SpectralGraph trait #38
Description
current
the SpectralGraph trait is absent from geonum
next
spectral graphs traditionally rely on eigendecomposition of graph matrices (adjacency, Laplacian) with O(n³) complexity
this is impractical for large-scale graphs
with geonum's angle-based representation, we can implement the same theoretical power with O(1) complexity per operation, enabling analysis of massive graphs
add a SpectralGraph trait to src/traits/spectral_graph.rs:
#[cfg(feature = "spectral_graph")]
pub trait SpectralGraph: Sized {
/// convert graph representation to multivector form
fn to_multivector(&self) -> Multivector;
/// compute graph spectrum directly through angle stability
fn spectrum(&self) -> Multivector;
/// perform spectral clustering via angle proximity
fn spectral_cluster(&self, k: usize) -> Vec<Vec<usize>>;
/// compute graph partitioning through angle thresholding
fn spectral_partition(&self) -> (Vec<usize>, Vec<usize>);
/// compute algebraic connectivity (Fiedler value) via angle gap
fn algebraic_connectivity(&self) -> f64;
/// compute stationary distribution of random walks via angle stability
fn stationary_distribution(&self) -> Multivector;
/// compute eigenvalue gaps for community detection
fn eigenvalue_gaps(&self) -> Vec<f64>;
/// compute diffusion distance between vertices
fn diffusion_distance(&self, i: usize, j: usize, t: f64) -> f64;
/// perform spectral embedding into k dimensions
fn spectral_embedding(&self, k: usize) -> Multivector;
/// compute graph conductance via angle flow
fn conductance(&self, partition: &[usize]) -> f64;
/// perform dimensionality reduction through spectral projection
fn spectral_dimension_reduction(&self, dim: usize) -> Multivector;
/// analyze community structure through angle patterns
fn community_structure(&self) -> Vec<Vec<usize>>;
/// compute vertex centrality through angle influence
fn spectral_centrality(&self) -> Vec<f64>;
/// perform Cheeger cut via angle separation
fn cheeger_cut(&self) -> (Vec<usize>, Vec<usize>);
/// perform heat kernel analysis
fn heat_kernel(&self, t: f64) -> Multivector;
/// returns the fiedler vector l2 eigenvector of the laplacian
fn fiedler_vector(&self) -> Multivector
/// returns the maximum spatial frequency in the graph’s structure
/// computed without matrices using geometric derivative estimates
/// consistent with ∇²ψ = λψ form where λ is the spectral radius
fn spectral_radius(&self) -> f64
/// performs the graph fourier transform of a signal over nodes
fn graph_fourier_transform(&self, signal: Multivector) -> Multivector
/// estimates the eigenvalue density using gaussian kernel smoothing
fn spectral_density(&self) -> Multivector
/// returns lower and upper bounds from cheeger inequality
fn cheeger_bound(&self) -> (f64, f64)
/// returns the directed laplacian for non-symmetric graphs
/// preserves directional flow using asymmetric edge angles
fn directed_laplacian(&self) -> Multivector
/// applies a spectral filter given a weight vector over frequencies
fn spectral_filter(&self, weights: &[f64]) -> Multivector
/// returns a low rank homological signature using laplacian rank collapse
/// captures persistent cycles as minimal energy modes in the graph geometry
fn homological_signature(&self) -> Multivector
}Impl for Multivector
example for Multivector:
#[cfg(feature = "spectral_graph")]
impl SpectralGraph for Multivector {
fn spectrum(&self) -> Multivector {
// extract graph structure (vertices and edges)
let vertices = self.vertices();
let edges = self.edges();
// use geometric product to identify angle-stable components
// these are equivalent to eigenvectors/eigenvalues without matrix operations
let eigencomponents = vertices.wedge(&edges);
// extract eigenvalues through length scaling factors
let eigenvalues = eigencomponents.0.iter().map(|g| {
Geonum {
length: g.length, // eigenvalue magnitude
angle: g.angle, // eigenvalue phase
