layout Plain Layout
Such as 
\begin_inset CommandInset href
LatexCommand href
name "neat-230"
target "https://libraries.io/npm/neat-230"
literal "false"

\end_inset


\end_layout

\end_inset

 and some irregular packages that list all their recursive dependencies
 (i.e.
\begin_inset space ~
\end_inset

the dependencies of their dependencies and so on) as their direct dependencies
\begin_inset Foot
status open

\begin_layout Plain Layout
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name "@ckeditor5-build-inline"
target "https://libraries.io/npm/@southcn\\%2Fckeditor5-build-inline"
literal "false"

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 and possibly 
\begin_inset CommandInset href
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name "react-misc-toolbox"
target "https://libraries.io/npm/react-misc-toolbox"
literal "false"

\end_inset


\end_layout

\end_inset

, so they have performed a transitive closure on their dependency networks.
 We chose not to omit these packages from the decomposition as we hoped
 that tensor decomposition might discover them as a community; it did not.
\end_layout

\begin_layout Standard
We believe that the presence of a rich and well-used standard library—the
 functions included in the language without any other dependencies—reduces
 the mean node degree
\begin_inset Foot
status open

\begin_layout Plain Layout
NB: the mean in- and out-degrees are always equal as was shown in section
 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Degree"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\end_inset

 of an ecosystem.
 Python (mean degree 
\begin_inset Formula $2.4$
\end_inset

) has such a library, while JavaScript (mean degree 
\begin_inset Formula $9.9$
\end_inset

) emphatically does not.
\end_layout

\begin_layout Standard
Elm (mean degree 
\begin_inset Formula $2.6$
\end_inset

) features a small standard library compared to Python, but its library
 is very rich within its domain of web application development.
\end_layout

\begin_layout Standard
Rust (mean degree 
\begin_inset Formula $4.4$
\end_inset

) features a small standard library by design (instead promoting and curating
 community packages).
 R (mean degree 
\begin_inset Formula $5.3$
\end_inset

) features a fairly small standard library and many users now prefer the
 community 
\family typewriter
tidyverse
\family default
 collection as a consistent community alternative.
\end_layout

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placement h!
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status open

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\begin_layout Plain Layout
Summary statistics for each ecosystem.
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Summary statistics for each software ecosystem
\end_layout

\end_inset

 Number of communities is the number we choose to plot and analyse.
 For elm-get, CRAN and PyPI, other numbers of communities also seemed reasonable.
 NRSS is not reported because it did not contribute to analysis.
 Communities are considered to have declined if their activity at the end
 of the study period is substantially lower than their peak activity.
 All numbers are provided to two significant figures.
 
\begin_inset Formula $k$
\end_inset

 refers to the in- and out-degrees, 
\begin_inset Formula $\sigma$
\end_inset

 is the standard deviation and 
\begin_inset Formula $\overline{k}$
\end_inset

 is the arithmetic mean of both the in- and out-degree distributions.
 
\begin_inset Formula $R$
\end_inset

 is the number of communities determined to be the optimum.
\end_layout

\end_inset


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name "tbl:results"

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\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Ecosystem-specific results
\end_layout

\begin_layout Standard
We first provide detailed examination of each ecosystem.
 A broader overview and comparison between ecosystems is given in section
 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Summary-and-inter-ecosystem"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Subsubsection
Elm
\end_layout

\begin_layout Standard
Figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:elmact"
plural "false"
caps "false"
noprefix "false"

\end_inset

 shows community activity over time.
 
\family typewriter
evancz/html
\family default
 is the precursor of 
\family typewriter
elm-lang/html
\family default
 and we can see an early community of packages using 
\family typewriter
evancz/html
\family default
 that decays over time as the newer community using 
\family typewriter
elm-lang/html
\family default
 emerges.
 We can also see the swift rise of a community of HTTP-using packages.
 This is perhaps particularly pronounced in Elm because the language is
 unusually restrictive (there was no official way to write HTTP packages
 before the package which became 
\family typewriter
elm-lang/http
\family default
) and young (there was a strong known demand for an obvious feature like
 HTTP support).
 In a more mature language, one might expect that the rise and fall of communiti
es would be less extreme.
\end_layout

\begin_layout Standard
The fall in activity of the 
\family typewriter
evancz/html
\family default
 community, shows that many earlier packages were abandoned and did not
 make the jump to Elm 0.17 and 
\family typewriter
elm-lang/html
\family default

\begin_inset Foot
status collapsed

\begin_layout Plain Layout

\family typewriter
evancz/html
\family default
 is deprecated in Elm 0.17
\end_layout

\end_inset

.
 Those that do make the jump appear in the 
\family typewriter
elm-lang/html
\family default
 community as they switch to depending on that package.
 New packages generally do not depend on them, nor depend on packages that
 depend on them, explaining the fall in activity.
\end_layout

\begin_layout Standard
Elm is a young and deliberately unconventional and experimental language,
 so it is to be expected that many packages will be abandoned as the community
 experiments with different approaches.
\end_layout

\begin_layout Standard
The rapid changes in activity indicate that the Elm ecosystem is not stable,
 that is, communities have experienced large relative changes in activity
 recently.
\end_layout

\begin_layout Standard
High AMI for communities two and four suggest that the Elm ecosystem is
 not diverse as it is well described by small numbers of communities.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/whole-langs/Elm-whole-indegree.pdf
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	width 50text%
	groupId sidebyside

\end_inset


\begin_inset Graphics
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	width 50text%
	groupId sidebyside

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Elm degree distributions
\end_layout

\end_inset

In- and out-degree distributions for the Elm package ecosystem
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:elmdist"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\bestrcaption}{ Relative activity of each community in each month. The communities are named by important packages as detailed in the Methods section }
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/results/Elm-best-r-time.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Elm community activity
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Elm community activity
\end_layout

\end_inset

: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
bestrcaption
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\end_inset


\end_layout

\end_inset


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name "fig:elmact"

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\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/results/Elm-ami-rss.pdf
	display false
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	groupId toowide

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Choosing the number of communities: Elm
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Choosing the number of communities: Elm
\end_layout

\end_inset

.
 Left: the mean pairwise adjusted mutual information of the community-node
 membership across repeated decompositions.
 
\begin_inset Formula $1$
\end_inset

 is a perfectly stable assignment to communities; 
\begin_inset Formula $0$
\end_inset

 is perfectly random.
 The shaded area corresponds to twice the standard error of the mean.
 Right: the mean normalised residual sum of squares
\begin_inset Index idx
status open

\begin_layout Plain Layout
residual sum of squares
\end_layout

\end_inset

 error for repeated decompositions on the Elm network against the number
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 The shaded area corresponds to twice the standard error of the mean above
 and below the line
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:Elm-ami-rss"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
Python
\end_layout

\begin_layout Standard
Figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:pypiact"
plural "false"
caps "false"
noprefix "false"

\end_inset

 demonstrates how these communities change over time: it is interesting
 that there are no communities which have declined particularly.
 One community, 
\family typewriter
django
\family default
, a web application framework, seems to have stagnated.
 Google Trends data suggest that interest in it has reached a plateau compared
 to frameworks in other languages such as 
\family typewriter
React
\family default
 for JavaScript.
\end_layout

\begin_layout Standard
The growth of communities, as shown in figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:pypiact"
plural "false"
caps "false"
noprefix "false"

\end_inset

, seems to be monotonic.
 There do not appear to be any major deprecated packages, which is surprising,
 given the impending deprecation of Python version 2, which is incompatible
 with Python version 3.
 This could be due to the popular package 
\family typewriter
six
\family default
 which provides compatibility helpers for Python 2 and 3.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/whole-langs/Pypi-whole-indegree.pdf
	display false
	width 50text%
	groupId sidebyside

\end_inset


\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/whole-langs/Pypi-whole-outdegree.pdf
	display false
	width 50text%
	groupId sidebyside

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PyPI degree distributions
\end_layout

\end_inset

In- and out-degree distributions for the PyPI package ecosystem
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:pypidist"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/results/Pypi-best-r-time.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
PyPI community activity: 
\begin_inset Argument 1
status collapsed

\begin_layout Plain Layout
PyPI community activity
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
bestrcaption
\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:pypiact"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/results/Pypi-ami-rss.pdf
	display false
	width 100text%
	groupId toowide

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Choosing the number of communities: PyPI
\end_layout

\end_inset

Choosing the number of communities: PyPI.
 Left: the mean pairwise adjusted mutual information of the community-node
 membership across repeated decompositions.
 
