Matrix Concentration for Random Signed Graphs and Community Recovery in the Signed Stochastic Block Model

TL;DR

This paper develops concentration inequalities for adjacency and Laplacian matrices of random signed graphs and applies these results to community detection in the signed stochastic block model, demonstrating spectral gap concentration and eigenvector-based community estimation.

Contribution

It introduces new concentration inequalities for signed graphs and analyzes spectral properties for community detection in the signed stochastic block model.

Findings

Spectral gap of signed Laplacian concentrates near 2s

Eigenvector sign provides a weakly consistent community estimator

Experimental validation supports theoretical results

Abstract

We consider graphs where edges and their signs are added independently at random from among all pairs of nodes. We establish strong concentration inequalities for adjacency and Laplacian matrices obtained from this family of random graph models. Then, we apply our results to study graphs sampled from the signed stochastic block model. Namely, we take a two-community setting where edges within the communities have positive signs and edges between the communities have negative signs and apply a random sign perturbation with probability $0< s <1/2$. In this setting, our findings include: first, the spectral gap of the corresponding signed Laplacian matrix concentrates near $2s$ with high probability; and second, the sign of the first eigenvector of the Laplacian matrix defines a weakly consistent estimator for the balanced community detection problem, or equivalently, the $\pm 1$ synchronization problem. We supplement our theoretical contributions with experimental data obtained from the models under consideration.