blade: 0, // scalar (grade 0) for eigenvalue
}
}).collect();
Multivector(eigenvalues)
}
fn algebraic_connectivity(&self) -> f64 {
// compute algebraic connectivity (Fiedler value) via direct angle gap
// traditional design: second smallest eigenvalue of Laplacian matrix
// in geonum, find the smallest non-zero angle gap
// between components of the graph spectrum
let spectrum = self.spectrum();
// sort spectrum by angle
let mut angles: Vec<f64> = spectrum.0.iter()
.map(|g| g.angle)
.collect();
angles.sort_by(|a, b| a.partial_cmp(b).unwrap_or(std::cmp::Ordering::Equal));
// find smallest non-zero gap (equivalent to Fiedler value)
let mut min_gap = std::f64::MAX;
for i in 1..angles.len() {
let gap = (angles[i] - angles[i-1]).abs();
if gap > 1e-10 && gap < min_gap {
min_gap = gap;
}
}
min_gap
}
fn spectral_cluster(&self, k: usize) -> Vec<Vec<usize>> {
// perform spectral clustering through angle proximity
// instead of computing eigenvectors and then k-means clustering
// 1. compute spectrum via angle stability
let spectrum = self.spectrum();
let vertices = self.vertices();
// 2. extract top-k eigencomponents by sorting
let mut eig_components: Vec<&Geonum> = spectrum.0.iter().collect();
eig_components.sort_by(|a, b| b.length.partial_cmp(&a.length).unwrap_or(std::cmp::Ordering::Equal));
let top_k = eig_components.iter().take(k).cloned().cloned().collect::<Vec<Geonum>>();
// 3. group vertices by angle proximity to eigenmodes
// This is equivalent to clustering in eigenspace
let mut clusters = vec![Vec::new(); k];
// for each vertex, find closest eigencomponent by angle
for (i, vertex) in vertices.0.iter().enumerate() {
let mut closest_idx = 0;
let mut min_angle_dist = std::f64::MAX;
for (j, eigen) in top_k.iter().enumerate() {
let angle_dist = vertex.angle_distance(eigen);
if angle_dist < min_angle_dist {
min_angle_dist = angle_dist;
closest_idx = j;
}
}
clusters[closest_idx].push(i);
}
clusters
}
fn spectral_partition(&self) -> (Vec<usize>, Vec<usize>) {
// use Fiedler vector (second eigenvector) for partitioning
// in geonum, use second angle component
// get spectrum ordered by stability (eigenvalue magnitude)
let spectrum = self.spectrum();
let mut components: Vec<Geonum> = spectrum.0.clone();
components.sort_by(|a, b| a.length.partial_cmp(&b.length).unwrap_or(std::cmp::Ordering::Equal));
// get second component (Fiedler vector equivalent)
let fiedler = if components.len() >= 2 {
components[1].clone()
} else {
// default value if not enough components
Geonum {
length: 1.0,
angle: 0.0,
blade: 0,
}
};
// partition vertices based on angle alignment with Fiedler component
let vertices = self.vertices();
let mut partition_a = Vec::new();
let mut partition_b = Vec::new();
for (i, vertex) in vertices.0.iter().enumerate() {
// compute angle alignment with Fiedler component
let alignment = vertex.dot(&fiedler);
if alignment >= 0.0 {
partition_a.push(i);
} else {
partition_b.push(i);
}
}
(partition_a, partition_b)
}
fn stationary_distribution(&self) -> Multivector {
// compute stationary distribution of random walks via angle stability
// traditional: leading eigenvector of transition matrix
// extract graph structure and adjacency information
let adjacency = self.adjacency();
// find the most stable angle component under adjacency transformation
// this is equivalent to the leading eigenvector
// apply adjacency operation until convergence
let mut state = Multivector(vec![
Geonum {
length: 1.0,
angle: 0.0,
blade: 1,
}
]);
// iterate a fixed number of times (in practice would use convergence check)
for _ in 0..10 {
// Transform state by adjacency (equivalent to matrix multiplication)
state = state.mul(&adjacency);
// Normalize state
let norm = state.0.iter().map(|g| g.length.powi(2)).sum::<f64>().sqrt();
state = Multivector(state.0.iter().map(|g| {
Geonum {
length: g.length / norm,
angle: g.angle,
blade: g.blade,
}
}).collect());
}
state
}
// added implementations for other methods...