\begin_inset Formula $1$
\end_inset

 is a perfectly stable assignment to communities; 
\begin_inset Formula $0$
\end_inset

 is perfectly random.
 The shaded area corresponds to twice the standard error of the mean.
 Right: the mean normalised residual sum of squares error for repeated decomposi
tions on the PyPI network against the number of components.
 The shaded area corresponds to twice the standard error of the mean above
 and below the line
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:Pypi-ami-rss"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
JavaScript
\end_layout

\begin_layout Standard
The AMI plot for npm is given in figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:npmami"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 We chose not to calculate the NRSS values due to computational constraints
 due to the far greater size of the dataset corresponding to npm compared
 to the other ecosystems we considered.
 The AMI for every number of communities we trialled is very low and although
 we use three communities in the time activity plot, figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:npmact"
plural "false"
caps "false"
noprefix "false"

\end_inset

, we are not particularly confident that the communities picked are meaningful.
 It is plausible that some larger number of communities would fit the data
 better, however we trialled up to 50 communities with no success.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/whole-langs/NPM-whole-indegree.pdf
	display false
	width 50text%
	groupId sidebyside

\end_inset


\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/whole-langs/NPM-whole-outdegree.pdf
	display false
	width 50text%
	groupId sidebyside

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
npm degree distributions
\end_layout

\end_inset

In- and out-degree distributions for the npm package ecosystem
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:npmdist"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
includegraphics[width=0.55
\backslash
paperwidth]{../ans-ml-cfp/paper/src/figures/results/NPM-best-r-time.pdf}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
npm community activity
\end_layout

\end_inset

npm community activity: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
bestrcaption
\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:npmact"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/results/NPM-ami.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
npm adjusted mutual information against components: 
\begin_inset Argument 1
status open

\begin_layout Plain Layout
npm adjusted mutual information against components
\end_layout

\end_inset

The pairwise adjusted mutual information of the community-node membership
 across repeated decompositions.
 
\begin_inset Formula $1$
\end_inset

 is a perfectly stable assignment to communities; 
\begin_inset Formula $0$
\end_inset

 is perfectly random.
 The shaded area corresponds to twice the standard error of the mean above
 and below the line
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:npmami"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
Rust
\end_layout

\begin_layout Standard
The AMI and NRSS plots for Rust are shown in figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Cargo-ami-rss"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 The AMI plot strongly indicates a first peak at 8 communities.
\end_layout

\begin_layout Standard
Figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:cargoact"
plural "false"
caps "false"
noprefix "false"

\end_inset

 demonstrates how these communities change over time.
 Of particular note are the communities centred around 
\family typewriter
rustc-serialize
\family default
 and 
\family typewriter
serde
\family default
.
 
\family typewriter
rustc-serialize
\family default
 has been deprecated and officially replaced with the 
\family typewriter
serde
\family default
 package.
 The decline in activity of the 
\family typewriter
rustc-serialize
\family default
 community and rise of the 
\family typewriter
serde
\family default
 community is quite distinct and clearly occurs at the same time.
\end_layout

\begin_layout Standard
The community labelled 
\family typewriter
winapi
\family default
 is also easily interpretable.
 The 
\begin_inset Formula $K$
\end_inset

 factor for this community is bimodal and the size of the higher strength
 cluster indicates that there are around 450 packages in this community.
 A brief review of these packages suggests that they are low-level cross-platfor
m packages for interacting with the operating system.
 These packages were amongst the first written and predate the package manager,
 which explains their steep initial rise.
\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: what Eva says below is quite valid.
 I don't have time to dig into the results for evidence of the exact packages
 (and, indeed, cargo predates winapi by ~6 months) :/
\end_layout

\begin_layout Plain Layout
Eva says:
\end_layout

\begin_layout Plain Layout
Where is this in your analysis? Support these conclusions with your analysis.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/whole-langs/Cargo-whole-indegree.pdf
	display false
	width 50text%
	groupId sidebyside

\end_inset


\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/whole-langs/Cargo-whole-outdegree.pdf
	display false
	width 50text%
	groupId sidebyside

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Cargo degree distributions
\end_layout

\end_inset

In- and out-degree distributions for the Cargo package ecosystem
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:cargodist"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/results/Cargo-best-r-time.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Cargo community activity
\end_layout

\end_inset

Cargo community activity: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
bestrcaption
\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:cargoact"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/results/Cargo-ami-rss.pdf
	display false
	width 100text%
	groupId toowide

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Choosing the number of communities: Cargo
\end_layout

\end_inset

Choosing the number of communities: Cargo.
 Left: the mean pairwise adjusted mutual information of the community-node
 membership across repeated decompositions.
 
\begin_inset Formula $1$
\end_inset

 is a perfectly stable assignment to communities; 
\begin_inset Formula $0$
\end_inset

 is perfectly random.
 The shaded area corresponds to twice the standard error of the mean.
 Right: the mean normalised residual sum of squares error for repeated decomposi
tions on the Cargo network against the number of components.
 The shaded area corresponds to twice the standard error of the mean above
 and below the line
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:Cargo-ami-rss"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
R
\end_layout

\begin_layout Standard
As shown in figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:CRAN-ami-rss"
plural "false"
caps "false"
noprefix "false"

\end_inset

, AMI suggests that decomposition into seven or eight communities is acceptable
 as this is where the peak is located.
 That CRAN can be stably decomposed into two communities suggests that the
 ecosystem is not very diverse, which is as we would expect from a language
 focused on scientific and statistical computing.
\end_layout

\begin_layout Standard
It can be seen in figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:cranact"
plural "false"
caps "false"
noprefix "false"

\end_inset

 that when decomposed into seven communities, distinctive waves of activity
 over time exist.
 We believe these waves are driven by CRAN's rolling release model which
 requires packages to be actively maintained or de-listed.
 This is a strong incentive for package authors to switch dependencies if
 e.g.
\begin_inset space \space{}
\end_inset

one of two testing libraries is actively maintained.
 The de-listing also means that older, unused packages are removed from
 the dependency graph over time, unlike any of the other studied ecosystems.
\end_layout

\begin_layout Standard
Visible in figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:cranact"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and an exemplar of this behaviour, 
\family typewriter
RUnit
\family default
 is an older unmaintained testing library and 
\family typewriter
testthat
\family default
 is a more modern replacement.
 Unlike any other long-established language in this analysis, there is a
 clear and steep decline of the earlier library in favour of the competitor
\begin_inset Foot
status open

\begin_layout Plain Layout
Package popularity indicated on METACRAN: 
\begin_inset CommandInset href
LatexCommand href
target "https://cranlogs.r-pkg.org/"
literal "false"

\end_inset


\end_layout

\end_inset

.
\end_layout

\begin_layout Standard
Despite R being an old language, some of its communities have grown at a
 rapid rate, specifically those relating to 
\family typewriter
knitr
\family default
 and 
\family typewriter
testthat
\family default
.
 This suggests that, despite being an older language, R's ecosystem is still
 subject to significant change.
\end_layout

\begin_layout Standard
Defining community lifetime as the full width at half maximum, we can see
 from figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:cranact"
plural "false"
caps "false"
noprefix "false"

\end_inset

 that communities tend to stay active for approximately 
\begin_inset Formula $4$
\end_inset

 to 
\begin_inset Formula $8$
\end_inset

 years.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/whole-langs/CRAN-whole-indegree.pdf
	display false
	width 50text%
	groupId sidebyside

\end_inset


\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/whole-langs/CRAN-whole-outdegree.pdf
	display false
	width 50text%
	groupId sidebyside

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
CRAN degree distributions
\end_layout

\end_inset

 In- and out-degree distributions for the CRAN package ecosystem
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:crandist"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/results/CRAN-best-r-time.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
CRAN community activity
\end_layout

\end_inset

CRAN community activity: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
bestrcaption
\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:cranact"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/results/CRAN-ami-rss.pdf
	display false
	width 100text%
	groupId toowide

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Choosing the number of communities: CRAN.
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Choosing the number of communities: CRAN
\end_layout

\end_inset

 Left: the mean pairwise adjusted mutual information of the community-node
 membership across repeated decompositions.
 
\begin_inset Formula $1$
\end_inset

 is a perfectly stable assignment to communities; 
\begin_inset Formula $0$
\end_inset

 is perfectly random.
 The shaded area corresponds to twice the standard error of the mean.
 Right: the mean normalised residual sum of squares error for repeated decomposi
tions on the CRAN network against the number of components.
 The shaded area corresponds to twice the standard error of the mean above
 and below the line
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:CRAN-ami-rss"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsubsection
Java
\end_layout

\begin_layout Standard
The AMI plot, figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Maven-ami-rss"
plural "false"
caps "false"
noprefix "false"

\end_inset

, has an unambiguous peak at five communities.
 AMI is very low for two communities and comparatively low for three and
 four, suggesting that Maven has a wider diversity of communities than CRAN
 or elm-get.
 These communities include a distinct Scala community, a kind of dialect
 of Java, centred around the Scala standard library.
\end_layout

\begin_layout Standard
Figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:mavenact"
plural "false"
caps "false"
noprefix "false"

\end_inset

 demonstrates how these communities change over time.
 There is a general upwards trend and some noise that may be spurious.
 As noted by 
\begin_inset CommandInset citation
LatexCommand citet
key "Decan2018h"
literal "false"

\end_inset

, there may be issues with the Maven dataset that are affecting these results.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/whole-langs/Maven-whole-indegree.pdf
	display false
	width 50text%
	groupId sidebyside

\end_inset


\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/whole-langs/Maven-whole-outdegree.pdf
	display false
	width 50text%
	groupId sidebyside

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Maven degree distributions
\end_layout

\end_inset

In- and out-degree distributions for the Maven package ecosystem
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:mavendist"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/results/Maven-best-r-time.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Maven community activity
\end_layout

\end_inset

Maven community activity: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
bestrcaption
\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:mavenact"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status collapsed

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
includegraphics[width=0.55
\backslash
paperwidth]{../ans-ml-cfp/paper/src/figures/results/Maven-ami-rss.pdf}
\end_layout

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Choosing the number of communities: Maven
\end_layout

\end_inset

Choosing the number of communities: Maven.
 Left: the mean pairwise adjusted mutual information of the community-node
 membership across repeated decompositions.
 
