This R package (which has been developed on top of the fantastic work that the Statnet team has done) implements estimation and simulation methods for Exponential Random Graph Models of small networks, in particular, up to 5 vertices for directed graphs and 7 for undirected networks. In the case of small networks, the calculation of the likelihood of ERGMs becomes computationally feasible, which allows us to avoid approximations and do exact calculations, ultimately obtaining MLEs directly.
Checkout the examples section, and specially the Using interaction effects example.
This material is based upon work support by, or in part by, the U.S. Army Research Laboratory and the U.S. Army Research Office under grant number W911NF-15-1-0577
Computation for the work described in this paper was supported by the University of Southern California’s Center for High-Performance Computing (hpcc.usc.edu).
When using ergmito, always cite this paper
Vega Yon G, Slaughter A, de la Haye K (2020). “Exponential Random Graph models for Little Networks.” Social Networks. doi:10.1016/j.socnet.2020.07.005 https://doi.org/10.1016/j.socnet.2020.07.005, in press.
Use this if you are using the GitHub version of ergmito:
Vega Yon G (????). ergmito: Exponential Random Graph Models for Small Networks. R package version 0.3-1 (wip), https://github.com/muriteams/ergmito.
Use this if you are using the CRAN version of ergmito:
Vega Yon G (????). ergmito: Exponential Random Graph Models for Small Networks. R package version 0.3-1, https://cran.r-project.org/package=ergmito.
To see these entries in BibTeX format, use ‘print(, bibtex=TRUE)’, ‘toBibtex(.)’, or set ‘options(citation.bibtex.max=999)’.
The development version from GitHub with:
# install.packages("devtools")
devtools::install_github("muriteams/ergmito")This requires compilation. Windows users can download the latest
compiled version from appveyor
here.
The file to download is the one named ergmito_[version number].zip.
Once downloaded, you can install typing the following:
install.packages("[path to the zipfile]/ergmito_[version number].zip", repos = FALSE)In the case of Mac users, and in particular, those with the Mojave version, they may need to install the following https://github.com/fxcoudert/gfortran-for-macOS/releases
In the following example, we simulate a small network with four vertices
and estimate the model parameters using ergm and ergmito. We start
by generating the graph
# Generating a small graph
library(ergmito)
library(ergm)
library(sna)
set.seed(12123)
n <- 4
net <- rbernoulli(n, p = .3)
gplot(net)
To estimate the model
model <- net ~ edges + istar(2)
# ERGMito (estimation via MLE)
ans_ergmito <- ergmito(model)# ERGM (estimation via MC-MLE)
ans_ergm <- ergm(
model, control = control.ergm(
main.method = "MCMLE",
seed = 444
)
)
# The ergmito should have a larger value
ergm.exact(ans_ergmito$coef, model) > ergm.exact(ans_ergm$coef, model)
#> [,1]
#> [1,] TRUE
summary(ans_ergmito)
#>
#> ERGMito estimates (MLE)
#> This model includes 1 networks.
#>
#> formula:
#> net ~ edges + istar(2)
#>
#> Estimate Std. Error z value Pr(>|z|)
#> edges -1.37749 1.00260 -1.3739 0.1695
#> istar2 0.50653 1.32701 0.3817 0.7027
#> AIC: 17.36312 BIC: 18.33294 (Smaller is better.)
summary(ans_ergm)
#> Call:
#> ergm(formula = model, control = control.ergm(main.method = "MCMLE",
#> seed = 444))
#>
#> Monte Carlo Maximum Likelihood Results:
#>
#> Estimate Std. Error MCMC % z value Pr(>|z|)
#> edges -1.3984 1.0326 0 -1.354 0.176
#> istar2 0.5927 1.4024 0 0.423 0.673
#>
#> Null Deviance: 16.64 on 12 degrees of freedom
#> Residual Deviance: 13.37 on 10 degrees of freedom
#>
#> AIC: 17.37 BIC: 18.34 (Smaller is better. MC Std. Err. = 0.01315)The following example shows the estimation of a dataset included in the
package, fivenets. This set of five networks was generated using the
new_rergmito function, which creates a function to draw random ERGMs
with a fixed set of parameters, in this case, edges = -2.0 and
nodematch("female") = 2.0
data(fivenets)
model1 <- ergmito(fivenets ~ edges + nodematch("female"))
summary(model1) # This data has know parameters equal to -2.0 and 2.0
#>
#> ERGMito estimates (MLE)
#> This model includes 5 networks.
#>
#> formula:
#> fivenets ~ edges + nodematch("female")
#>
#> Estimate Std. Error z value Pr(>|z|)
#> edges -1.70475 0.54356 -3.1363 0.001711 **
#> nodematch.female 1.58697 0.64305 2.4679 0.013592 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> AIC: 73.34109 BIC: 77.52978 (Smaller is better.)We can also compute GOF
fivenets_gof <- gof_ergmito(model1)
fivenets_gof
#>
#> Goodness-of-fit for edges
#>
#> obs min mean max lower upper lower prob. upper prob.
#> net 1 2 0 3.746607 12 0 6 0.008080856 0.9635261
#> net 2 7 0 3.746607 12 0 6 0.008080856 0.9635261
#> net 3 4 0 3.113122 12 0 6 0.020642814 0.9881036
#> net 4 5 0 5.647059 12 2 8 0.030934588 0.9514908
#> net 5 2 0 3.746607 12 0 6 0.008080856 0.9635261
#>
#>
#> Goodness-of-fit for nodematch.female
#>
#> obs min mean max lower upper lower prob. upper prob.