}add doc.rs comments and unit test coverage
feature flag
add to Cargo.toml:
[features]
spectral_graph = []module integration
add spectral_graph module to src/traits/mod.rs:
#[cfg(feature = "spectral_graph")]
pub mod spectral_graph;
#[cfg(feature = "spectral_graph")]
pub use spectral_graph::SpectralGraph;re-export from lib.rs:
#[cfg(feature = "spectral_graph")]
pub use traits::SpectralGraph;test implementation
add tests/spectral_graph_test.rs with the following test:
#[cfg(feature = "spectral_graph")]
mod tests {
use geonum::*;
use std::f64::consts::PI;
// include O(1) performance assertions in tests:
// // spectral embedding & structure
// it_embeds_graph_in_low_dimensions();
// it_discovers_latent_structure_with_low_rank_spectrum();
// it_recovers_shape_signatures_from_cycle_basis();
// // spectral operators & transforms
// it_performs_graph_fourier_transform_using_geometric_angles();
// it_computes_diffusion_distances_between_vertices();
// it_models_diffusion_processes_with_angle_weighted_steps();
// it_estimates_spectral_radius_without_matrix_factorization();
// it_derives_spectral_density_from_low_rank_modes();
// it_filters_graph_signal_in_frequency_domain();
// // topology & homology
// it_extracts_homological_cycles_from_laplacian_rank_collapse();
// it_identifies_homology_through_spectral_degeneracy();
// it_tests_duality_between_energy_and_topology();
// // flow & directionality
// it_constructs_directed_laplacian_preserving_flow_directions();
// it_detects_flow_asymmetry_through_directed_blades();
// // optimization & inference
// it_solves_minimum_cut_problems();
// it_bounds_cheeger_constant_using_geometric_edges();
// it_identifies_bottlenecks_using_fiedler_geometry();
// // signal processing
// it_improves_graph_signals_with_directional_spectral_filter();
// it_establishes_graph_similarity_via_fourier_signatures();
// // centrality, community, conductance
// it_measures_graph_conductance();
// it_identifies_communities_in_social_network();
// it_computes_centrality_measures();
// // misc geometry
// it_encodes_graph_topology_with_bivector_rotation();
// it_visualizes_high_frequency_modes_as_topological_noise();
// it_analyzes_path_structures_through_angles();
#[test]
fn its_a_graph_spectrum() {
// create a simple graph with 4 vertices and 5 edges
// 0 -- 1 -- 2
// | |
// +----3----+
// represent vertices as geometric numbers
let vertices = Multivector(vec![
Geonum {
length: 1.0,
angle: 0.0,
blade: 1,
},
Geonum {
length: 1.0,
angle: PI/2.0,
blade: 1,
},
Geonum {
length: 1.0,
angle: PI,
blade: 1,
},
Geonum {
length: 1.0,
angle: 3.0*PI/2.0,
blade: 1,
},
]);
// represent edges as geometric numbers
let edges = Multivector(vec![
// edge 0-1
Geonum {
length: 1.0,
angle: PI/4.0, // halfway between vertices 0 and 1
blade: 2, // bivector (grade 2) representing edge between vertices
},
// edge 1-2
Geonum {
length: 1.0,
angle: 3.0*PI/4.0,
blade: 2,
},
// edge 2-3
Geonum {
length: 1.0,
angle: 5.0*PI/4.0,
blade: 2,
},
// edge 3-0
Geonum {
length: 1.0,
angle: 7.0*PI/4.0,
blade: 2,
},
// edge 0-2 (diagonal)
Geonum {
length: 1.0,
angle: PI/2.0,
blade: 2,
},
]);
// create graph as combined multivector
let mut graph_components = Vec::new();
graph_components.extend(vertices.0.clone());
graph_components.extend(edges.0.clone());
let graph = Multivector(graph_components);
// compute spectrum
let spectrum = graph.spectrum();
// basic test - verify we get non-empty spectrum
assert!(!spectrum.0.is_empty());
// perform spectral partition
let (part_a, part_b) = graph.spectral_partition();
// the graph should be partitioned into two approximately equal groups
assert!(!part_a.is_empty());
assert!(!part_b.is_empty());
// find algebraic connectivity
let conn = graph.algebraic_connectivity();
// test non-zero for a connected graph
assert!(conn > 0.0);
// stationary distribution has entries for all vertices
let stationary = graph.stationary_distribution();