\begin_inset Formula $1$
\end_inset

 is a perfectly stable assignment to communities; 
\begin_inset Formula $0$
\end_inset

 is perfectly random.
 The shaded area corresponds to twice the standard error of the mean.
 Right: the mean normalised residual sum of squares error for repeated decomposi
tions on the Maven network against the number of components.
 The shaded area corresponds to twice the standard error of the mean above
 and below the line.
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:Maven-ami-rss"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Summary and inter-ecosystem comparison
\begin_inset CommandInset label
LatexCommand label
name "subsec:Summary-and-inter-ecosystem"

\end_inset


\end_layout

\begin_layout Standard
In this section we provide a brief summary of the results and compare the
 ecosystems broadly.
\end_layout

\begin_layout Subsubsection
Number of communities
\begin_inset CommandInset label
LatexCommand label
name "subsec:Number-of-communities"

\end_inset


\end_layout

\begin_layout Standard
Figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Elm-ami-rss"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Pypi-ami-rss"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:npmami"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Cargo-ami-rss"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:CRAN-ami-rss"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Maven-ami-rss"
plural "false"
caps "false"
noprefix "false"

\end_inset

 show the mean AMI and NRSS for each number of communities we evaluated
 for each ecosystem.
 We computed 10 decompositions for each number of communities we tested
 in each ecosystem.
\end_layout

\begin_layout Standard
Identifying a suitable number of communities was easiest by looking for
 early peaks in the AMI plots, which correlated well with visual inspection
 of decomposition stability mentioned in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Eyeballing-it"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and our manual checks on package names.
 The NRSS plots have no obvious elbows and contributed nothing to our analysis;
 we include them as a relevant negative result showing that NRSS is not
 useful on some real-world networks.
\end_layout

\begin_layout Standard
For Elm, CRAN and PyPI, AMI indicates more than one 
\begin_inset Quotes xld
\end_inset

good
\begin_inset Quotes xrd
\end_inset

 number of communities to split the ecosystem into.
 We have chosen the smallest good number except when that number was two.
\end_layout

\begin_layout Standard
We observed interesting differences between ecosystems in this metric: Elm
 and R have relatively high AMIs at relatively low numbers of components
 while other languages such as JavaScript had low AMIs for all of the numbers
 of components we tried, suggesting that a more representative number of
 components might be much higher.
 This tells us something about the diversity of an ecosystem: the more communiti
es it has, the more diverse it must be, suggesting that R and Elm are less
 diverse than the other language considered.
 This matches up neatly with our domain knowledge that Elm is almost exclusively
 used for creating websites and R for performing statistical analysis, whereas
 the other languages are much more general purpose.
\end_layout

\begin_layout Subsubsection
Activity over time
\end_layout

\begin_layout Standard
Figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:elmact"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:pypiact"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:npmact"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:cargoact"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:cranact"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:mavenact"
plural "false"
caps "false"
noprefix "false"

\end_inset

 show the relative activity of communities in each ecosystem over time for
 a representative decomposition; i.e.
\begin_inset space \space{}
\end_inset

they are plots of 
\begin_inset Formula $\tau_{rt}$
\end_inset

 from equation
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eqn:decomp"
plural "false"
caps "false"
noprefix "false"

\end_inset

 against time, 
\begin_inset Formula $t$
\end_inset

.
\end_layout

\begin_layout Standard
The labels are reasonably stable across decompositions when AMI is high
 and they appear to be informative in most ecosystems.
 The relative size and activity over time of the communities can often be
 matched with what we know about the their constituent packages and the
 ecosystem from other sources.
\end_layout

\begin_layout Standard
The community activity plots show clear differences between the various
 ecosystems.
 Elm, R and Rust all contain communities that decline in activity over time.
 Some languages such as Elm, R and Rust have 
\begin_inset Quotes xld
\end_inset

peaky
\begin_inset Quotes xrd
\end_inset

 community activity where packages have been replaced by others.
 Java, Python and JavaScript do not exhibit such behaviour; suggesting that
 their communities are more stable—their composition does not change much
 over time in terms of nodes or links between those nodes.
 The 
\begin_inset Quotes xld
\end_inset

% declined
\begin_inset Quotes xrd
\end_inset

 column in table
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "tbl:results"
plural "false"
caps "false"
noprefix "false"

\end_inset

 summarises this behaviour.
\end_layout

\begin_layout Subsubsection
Community composition
\end_layout

\begin_layout Standard
Comparing ecosystems, we observe that Python has communities centred around
 making websites (
\family typewriter
django
\family default
), scientific computing (
\family typewriter
numpy
\family default
), or interpreting configuration files (
\family typewriter
pyyaml
\family default
).
 By contrast, Elm's communities are all to do with building web applications
 and R's communities are mostly focused towards scientific computing.
 This suggests, uncontroversially, that Elm and R are less general purpose
 languages, as their communities are full of packages doing similar things,
 than Python, where the communities do very different things.
\end_layout

\begin_layout Section
Conclusions
\begin_inset CommandInset label
LatexCommand label
name "sec:Conclusions"

\end_inset


\end_layout

\begin_layout Standard
We have found a large amount of variation between different software ecosystems;
 some, such as Python, are stable and long-lived and others, such as Elm,
 have packages that have been important but short-lived due to deprecation.
 It is our suspicion that a large amount of the variation between languages
 is due to the size and usefulness of their standard libraries.
 However, some of the differences, especially those found by the tensor
 decompositions, could be because early communities which fall out of use
 are ignored by the decomposition in favour of later, larger communities.
 It would be interesting to investigate whether this effect is real.
\end_layout

\begin_layout Standard
Based on our analysis, we have the following answers to our research questions:
\end_layout

\begin_layout Itemize
\begin_inset Formula $RQ_{1}$
\end_inset

 What are the differences between different software ecosystems? 
\end_layout

\begin_layout Standard
There are differences in the number of communities of software packages,
 the rate at which communities of software packages gain and lose popularity
 as measured by how often they are dependant on each other and in the overall
 trajectory of the growth of these communities.
\end_layout

\begin_layout Itemize
\begin_inset Formula $RQ_{2}$
\end_inset

 What do communities of commonly-used-together software packages look like?
 
\end_layout

\begin_layout Standard
Similar to the languages as a whole, the communities have fairly homogeneous
 in-degree distributions but heterogeneous power-law out-degree distributions.
\end_layout

\begin_layout Itemize
\begin_inset Formula $RQ_{3}$
\end_inset

 How do these communities change with time? 
\end_layout

\begin_layout Standard
The general rule is 
\begin_inset Quotes xld
\end_inset

up and to the right
\begin_inset Quotes xrd
\end_inset

: all of the ecosystems we considered increased in activity as time went
 on, in agreement with 
\begin_inset CommandInset citation
LatexCommand citet
key "Decan2018h"
literal "false"

\end_inset

.
 This is unsurprising as it is relatively rare for a package to lose dependencie
s—it can fall out of contemporary use in that new packages will not choose
 to depend on it, but the packages that previously depended on it will tend
 to continue to depend on it as rewriting them to use alternative packages
 often requires significant effort.
\end_layout

\begin_layout Standard
Our results agree with that found in other works, especially that of 
\begin_inset CommandInset citation
LatexCommand citet
key "Decan2018h"
literal "false"

\end_inset

.
 Figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:cranact"
plural "false"
caps "false"
noprefix "false"

\end_inset

 in particular shows the effect of CRAN's strict rolling-release model where
 out-of-date packages are archived (meaning that they are no longer as easy
 to install); figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:cargoact"
plural "false"
caps "false"
noprefix "false"

\end_inset

 shows how the Rust ecosystem seems to start very suddenly as the package
 manager came relatively late into the language's life.
\end_layout

\begin_layout Standard
For Java specifically, our work does not show much overlap with the communities
 found by 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citet{Subelj2011b}
\end_layout

\end_inset

.
 This could be due to differences between their static and our dynamic community
 detection, but it could also be that the validity of our results is questionabl
e for Maven as the 
\family typewriter
Libraries.io
\family default
 data for Maven is incomplete 
\begin_inset CommandInset citation
LatexCommand citep
key "Decan2018h"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard
The absolute level of the adjusted mutual information at low numbers of
 communities seems to reflect the diversity of an ecosystem: R and elm,
 both domain specific languages, have high levels of AMI for 
\begin_inset Formula $2$
\end_inset

 and 
\begin_inset Formula $3$
\end_inset

, whereas all of the general purpose languages we consider do not.
\end_layout

\begin_layout Standard
Methodologically, we found that using adjusted mutual information as a metric
 by which to choose the number of communities led to (predominantly) explainable
 communities that appeared to be stable across multiple runs of the decompositio
ns and across multiple values of the number of communities chosen.
 Our labelling algorithm, equation
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "eqn:labelling"
plural "false"
caps "false"
noprefix "false"

