Tools for spectral analysis of complex networks: Laplacian operators, eigen-spectra, heat kernels, diffusion distances, spectral dimension scaling, and network embeddings—plus metrics (BH, shortest paths), partition similarity, geographic distances, and plotting helpers.
Inspired by: Diffusion Geometry Unravels the Emergence of Functional Clusters in Collective Phenomena
M. De Domenico, Phys. Rev. Lett.118, 168301 (2017). Open access: https://arxiv.org/abs/1704.07068
# install.packages("devtools")devtools::install_github("yourname/SpectralGeometry")Requires R ≥ 4.2.0. Core imports: igraph, Matrix, RSpectra, expm, stats, utils.
Optional (used in examples/plots): ggplot2, ggdendro, RColorBrewer, plot3D, MBA, dendextend, fossil, reshape2, geosphere.
Random walks and diffusion processes reveal functional proximity beyond pure topology. The (normalized / symmetric) Laplacian’s spectrum shapes return probabilities, diffusion distances, and low-dimensional embeddings that highlight mesoscale structure and functional clusters.
library(SpectralGeometry)
library(igraph)
# A simple graphg<- make_ring(50)
# 1) LaplaciansLs<- get_laplacians(g)
L<- getLaplacianMatrix(g, type="Laplacian") # combinatorialLrw<- getLaplacianMatrix(g, type="Normalized Laplacian") # random-walkLsym<- getLaplacianMatrix(g, type="Quantum Laplacian") # symmetric# 2) Spectral dimension scaling (average return probability)taus<-10^seq(-2, 1, length.out=10)
sd<- getSpectralDimensionScaling(g, taus) # uses Lrw by default# 3) Diffusion distance at time tauD<- getDistanceMatrix(g, tau=1.0, type="Normalized Laplacian")
# 4) Embedding (MDS on diffusion distances)coords<- getEmbeddingCoordinates(g, tau=1.0, dimension=2, method="MDS")
plot(coords[,1], coords[,2], pch=19, xlab="x1", ylab="x2",
main="Diffusion embedding (MDS)")
# 5) Partition similarity (VI/NMI/RAND/ARAND)m1<- rep(1:2, each=25)
m2<- sample(m1)
getPartitionSimilarity(m1, m2, method="NMI")library(SpectralGeometry); library(igraph)
set.seed(1)
g<- sample_smallworld(1, size=60, nei=2, p=0.05)
# Diffusion distance (functional)D_diff<- getDistanceMatrix(g, tau=0.8, type="Normalized Laplacian")
# Shortest path (topological)D_sp<- getDistanceMatrixSP(g)
# Compare 2D embeddingsemb_diff<- getEmbeddingCoordinatesFromDistanceMatrix(D_diff, 2, "MDS")
emb_sp<- getEmbeddingCoordinatesFromDistanceMatrix(D_sp, 2, "MDS")
par(mfrow= c(1,2))
plot(emb_sp[,1], emb_sp[,2], pch=19, main="MDS on shortest paths", xlab="x1", ylab="x2")
plot(emb_diff[,1], emb_diff[,2], pch=19, main="MDS on diffusion distance", xlab="x1", ylab="x2")
par(mfrow= c(1,1))# Two identical layers should match single-layer distanceg1<- make_ring(30)
g2<- graph_from_edgelist(as_edgelist(g1), directed=FALSE)
D1<- getDistanceMatrix(g1, tau=0.6)
D2<- getDistanceMatrixEdgeColored(list(g1, g2), tau=0.6)
max(abs(D1-D2)) # ~ 0If you use this package in academic work, please cite the original paper:
@article{DeDomenico2017PRL,
title = {Diffusion Geometry Unravels the Emergence of Functional Clusters in Collective Phenomena},
author = {De Domenico, Manlio},
journal = {Physical Review Letters},
year = {2017},
volume = {118},
number = {16},
pages = {168301},
doi = {10.1103/PhysRevLett.118.168301},
url = {https://arxiv.org/abs/1704.07068}
}
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