assert!(stationary.0.len() > 0);
// spectral clustering - test with k=2
let clusters = graph.spectral_cluster(2);
assert_eq!(clusters.len(), 2);
// all vertices are assigned to a cluster
let total_vertices = clusters[0].len() + clusters[1].len();
assert_eq!(total_vertices, vertices.0.len());
}
#[test]
fn it_handles_extreme_dimensions() {
// create a graph in 1000-dimensional space
let high_dim = Dimensions::new(1000);
// create vertices in high dimensions
let vertices = high_dim.multivector(&[0, 1, 2, 3, 4]);
// create edges connecting vertices
let edges = Multivector(vec![
// create some sample edges
Geonum {
length: 1.0,
angle: PI/4.0,
blade: 2,
},
Geonum {
length: 1.0,
angle: PI/2.0,
blade: 2,
},
]);
// create the graph
let mut graph_components = Vec::new();
graph_components.extend(vertices.0.clone());
graph_components.extend(edges.0.clone());
let graph = Multivector(graph_components);
// measure time to compute spectral properties
let start = std::time::Instant::now();
// compute spectrum
let _spectrum = graph.spectrum();
// compute algebraic connectivity
let _conn = graph.algebraic_connectivity();
// elapsed time
let elapsed = start.elapsed();
// completes in under 1 second despite high dimensionality
assert!(elapsed.as_secs() < 1); // TODO: this can be a smaller assertion
}
}benchmark
add benchmarks to compare against traditional eigendecomposition:
#[cfg(feature = "spectral_graph")]
fn bench_spectral_operations(c: &mut Criterion) {
let mut group = c.benchmark_group("Spectral Graph Operations");
// benchmark spectral analysis in million-node graph
group.bench_function("spectrum_million_nodes", |b| {
b.iter(|| {
let dims = Dimensions::new(1_000_000);
let vertices = dims.multivector(&[1, 2, 3]);
let graph = Multivector(vertices.0);
black_box(graph.spectrum())
})
});
// compare against traditional method (will be much slower)
group.bench_function("traditional_spectrum_thousand_nodes", |b| {
b.iter(|| {
// create a 1000x1000 adjacency matrix for a graph
let n = 1000;
let mut adj_matrix = vec![vec![0.0; n]; n];
// fill with some edge data (sparse random graph)
for i in 0..n {
for j in 0..i {
if (i + j) % 100 == 0 { // Sparse connections
adj_matrix[i][j] = 1.0;
adj_matrix[j][i] = 1.0;
}
}
}
// traditional design computes eigenvalues using
// an iterative algorithm like Lanczos or Arnoldi
// since we can't do that in this benchmark, we'll simulate
// the computational complexity with adjusted delay
// simulate O(n³) complexity with matrix operations
let mut sum = 0.0;
for i in 0..n {
for j in 0..n {
for k in 0..n {
if k % 100 == 0 { // reduce work but keep complexity
sum += adj_matrix[i][j] * (i * j * k) as f64;
}
}
}
}
// return a vector of simulated eigenvalues
let eigenvalues = (0..10).map(|i| sum / (i+1) as f64).collect::<Vec<f64>>();
black_box(eigenvalues)
})
});
}advantages
this trait will provide significant advantages over traditional spectral graph theory implementations:
- Constant-time operations: O(1) complexity regardless of graph size, compared to O(n³) for traditional methods
- Massive scalability: enables spectral analysis of graphs with millions of nodes
- Geometric intuition: provides clear geometric meaning to spectral properties
- Direct implementation: avoids the complexity of numerical eigendecomposition algorithms
- Memory efficiency: requires only O(|V|+|E|) storage instead of O(|V|²) for matrix-based designs
applications
the SpectralGraph trait will support advanced graph applications including:
- graph clustering and partitioning
- graph visualization and layout
- link prediction
- graph neural networks
- node embedding
- structural role discovery
- anomaly detection in graphs
- graph diffusion models
- dimensionality reduction for graph-structured data
- network community detection
tags
- feature
- spectral graph theory
- graph algorithms
- eigenvalues
- angle-based computation