\end_inset

 worked well for our application, giving plausible and comprehensible labels
 and could plausibly work well for any communities in networks with power-law
 degree distributions.
 The stability of an ecosystem can be seen from our activity over time plots,
 i.e.
\begin_inset space \space{}
\end_inset

the 
\begin_inset Formula $\boldsymbol{\tau}_{r}$
\end_inset

 vectors.
\end_layout

\begin_layout Standard
Further work could involve investigating larger components for languages
 for which we currently have fairly low AMI scores.
 It would also be fairly straight-forward to extend our technique to consider
 the co-authorship network that creates the software as another layer in
 the network.
 Knowledge of how the ecosystem evolves organically could be used to detect
 fraudulent packages, especially by unknown authors.
 The creation of models of software ecosystem evolution from simple sets
 of rules (for example, 10% of packages are deprecated every 6 months) to
 try to replicate our results synthetically could also prove insightful.
\end_layout

\begin_layout Standard
Our work could have applications in helping designers of software ecosystems
 to make informed choices; it is clear that, for example, CRAN's rolling
 release policy has a big impact on its software communities.
 One could imagine using clusterings like those we have created to determine
 whether part of a software ecosystem was on the wane and was likely to
 be replaced soon.
 Such considerations would make sense if one was choosing dependencies for
 a project that was intended to last many years.
 Additionally, if one knows how communities form naturally, it becomes easier
 to pick out outliers, as we have seen with the npm 
\begin_inset Quotes xld
\end_inset

joke
\begin_inset Quotes xrd
\end_inset

 packages.
\end_layout

\begin_layout Standard
In conclusion, we have described the long term evolution of several software
 ecosystems by breaking them up into their constituent communities.
 We have been able to spot ground-truth events such as the deprecation of
 major software packages in the temporal activities of these communities.
 We have demonstrated striking differences between different software ecosystems
, particularly with the rolling release cycle of R.
 Previous work has not had as much success highlighting these differences;
 we have demonstrated the use of applying state-of-the-art network science
 techniques for analysing software dependency networks.
 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "foss"
symbol "FOSS"
description "free and open-source software"
literal "false"

\end_inset

 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "ami"
symbol "AMI"
description "adjusted mutual information"
literal "false"

\end_inset


\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "cran"
symbol "CRAN"
description "Comprehensive R Archive Network"
literal "false"

\end_inset

 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "cp"
symbol "CP"
description "CANDECOMP/PARAFAC"
literal "false"

\end_inset

 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "cande"
symbol "CANDECOMP"
description "canonical decomposition"
literal "false"

\end_inset

 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "parafac"
symbol "PARAFAC"
description "parallel factor analysis"
literal "false"

\end_inset

 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "lida"
symbol "LIDA"
description "Leeds Institute for Data Analytics"
literal "false"

\end_inset

 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "nrss"
symbol "NRSS"
description "normalised residual sum of squares"
literal "false"

\end_inset

 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "http"
symbol "HTTP"
description "hypertext transfer protocol"
literal "false"

\end_inset

 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "pypi"
symbol "PyPI"
description "Python Package Index"
literal "false"

\end_inset

 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "dom"
symbol "DOM"
description "document object model"
literal "false"

\end_inset


\end_layout

\begin_layout Standard
In the next chapter we explore whether we can repeat our success in applying
 tensor decompositions to real-world networks in a completely different
 field: we explore applying them to networks representing human languages,
 specifically the English language as recorded by 
\begin_inset CommandInset citation
LatexCommand citet
key "Coleridge2019"
literal "false"

\end_inset

.
\end_layout

\begin_layout Standard

\end_layout

\begin_layout Section*
\begin_inset Newpage newpage
\end_inset

Bibliography
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "monorefs"
options "authordate4"

\end_inset


\end_layout

\begin_layout Chapter
Linguistic evolution
\begin_inset CommandInset label
LatexCommand label
name "chap:Linguistic-evolution"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
epigraph{``
\backslash
textup{Gaéð á wyrd swá hío scel.}
\backslash

\backslash
Fate goes always as She must"}{--- 
\backslash
textup{Unknown author,} Beowulf c.
 990
\backslash

\backslash
and modern translation by B Slade}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
MyLettrine{T}{he English}
\end_layout

\end_inset

 language has existed for approximately 1500 years and has therefore been
 exposed to massive societal and technological change.
 Previous work has shown that this exogenous change is visible in the language
 
\begin_inset CommandInset citation
LatexCommand citep
key "Perc2012,Cuskley2014"
literal "false"

\end_inset

.
 Our work conducted in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Software"
plural "false"
caps "false"
noprefix "false"

\end_inset

 proved that tensor decompositions can be applied to study communities in
 temporal real-world networks.
 In this chapter, we hope to understand the change in the English language
 more clearly through representing the language as a temporal network and
 performing temporal clustering on it.
\end_layout

\begin_layout Standard
As shown in figures 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:oed-words-born"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:oed-words-over-time"
plural "false"
caps "false"
noprefix "false"

\end_inset

, the English language has evolved in a fitful manner; its growth rate has
 been far from constant with many periods in which expansion was above historica
l norms.
 This chapter hopes to explore the causes of these expansions by identifying
 groups of words which appeared in the language at around the same time.
 We note that naive approaches, such as splitting the language up into regions,
 do not give a satisfactory answer except, perhaps, for the last 200 years
 which may have been driven by colonisation, as may be seen in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:oed-words-over-time-per-region"
plural "false"
caps "false"
noprefix "false"

\end_inset

 where the growth of words increases in regions colonised by the British
 around the 18
\begin_inset Formula $^{\text{th}}$
\end_inset

 century.
\end_layout

\begin_layout Standard
We hope to be able to see the rise and fall of terminology related to specific
 fields, such as those used in sailing, the automotive industry and computing;
 we also hope to see changes from exposure to foreign influence, such as
 the Columbian exchange.
 This work has the potential to improve our understanding of how groups
 of words enter and leave the English language and how the rate at which
 that evolution occurs has changed over time.
\end_layout

\begin_layout Standard
This chapter is split into five sections: section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Background"
plural "false"
caps "false"
noprefix "false"

\end_inset

 provides a brief introduction to the surrounding literature; section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Methods-1"
plural "false"
caps "false"
noprefix "false"

\end_inset

 presents the manner in which we present the network and decompose it; section
 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Preliminary-results"
plural "false"
caps "false"
noprefix "false"

\end_inset

 displays and discusses our results so far; section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Further-work"
plural "false"
caps "false"
noprefix "false"

\end_inset

 explores how we might improve our results; and finally section 
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Conclusions-1"
plural "false"
caps "false"
noprefix "false"

\end_inset

 compares our work to previous work, providing a short summary.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/oed_word_births.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Word-meanings entering the English language over time
\end_layout

\end_inset

The number of lemmas entering the English language over time as recorded
 by the Oxford English Dictionary (OED, 
\begin_inset CommandInset citation
LatexCommand citealp
key "Coleridge2019"
literal "false"

\end_inset

).
 The large spike at the origin is due to how the OED records words for which
 there is no known date of first use—it assumes 0AD
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:oed-words-born"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ccs19singapore/code/oed_words_per_year.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Size of English language over time
\end_layout

\end_inset

The size of the English language over time: the number of lemmas marked
 as 
\begin_inset Quotes xld
\end_inset

active
\begin_inset Quotes xrd
\end_inset

 or 
\begin_inset Quotes xld
\end_inset

inactive
\begin_inset Quotes xrd
\end_inset

 in the Oxford English Dictionary 
\begin_inset CommandInset citation
LatexCommand citep
key "Coleridge2019"
literal "false"

\end_inset

 over time
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:oed-words-over-time"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ccs19singapore/code/oed_words_per_year_region.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Size of English language over time per region
\end_layout

\end_inset

The size of the English language over time per region: the number of lemmas
 marked as 
\begin_inset Quotes xld
\end_inset

active
\begin_inset Quotes xrd
\end_inset

 or 
\begin_inset Quotes xld
\end_inset

inactive
\begin_inset Quotes xrd
\end_inset

 in the Oxford English Dictionary 
\begin_inset CommandInset citation
LatexCommand citep
key "Coleridge2019"
literal "false"

\end_inset

 over time, marked as predominantly used in the given region.
 NB: the given region corresponds to modern usage, so old words which are
 predominantly used in a different region to where they were first introduced
 will show up in the 
\begin_inset Quotes xld
\end_inset

wrong
\begin_inset Quotes xrd
\end_inset

 region; this explains why North America has many words listed for centuries
 before a word of English was spoken on that continent
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:oed-words-over-time-per-region"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Background
\begin_inset CommandInset label
LatexCommand label
name "sec:Background"