#> net 1 2 0 2.823530 6 0 5 0.02201716 0.9891396
#> net 2 5 0 2.823530 6 0 5 0.02201716 0.9891396
#> net 3 3 0 1.882353 4 0 3 0.07855507 0.9509584
#> net 4 5 0 5.647059 12 2 8 0.03093459 0.9514908
#> net 5 1 0 2.823530 6 0 5 0.02201716 0.9891396
#>
#> Note: Exact confidence intervals where used. This implies that the requestes CI may differ from the one used (see ?gof_ergmito).
plot(fivenets_gof)
The pooled model can be compared to a block-diagonal ERGM. The package
includes three functions to help with this task: blockdiagonalize,
splitnetwork, and ergm_blockdiag.
data("fivenets")
# Stacking matrices together
fivenets_blockdiag <- blockdiagonalize(fivenets, "block_id")
fivenets_blockdiag # It creates the 'block_id' variable
#> Network attributes:
#> vertices = 20
#> directed = TRUE
#> hyper = FALSE
#> loops = FALSE
#> multiple = FALSE
#> bipartite = FALSE
#> total edges= 20
#> missing edges= 0
#> non-missing edges= 20
#>
#> Vertex attribute names:
#> block_id female name_original vertex.names
#>
#> No edge attributes# Fitting the model with ERGM
ans0 <- ergm(
fivenets_blockdiag ~ edges + nodematch("female"),
constraints = ~ blockdiag("block_id")
)ans1 <- ergm_blockdiag(fivenets ~ edges + nodematch("female"))# Now with ergmito
ans2 <- ergmito(fivenets ~ edges + nodematch("female"))
# All three are equivalent
cbind(
ergm = coef(ans0),
ergm_blockdiag = coef(ans1),
ergmito = coef(ans2)
)
#> ergm ergm_blockdiag ergmito
#> edges -1.704748 -1.704748 -1.704748
#> nodematch.female 1.586965 1.586965 1.586965The benefit of ergmito:
t_ergm <- system.time(ergm(
fivenets_blockdiag ~ edges + nodematch("female") + ttriad,
constraints = ~ blockdiag("block_id")
))
t_ergmito <- system.time(
ergmito(fivenets ~ edges + nodematch("female") + ttriad)
)# Relative elapsed time
(t_ergm/t_ergmito)[3]
#> elapsed
#> 110.6667Suppose that we have a large sample of small networks (ego from Facebook, Twitter, etc.), 20,000 which account for 80,000 vertices:
set.seed(123)
bignet <- rbernoulli(sample(3:5, 20000, replace = TRUE))
# Number of vertices
sum(nvertex(bignet))
#> [1] 80089We can fit this model in a memory-efficient way.
system.time(ans0 <- ergmito(bignet ~ edges + mutual))
#> user system elapsed
#> 9.314 0.084 9.400
summary(ans0)
#>
#> ERGMito estimates (MLE)
#> This model includes 20000 networks.
#>
#> formula:
#> bignet ~ edges + mutual
#>
#> Estimate Std. Error z value Pr(>|z|)
#> edges -0.0020193 0.0068675 -0.2940 0.7687
#> mutual -0.0018160 0.0112247 -0.1618 0.8715
#> AIC: 352097.2 BIC: 352118.1 (Smaller is better.)One advantage of using exact statistics is the fact that we have
significantly more flexibility when it comes to specifying sufficient
statistics. Just like one would do when working with Generalized Linear
Models in R (the glm function), users can alter the specified formula
by adding arbitrary offsets (using the offset function) or creating new
terms by using the “I” function. In this brief example, where we
estimate a model that includes networks of size four and five, we will
add an interaction effect between the edge-count statistic and the
indicator function that equals one if the network is of size 5. This
way, while poling the data, we can still obtain different edge-count
estimates depending on the number of vertices in the graph.
# Simulating networks of different sizes
set.seed(12344)
nets <- rbernoulli(c(rep(4, 10), rep(5, 10)), c(rep(.2, 10), rep(.1, 10)))Fitting an ergmito under the Bernoulli model
ans0 <- ergmito(nets ~ edges)
summary(ans0)
#>
#> ERGMito estimates (MLE)
#> This model includes 20 networks.
#>
#> formula:
#> nets ~ edges
#>
#> Estimate Std. Error z value Pr(>|z|)
#> edges -1.68640 0.15396 -10.954 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> AIC: 279.3753 BIC: 283.1436 (Smaller is better.)Fitting the model with a reference term for networks of size five. Notice that the variable -n- and other graph attributes can be used with -model_update-.
ans1 <- ergmito(nets ~ edges, model_update = ~ I(edges * (n == 5)))
summary(ans1)
#>
#> ERGMito estimates (MLE)
#> This model includes 20 networks.
#>
#> formula:
#> nets ~ edges + I(edges * (n == 5))
#>
#> Estimate Std. Error z value Pr(>|z|)
#> edges -1.18958 0.21583 -5.5116 3.556e-08 ***
#> I(edges * (n == 5)) -0.90116 0.31250 -2.8837 0.00393 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> AIC: 272.9916 BIC: 280.5282 (Smaller is better.)The resulting parameter for the edge count is smaller for networks of size five
plogis(coef(ans1)[1])
#> edges
#> 0.2333333
plogis(sum(coef(ans1)))
#> [1] 0.11We can see that the difference in edge count matters.
library(lmtest)
lrtest(ans0, ans1)
#> Likelihood ratio test
#>
#> Model 1: nets ~ edges
#> Model 2: nets ~ edges + I(edges * (n == 5))
#> #Df LogLik Df Chisq Pr(>Chisq)
#> 1 1 -138.69
#> 2 2 -134.50 1 8.3837 0.003786 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The ‘ergmito’ project is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.