\end_inset


\end_layout

\begin_layout Standard
As discussed in section 
\begin_inset CommandInset ref
LatexCommand vref
reference "subsec:Semantic-networks"
plural "false"
caps "false"
noprefix "false"

\end_inset

, there is some related work in this area where authors have applied tensor
 decompositions to semantic networks 
\begin_inset CommandInset citation
LatexCommand citep
key "Nakatsuji2016a"
literal "false"

\end_inset

.
 The applications are very different, however; the authors are trying to
 predict how users feel about products and activities and there is no temporal
 factor.
 There are also works utilising non-negative tensor decomposition to probe
 the syntax of language albeit again without any temporal factor 
\begin_inset CommandInset citation
LatexCommand citep
key "Anisimov2014"
literal "false"

\end_inset

.
 There is some literature based on data similar to that which we use but
 again for completely different purposes; 
\begin_inset CommandInset citation
LatexCommand citet
key "Piao2017"
literal "false"

\end_inset

, for example, built a programme to tag words to specific lemmas, single
 meanings of words as was explained in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Semantic-networks"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Similar data is available 
\begin_inset CommandInset citation
LatexCommand citep
key "kay2009historical"
literal "false"

\end_inset

 and some work has been done on visualising it 
\begin_inset CommandInset citation
LatexCommand citep
key "alexander2012patchworks"
literal "false"

\end_inset

 and looking at the evolution of specific words over time 
\begin_inset CommandInset citation
LatexCommand citep
key "alexander2014spread"
literal "false"

\end_inset

.
 There has also been interesting work on the evolution of languages as a
 syntactic network both from a theoretical 
\begin_inset CommandInset citation
LatexCommand citep
key "Dorogovtsev2001"
literal "false"

\end_inset

 and empirical viewpoint 
\begin_inset CommandInset citation
LatexCommand citep
key "Sole2010"
literal "false"

\end_inset

.
 For a thorough survey of detecting (rather than analysing) semantic change
 see 
\begin_inset CommandInset citation
LatexCommand citet
key "Tahmasebi2018"
literal "false"

\end_inset

.
 We are unaware of any work applying tensor decompositions to temporal semantic
 networks in an attempt to better understand their evolution.
\end_layout

\begin_layout Section
Methods
\begin_inset CommandInset label
LatexCommand label
name "sec:Methods-1"

\end_inset


\end_layout

\begin_layout Standard
The bulk of this chapter follows the methods introduced in chapters 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Locally-optimal-numbers"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Software"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 We made several application specific adjustments.
\end_layout

\begin_layout Subsection
Building the network
\end_layout

\begin_layout Standard
We present a subset of the language from the Oxford English Dictionary (OED,
 
\begin_inset CommandInset citation
LatexCommand citealp
key "Coleridge2019"
literal "false"

\end_inset

) as a temporal network with an undirected adjacency tensor
\begin_inset Index idx
status open

\begin_layout Plain Layout
network representations!adjacency tensor
\end_layout

\end_inset

 
\begin_inset Formula $A$
\end_inset

 which comprises
\begin_inset Formula 
\[
a_{ijt}=\alpha_{i}(t)\cdot\alpha_{j}(t)\cdot\begin{cases}
1 & \text{if lemma }i\text{ is a synonym or near-synonym of lemma }j\text{,}\\
0 & \text{otherwise,}
\end{cases}
\]

\end_inset

where
\begin_inset Formula 
\[
\alpha_{i}(t)=\begin{cases}
1 & \text{if lemma }i\text{ is in usage at time }t\text{,}\\
0 & \text{otherwise.}
\end{cases}
\]

\end_inset

We define near-synonymy as any words no more than one level above or below
 each other in the OED's hierarchical thesaurus; in practice this means
 that things like adverbs and adjectives with the same root (e.g
\begin_inset space ~
\end_inset

quickly, quick) are linked as well as true synonyms (e.g.
\begin_inset space ~
\end_inset

fast, quick).
 This is clearly a network with many disconnected components.
 There are approximately 800,000 lemmas in the dataset, although only approximat
ely 600,000 of these have any links.
 There are 12000 lemmas which are so old that there is no estimate of their
 age; these are assumed to be in usage from 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
liningnums{0}
\end_layout

\end_inset

AD, as shown in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:oed-words-born"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 The OED aims to be a comprehensive dictionary for all words in use since
 1150 and so we only consider years after that in our analysis.
 We chose a temporal resolution of 10 years to speed up the computation
 time as we noted that that was approximately the timescale at which large
 changes occurred in the language by referencing figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:oed-words-over-time"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
We perform non-negative tensor factorisation 
\begin_inset CommandInset citation
LatexCommand citep
key "Gauvin2014b"
literal "false"

\end_inset

 using the APG-TF algorithm 
\begin_inset CommandInset citation
LatexCommand citep
key "Xu2013b"
literal "false"

\end_inset

 on this network to extract 
\begin_inset Formula $R$
\end_inset

 temporal communities of words.
 We decompose the the network such that
\begin_inset Formula 
\[
a_{ijt}\approx\sum_{r=1}^{r=R}\kappa_{ir}\kappa_{jr}\tau_{tr}\text{,}
\]

\end_inset

where 
\begin_inset Formula $K=[\kappa_{ir}]$
\end_inset

 signifies strength of lemma membership to communities and 
\begin_inset Formula $T=[\tau_{tr}]$
\end_inset

 gives the temporal activity of the communities.
 Note that here, unlike chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Software"
plural "false"
caps "false"
noprefix "false"

\end_inset

, we only have two distinct factors rather than three as the network is
 undirected: 
\begin_inset Formula $\forall i,j,t:a_{ijt}=a_{jit}$
\end_inset

.
\end_layout

\begin_layout Subsection
Naming the communities
\begin_inset CommandInset label
LatexCommand label
name "subsec:Naming-the-communities"

\end_inset


\end_layout

\begin_layout Standard
We take a similar approach to that introduced in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Labelling-soft-communities"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 However, because lemmas with the strongest memberships are those with the
 most synonyms, they may not necessarily be very well known.
 To aid the ease of interpretation we therefore take the top 10 lemmas with
 the strongest memberships to each community and sort them in order of decreasin
g number of quotations listed in the 
\begin_inset CommandInset nomenclature
LatexCommand nomenclature
prefix "OED"
symbol "OED"
description "Oxford English Dictionary"
literal "false"

\end_inset

OED, which we use as a crude proxy for how common words are.
 We found that naming each community by the top three names in these lists
 was sufficient for us to distinguish between homonyms, i.e.
\begin_inset space ~
\end_inset

words with different meanings which are spelt the same, bypass uncommon
 words and get some idea what the community represented.
\end_layout

\begin_layout Section
Preliminary results
\begin_inset CommandInset label
LatexCommand label
name "sec:Preliminary-results"

\end_inset


\end_layout

\begin_layout Standard
We found that all choices for the numbers of communities between two and
 fifty produced very noisy activity plots.
 This was unexpected, but it could be because the network is split into
 so many disconnected components.
 We therefore did not perform the analysis from chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Locally-optimal-numbers"
plural "false"
caps "false"
noprefix "false"

\end_inset

 of the optimum number of communities as there was clearly something wrong
 with our approach.
 We discuss possibilities for other approaches in the next section, but
 we did not have time to perform any of them in time for inclusion in this
 work.
\end_layout

\begin_layout Standard
However, we noticed that with a moving arithmetic average corresponding
 to 15 time intervals, i.e.
\begin_inset space ~
\end_inset

150 years, some pattern emerged.
 We present this smoothed temporal activity for the decomposition with in
 figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:oed-decomp"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 There is some promise here: words in the same community appear to be related,
 with perhaps the exception of the 
\begin_inset Quotes xld
\end_inset

farther, absinth, flea-dock
\begin_inset Quotes xrd
\end_inset

 community.
 It is not clear whether the temporal activities have any bearing in reality;
 cross-referencing with Google Ngrams it is clear that 
\begin_inset Quotes xld
\end_inset

mickle
\begin_inset Quotes xrd
\end_inset

, for example, has been in declining usage, but 
\begin_inset Quotes xld
\end_inset

full
\begin_inset Quotes xrd
\end_inset

 and 
\begin_inset Quotes xld
\end_inset

more
\begin_inset Quotes xrd
\end_inset

 have been flat and it is not clear that words relating to 
\begin_inset Quotes xld
\end_inset

full
\begin_inset Quotes xrd
\end_inset

 and 
\begin_inset Quotes xld
\end_inset

more
\begin_inset Quotes xrd
\end_inset

 have dropped out of the language in the last hundred years or so—although
 it is quite possible that they have and that this author is just unaware
 of it.
 A more thorough study would be required.
\end_layout

\begin_layout Standard
Additionally, many of the communities are centred around flowers (
\begin_inset Quotes xld
\end_inset

argrimony, water lily, woodruff
\begin_inset Quotes xrd
\end_inset

) or insults (
\begin_inset Quotes xld
\end_inset

dully, sap-headed, thickheadedness
\begin_inset Quotes xrd
\end_inset

) which may point to a fatal flaw in our analysis: we heavily rely on the
 number of synonyms a word has—words are given community strengths based
 on whether they are synonymous with the same words and whether their links
 appear at the same time.
 This appears to be a poor approach as the number of synonyms appears to
 be skewed in favour of plants, for which there are many scientific and
 common names for similar plants which the OED classes as synonyms, and
 insults.
 Our desire to spot the rise of new technologies and cultural exchange has
 gone unmet.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ccs19singapore/code/oed_decomp_firstgo.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Relative activity of each found community in the English language over time
\end_layout

\end_inset


\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none
Relative activity of each community at each time interval.
 The communities are named as detailed in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Naming-the-communities"
plural "false"
caps "false"
noprefix "false"

\end_inset


\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:oed-decomp"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Further work
\begin_inset CommandInset label
LatexCommand label
name "sec:Further-work"

\end_inset


\end_layout

\begin_layout Standard
The OED has a hierarchical thesaurus where words are linked to categories,
 and categories are linked to parent categories in a hierarchical fashion;
 e.g.
\begin_inset space ~
\end_inset


\begin_inset Quotes xld
\end_inset

cat
\begin_inset Quotes xrd
\end_inset

 and 
\begin_inset Quotes xld
\end_inset

pussycat
\begin_inset Quotes xrd
\end_inset

 both belong to the 
\begin_inset Quotes xld
\end_inset

felis domesticus (cat)
\begin_inset Quotes xrd
\end_inset

 category which itself belongs to the 
\begin_inset Quotes xld
\end_inset

family Felidae (feline)
\begin_inset Quotes xrd
\end_inset

 category.
 We can present this as multi-layer temporal directed network with adjacency
 tensor
\begin_inset Formula 
\[
a_{i\leftarrow j}^{\alpha t}=\begin{cases}
1 & \text{if node }i\text{ is linked to node }j\text{ at time }t\text{ in layer }\alpha\\
0 & \text{otherwise,}
\end{cases}
\]

\end_inset

where all other symbols retain their previous meaning and 
\begin_inset ERT
status open

\begin_layout Plain Layout

$
\backslash
alpha 
\backslash
in 
\backslash
{
\backslash
text{synonymy},
\backslash
text{hierarchical}
\backslash
}$
\end_layout

\end_inset

 denotes specify the type of relationship between two nodes.
 The hierarchical layer could be organised such that
\begin_inset Formula 
\[
a_{i\leftarrow j}^{\text{hierarchical, }t}=\chi_{i}(t)\cdot\begin{cases}
1 & \text{if node }i\text{ is a child of node }j\text{ in the OED hierarchical thesaurus}\\
0 & \text{otherwise,}
\end{cases}
\]

\end_inset

where 
\begin_inset Formula 
\[
\chi_{i}(t)=\begin{cases}
1 & \text{if node }i\text{ is a lemma node or it has a path to one at time }t\\
0 & \text{otherwise,}
\end{cases}
\]

\end_inset

a recursive definition which can be resolved by iterating up from the lemmas.
 We hope that this representation may prove more fruitful than the previous
 one as it relies less on synonymy and puts more weight on hierarchy.
\end_layout

\begin_layout Standard
Another possibility is to attempt to vary link strengths based on the rarity
 of words at each time interval; links in the hierarchy layer could have
 weights according to the mean rarity of all of their children and links
 in the synonymy layer could have links with weights based on the minimum
 rarity of the two nodes to which they are attached.
 The OED has some static data on rarity but it is not currently in the dataset
 we possess.
 There are detailed data on the frequency of words written in printed books
 over time for approximately the last two hundred years available from Google
 Ngrams, but these do not account for homonymy.
 
\end_layout

\begin_layout Section
Conclusions
\begin_inset CommandInset label
LatexCommand label
name "sec:Conclusions-1"

\end_inset


\end_layout

\begin_layout Standard
This is a much more wide-ranging analysis than previous work, e.g.
\begin_inset space \space{}
\end_inset


\begin_inset CommandInset citation
LatexCommand citet
key "alexander2014spread"
literal "false"

\end_inset

 where only a single category of words was considered (equivalent to our
 work if we had plotted the in-degree of semantic classes over time).
 However, we have not been able to extract much meaning out of our analysis,
 despite some promising preliminary results.
 We have presented suggestions for how this research could be taken forwards.
\end_layout

\begin_layout Section*
\begin_inset Newpage newpage
\end_inset

Bibliography
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "monorefs"
options "authordate4"

\end_inset


\end_layout

\begin_layout Part
Conclusions
\end_layout

\begin_layout Chapter
Summary and discussion
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
epigraph{``Nothing except a battle lost can be half so melancholy as a battle
 won.
\begin_inset Quotes xrd
\end_inset

}{--- 
\backslash
textup{The Duke of Wellington,}
\backslash

\backslash
letter from the field of Waterloo}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
MyLettrine{T}{hroughout this}
\end_layout

\end_inset

 thesis, we have shown the value of characterisation at many different scales
 in networks.
 This final part will try to wrap up our work and suggest future avenues
 we or others might take.
 Here, we provide a recapitulation, highlighting some of our key findings,
 and explore areas where we think improvements could be made.
 This chapter is split into four sections: one for each of the three main
 parts of the thesis and a final section presenting our overall conclusions.
\end_layout

\begin_layout Section
Node characterisation
\end_layout

\begin_layout Standard
Our first area of contribution in the thesis was in node characterisation.
 Building on previous work that had shown that the use of ego-networks
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego-network
\end_layout

\end_inset

 to characterise nodes was a promising technique, we demonstrated the benefits
 of considering where the ego
\begin_inset Index idx
status open

\begin_layout Plain Layout
ego
\end_layout

\end_inset

 lay in these networks through the use of colour
\begin_inset Index idx
status open

\begin_layout Plain Layout
graph colour
\end_layout

\end_inset

.
 In an academic co-authorship network, we found that the proportion of coloured
 graphlets of a certain kind—a star centred on the ego with 3 links to alters—we
re the second strongest predictors of career length of all the topological
 measures we tried, and that there were no statistically significant links
 between monochromatic graphlets and career length.
\end_layout

\begin_layout Standard
There is an obvious criticism here in that we did not include other real-world
 properties of the authors which could have confounded our analysis.
 Unfortunately, the disambiguated network we use did not contain that informatio
n and the authors were irreversibly anonymous.
 It would be interesting to see if the result held if 
\begin_inset Quotes xld
\end_inset

total papers published
\begin_inset Quotes xrd
\end_inset

 was also included in our analysis; to our eyes the 4.3b graphlet, shown
 again in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:4.3bfinale"
plural "false"
caps "false"
noprefix "false"

\end_inset

, looks suspiciously like it could simply be a proxy for the number of papers
 published as it is the graphlet which requires the most papers published
 (if there was a link between any of the alters, that would reduce the number
 of papers published).
 The topology only explains a small fraction of research career length,
 so academic career length isn't solely determined by the topology of the
 co-authorship network, which is to be expected.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
centering
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% % % % 
\backslash
% includegraphics[width=0.1% 
\backslash
% textwidth]{{{figures/graphlet_menagerie/3.2b}}}% 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset Graphics
	filename ../complexcam/paper/figures/graphlet_menagerie/4.3b.pdf
	width 20text%

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
Graphlet 4.3b
\end_layout

\end_inset

Graphlet 4.3b from figure
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "fig:importantgraphlets"
plural "false"
caps "false"
noprefix "false"

\end_inset

, the only graphlet found to have a significant link with academic career
 length, after controlling for many standard node centralities
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% not any more.
 reword.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "fig:4.3bfinale"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Network characterisation
\end_layout

\begin_layout Standard
We have shown that so-called 
\begin_inset Quotes xld
\end_inset

unbiased
\begin_inset Quotes xrd
\end_inset

 descriptions of networks can be used as an input to matrix decompositions
 of networks to highlight similarities and differences to them.
 We provided some indications that these 
\begin_inset Quotes xld
\end_inset

unbiased
\begin_inset Quotes xrd
\end_inset

 inputs were about as good as 
\begin_inset Quotes xld
\end_inset

biased
\begin_inset Quotes xrd
\end_inset

 ones but did not test it very rigorously.
 We showed that our technique can successfully distinguish between configuration
 models and original networks.
 An example of using the technique to track network damage was given.
 The similarities and dissimilarities between some real-world networks and
 synthetic networks were shown, repeated here in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:Degree-correlation-matrix-1"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement document
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename figures/degree/GFPCA_degree.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
PCA applied to the degree mixing matrices of various networks
\begin_inset Argument 1
status open

\begin_layout Plain Layout
PCA applied to the degree mixing matrices of various networks
\end_layout

\end_inset

, as detailed in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Construction-of-data"
plural "false"
caps "false"
noprefix "false"

\end_inset

, projected onto their first two principal components.
 Each point corresponds to a single network
\begin_inset CommandInset label
LatexCommand label
name "fig:Degree-correlation-matrix-1"

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
We built on our work in the first part and again showed that coloured graphlets
 have a far greater richness to them than monochromatic graphlets.
 The figures relating to the academic co-authorship network
\begin_inset Index idx
status open

\begin_layout Plain Layout
real-world networks!co-authorship network
\end_layout

\end_inset

 are hard to interpret.
 We suspect this is because, as was found in the first part, the topology
 only accounts for a small amount of variation in career length and career
 length is hardly the only variable describing academic researchers.
 An interactive version of the plot, where one could hover over outliers
 and see who they were, could be interesting.
\end_layout

\begin_layout Section
Community characterisation
\end_layout

\begin_layout Standard
In the previous part, we explored the use of tensor decomposition to characteris
e communities in large, real-world temporal networks.
 We found that existing techniques for determining sensible numbers of communiti
es to decompose these networks into were not suited to the networks we considere
d and devised a novel one.
 This technique uses adjusted mutual information to determine which numbers
 of communities lead to the formation of similar communities on repeated
 runs, with the logic that if an optimising method with random elements
 finds similar communities each time, then these are plausible.
\end_layout

\begin_layout Standard
We applied this technique to several software dependency networks
\begin_inset Index idx
status open

\begin_layout Plain Layout
real-world networks!software dependency network
\end_layout

\end_inset

 and found we could see interesting real-world behaviour reflected in the
 activity of these communities.
 Particularly, we could see the deprecation of packages and their effect
 on the network and we could observe how some of the rules governing some
 software dependency ecosystems shaped the activity of their communities
 over time, with the rolling-release cycle of R especially well-reflected,
 shown again in figure 
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:cranact-1"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\begin_inset Graphics
	filename /home/olie/Dropbox/working_dir/ans-ml-cfp/paper/src/figures/results/CRAN-best-r-time.pdf
	display false
	height 20pheight%
	groupId scatterplots

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
\begin_inset Argument 1
status open

\begin_layout Plain Layout
CRAN community activity
\end_layout

\end_inset

CRAN community activity: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
bestrcaption
\end_layout

\end_inset


\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "fig:cranact-1"

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
We also demonstrated how this technique could be applied to semantic networks:
 we investigated how we would go about applying it to study the evolution
 of the English language.
 We hoped we would be able to see the impact of new technologies and societal
 change in the language but our results so far have been hard to interpret.
 We have provided suggestions for how we might proceed.
\end_layout

\begin_layout Section
Summary
\end_layout

\begin_layout Standard
We have made contributions at three scales of characterisation: node (local),
 community (mid-range) and whole network (global).
 Our exploration of graphlet profiles to characterise nodes through their
 ego-networks led us to consider how we could characterise whole networks
 through their graphlet profiles; our investigation of the use of matrix
 decomposition on graphlet profiles led neatly to the concept of tensor
 decompositions, albeit applied to networks in a totally different way.
\end_layout

\begin_layout Standard
We hope that our work encourages more people to use tensor decompositions
 to analyse large real-world networks; our technique should help others
 to have the confidence to use decompositions in the knowledge that there
 are metrics by which they can evaluate their choice of the number of communitie
s.
 Regarding our applications, we believe that there are many interesting
 problems that may have more light shed on them through applying network
 science techniques to software dependency networks.
 
\end_layout

\begin_layout Standard
We feel that we have only scratched the surface of what is possible.
 In our next and final chapter we shall enumerate some of these possibilities.
\end_layout

\begin_layout Chapter
Further work
\end_layout

\begin_layout Standard

\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
epigraph{``Adapt or perish, now as ever, is Nature's inexorable imperative."}{---
 
\backslash
textup{HG Wells,} The Mind at the End of its Tether}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
MyLettrine{T}{here is}
\end_layout

\end_inset

 room to expand on almost every part of this thesis.
 Each discovery has led us to more questions.
 In this final chapter, we explore some of these areas we think would be
 interesting for us and perhaps other researchers to explore.
\end_layout

\begin_layout Section
Network characterisation
\end_layout

\begin_layout Standard
There are three main directions where our network characterisation could
 be improved.
\end_layout

\begin_layout Subsection
Other uses of data matrices
\end_layout

\begin_layout Standard
As was mentioned in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Construction-of-data"
plural "false"
caps "false"
noprefix "false"

\end_inset

, many different machine learning techniques can be applied to data matrices.
 DeepGL 
\begin_inset CommandInset citation
LatexCommand citep
key "Rossi2017"
literal "false"

\end_inset

 is one example of work that has already attempted to do something similar
 by applying deep-learning techniques to networks.
 We are somewhat reluctant to go so far as our technique was able to distinguish
 well between many types of network without requiring labels.
 Indeed, we think the best way forward for our technique is to apply it
 to as many real-world networks as possible to see what sort of clusters
 emerge, if any.
\end_layout

\begin_layout Subsection
Construction of synthetic networks
\end_layout

\begin_layout Standard
Any point in the transformed co-ordinates, 
\begin_inset Formula $\hat{\boldsymbol{t}}$
\end_inset

, from PCA can be reprojected back into the original space with
\begin_inset Formula 
\[
\hat{\boldsymbol{x}}=\sum_{k}t_{k}\boldsymbol{w}_{k}\text{.}
\]

\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: this sounds wrong, but it's right: W^-1=W' as it's orthogonal (nb:
 only if we retain all components), which leads to that equation
\end_layout

\end_inset

If we apply this to the principal components of adjacency matrices or Laplacian
 matrices, we can create synthetic weighted & directed networks from any
 point in the reduced space.
 We can therefore create a network 
\begin_inset Quotes xld
\end_inset

midway
\begin_inset Quotes xrd
\end_inset

 between two other kinds of network.
 It would be interesting to see what those look like; in the facial recognition
 subfield of computer vision these reconstructed examples are called 
\begin_inset Quotes xld
\end_inset

eigenfaces
\begin_inset Quotes xrd
\end_inset

 
\begin_inset CommandInset citation
LatexCommand citep
key "turk1991eigenfaces"
literal "false"

\end_inset

.
 With absolutely no imagination required, we would dub them eigennetworks.
\end_layout

\begin_layout Subsection
Quantitative comparison of our work with related work
\end_layout

\begin_layout Standard
There have been several related attempts to characterise networks through
 dimension reductions 
\begin_inset CommandInset citation
LatexCommand citep
key "Costa2007,bounova_overview_2012,Ronqui2015,schieber2017quantification"
literal "false"

\end_inset

.
 There has not yet been an attempt to quantify the success or otherwise
 of these techniques or to compare them in a thorough manner.
 An experiment could be devised with a straightforward collection of synthetic
 and real networks from several categories.
 A clustering algorithm such as DBSCAN 
\begin_inset CommandInset citation
LatexCommand citep
key "ester1996density"
literal "false"

\end_inset

 would then be run in the reduced spaces and the adjusted mutual information
 (AMI) between the clustering and the ground-truth labels could be determined.
 The dimension reduction with the highest AMI could be considered the best.
\end_layout

\begin_layout Section
Graphlets
\end_layout

\begin_layout Standard
There are two main areas where we feel our work on graphlets could be extended.
\end_layout

\begin_layout Subsection
Graphlet profile normalisation performance
\end_layout

\begin_layout Standard
Graphlets may appear in networks for uninteresting reasons: for example,
 in a denser graph, denser graphlets are likely to appear.
 We need some kind of benchmark by which we can tell how unusual a graphlet
 is.
 Most of the current approaches in the literature use the so-called z-score
 
\begin_inset CommandInset citation
LatexCommand citep
key "milo_network_2002"
literal "false"

\end_inset

: the number of standard deviations away from the mean the number of appearances
 of a graphlet are compared to an ensemble of 
\begin_inset Quotes xld
\end_inset

similar
\begin_inset Quotes xrd
\end_inset

 random networks.
 Kavosh and acc-Motif 
\begin_inset CommandInset citation
LatexCommand citet
key "kashani_kavosh:_2009,meira_acc-motif:_2014"
literal "false"

\end_inset

 use an ensemble of configuration models, as described in section 
\begin_inset CommandInset ref
LatexCommand ref
reference "subsec:Configuration-model"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 These routines are predominantly designed for use on a few networks; the
 authors are not particularly concerned about the performance of calculating
 graphlet profiles for many networks.
 They recommend generating a thousand similar graphs per graph.
\end_layout

\begin_layout Standard
That is too slow for many purposes where one wishes to compare many hundreds
 or thousands of real networks.
 We propose a faster, approximate technique: instead generate ten random
 graphs for each decile of density and interpolate the number of graphlets
 between them and then calculate the z-scores for each graphlet in a real
 network,
\begin_inset Formula 
\[
z_{i}=\frac{\mu_{i}(\rho)-x_{i}}{\sigma_{i}(\rho)},
\]

\end_inset

 where 
\begin_inset Formula 
\begin{align}
\mu_{i}(\rho) & =w_{-}(\rho)\overline{x_{i\rho_{-}}}-w_{+}(\rho)\overline{x_{i\rho_{+}}}\text{,}\nonumber \\
\sigma_{i}(\rho) & =w_{-}(\rho)\sigma(x_{i\rho_{-}})-w_{+}(\rho)\sigma(x_{i\rho_{+}})\text{,}\nonumber \\
w_{+}(\rho) & =\frac{\rho-\rho_{+}}{\rho_{+}-\rho_{-}}\text{,}\nonumber \\
w_{-}(\rho) & =\frac{\rho-\rho_{-}}{\rho_{+}-\rho_{-}}\text{,}\nonumber \\
x_{i} & =\frac{n_{i}+1}{(\Sigma_{j}n_{j})+\text{max}(j)}\text{,}\label{eq:ruleofsuccession}
\end{align}

\end_inset


\end_layout

\begin_layout Standard
where 
\begin_inset Formula $z_{i}$
\end_inset

 is the z-score of the 
\begin_inset Formula $i^{\text{th}}$
\end_inset

 graphlet which occurs 
\begin_inset Formula $n_{i}$
\end_inset

 times in a network of density 
\begin_inset Formula $\rho$
\end_inset

, 
\begin_inset Formula $\rho_{+}$
\end_inset

 and 
\begin_inset Formula $\rho_{-}$
\end_inset

 are the densities of the closest ensembles of random graphs, 
\begin_inset Formula $\overline{x}$
\end_inset

 is the arithmetic mean of the set 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $\sigma(x)$
\end_inset

 is its standard deviation and 
\begin_inset Formula $x_{i\rho_{+}}$
\end_inset

 and 
\begin_inset Formula $x_{i\rho_{-}}$
\end_inset

 are sets corresponding to 
\begin_inset Formula $x_{i}$
\end_inset

 in each of the ten random graphs in the ensembles for the closest densities
 matching the graph of interest.
 A small but important detail that has not been covered by the existing
 literature is how to calculate the z-score of a graphlet that has never
 appeared in the ensemble.
 To do this, we suggest using the rule of succession 
\begin_inset CommandInset citation
LatexCommand citep
key "zabell1989rule"
literal "false"

\end_inset

 which leads to equation 
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:ruleofsuccession"
plural "false"
caps "false"
noprefix "false"

\end_inset

, assuming 
\shape italic
a priori
\shape default
 that each graphlet is equally likely to appear and updating that assumption
 in the face of additional evidence.
\end_layout

\begin_layout Standard
Any research investigating this novel graphlet normalisation procedure should
 carefully consider the implications of controlling only for density where
 existing methods control for degree sequences.
 It should be noted that in the limit of a large number of networks, this
 procedure is 
\begin_inset Formula $N$
\end_inset

 times faster than the existing procedure, where 
\begin_inset Formula $N$
\end_inset

 is the number of configuration models generated per graph, commonly a thousand.
\end_layout

\begin_layout Subsection
Academic career length from collaboration network improvements
\end_layout

\begin_layout Standard
As we discussed in chapter 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Career-length-in"
plural "false"
caps "false"
noprefix "false"

\end_inset

, the graphlet most related to career length was the star graphlet, 4.3b,
 with no links between the co-authors of the author of interest.
 This graphlet is the one which requires the highest number of papers to
 be authored.
 A clear improvement, then, would be to include the number of papers written
 alongside other characteristics such as citation count to see whether the
 graphlet is still related to career length, taking those other factors
 into account.
 However, this could all be of limited interest, as the variance in that
 graphlet can only explain approximately a year of difference in career
 length.
\end_layout

\begin_layout Standard
Additionally, the network lends itself quite naturally for analysis through
 the lens of tensor decomposition.
 An additional layer representing papers, with links between the author
 and paper layer denoting authorship and links between papers denoting citation
 could also prove interesting to analyse; one could potentially spot fields
 forming and fading away.
\end_layout

\begin_layout Section
Community characterisation
\end_layout

\begin_layout Standard
Our analysis of software ecosystems could be deepened by adding a layer
 relating authors to the packages to which they have contributed.
 Most of these packages are developed in the open, so these contributions
 could be weighted by the number of patches applied.
 It would be particularly interesting to see what happens to authors when
 they cease contributing to one package.
 Authors often contribute to more than one language ecosystem, so another
 enhancement could be to consider all software ecosystems together to try
 to analyse how authors move between them.
 The patterns we observed, particularly the rolling-release cycle of R,
 could lend themselves to simple theoretical models of ecosystems.
 Such models could be of great interest to those creating software ecosystems.
\end_layout

\begin_layout Standard
Similarly, our linguistic network could be expanded to consider multiple
 languages, with a layer for each language, where links between layers denote
 synonymy.
 One could then potentially observe the cross-pollination between languages
 which would be a fascinating endeavour.
\end_layout

\begin_layout Standard
In general, we feel that there is a large room for tensor decompositions
 to be applied to more data.
 For example, 
\begin_inset CommandInset citation
LatexCommand citet
key "reale_khelfaoui_montiglio_gingras_2019"
literal "false"

\end_inset

 provide a fascinating exploration of the evolution of the study of ecological
 networks, but they study the network as a series of piecewise static networks.
 We believe that tensor decompositions would have greater insight in cases
 such as these.
\end_layout

\begin_layout Standard
A further symptom of the immaturity of tensor decompositions
\begin_inset Index idx
status open

\begin_layout Plain Layout
tensor decomposition
\end_layout

\end_inset

 in network science is the lack of polished software.
 Multilayer
\begin_inset Index idx
status open

\begin_layout Plain Layout
multilayer network
\end_layout

\end_inset

 and temporal networks
\begin_inset Index idx
status open

\begin_layout Plain Layout
temporal networks
\end_layout

\end_inset

 are already somewhat neglected and tensor decompositions are no exception.
 Reducing the barrier to entry so that non-specialists could apply these
 dynamic community detection algorithms to their own networks could yield
 dividends in terms of the amount of output produced.
\end_layout

\begin_layout Standard
Lastly, it would be beneficial if there was a theoretical basis as to the
 soundness of using adjusted mutual information to choose the number of
 components into which a tensor should be decomposed.
 Our current arguments have only really suggested that it is plausible,
 which is supported by our results.
 Further work could also consider why CORCONDIA 
\begin_inset CommandInset citation
LatexCommand citep
key "Bro2003b"
literal "false"

\end_inset

 was not useful for picking the number of components in our applications,
 which appears to be at odds with other work 
\begin_inset CommandInset citation
LatexCommand citep
key "Sapienza2018d"
literal "false"

\end_inset

.
\end_layout

\begin_layout Section
Characterisation of complex networks
\end_layout

\begin_layout Standard
There is a wonderful survey of the characterisation of complex networks
 
\begin_inset CommandInset citation
LatexCommand citep
key "Costa2007"
literal "false"

\end_inset

.
 At the time of writing, it is 12 years old and so misses out almost all
 of the works cited in this thesis.
 It is particularly silent on the application of machine learning techniques
 to networks, many of which had not been popularised at the time of its
 publication, although there are more recent reviews specifically on characteris
ing nodes for machine learning 
\begin_inset CommandInset citation
LatexCommand citep
key "hamilton2017representation,goyal2018graph"
literal "false"

\end_inset

 they do not consider network or community characterisation.
 The work of 
\begin_inset CommandInset citation
LatexCommand citeauthor
key "Costa2007"
literal "false"

\end_inset

 is well due an update.
\end_layout

\begin_layout Section
Interplay between topology and node dynamics
\end_layout

\begin_layout Standard
As we have intimated in previous chapters, we believe that complex networks
 are interesting predominately because they reflect real-world networks.
 In these real-world networks, we rarely if ever care directly about the
 network topology: it is the topology's impact on the observable properties
 of the system, such as the amount of time it takes to travel from one place
 to another, or the diversity of an ecosystem, that we want to optimise.
 It is therefore crucial to link the topology to these observable properties,
 which is what we have done in chapters 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Career-length-in"
plural "false"
caps "false"
noprefix "false"

\end_inset

, 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Software"
plural "false"
caps "false"
noprefix "false"

\end_inset

 and 
\begin_inset CommandInset ref
LatexCommand ref
reference "chap:Linguistic-evolution"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 What we have not done is linked the topology to models of the dynamics
 that generate these observable properties.
 This is a very active field—see 
\begin_inset CommandInset citation
LatexCommand citet
key "Menck2013a"
literal "false"

\end_inset

 and 
\begin_inset CommandInset citation
LatexCommand citet
key "Gao2016"
literal "false"

\end_inset

 for two examples—but little work has been done in terms of characterising
 networks by considering what sort of dynamics they support or encourage
\begin_inset Note Note
status open

\begin_layout Plain Layout
Viva: this is a pretty bold claim.
\end_layout

\end_inset

.
 We would like to see how well methods of characterising networks correlate
 with dynamical properties such as the resilience of the system.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{flushright}$
\backslash
blacksquare$
\backslash
end{flushright}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Newpage newpage
\end_inset


\end_layout

\begin_layout Section*
Bibliography
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
btprint "btPrintCited"
bibfiles "monorefs"
options "authordate4"

\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset index_print
LatexCommand printindex
type "idx"
name "Index"

\end_inset


\end_layout

\end_body
\end_document