Robustness of Community Detection to Random Geometric Perturbations

TL;DR

This paper investigates how spectral community detection methods perform when the underlying graph is perturbed by a latent random geometric component, establishing regimes where recovery remains effective.

Contribution

It provides a rigorous analysis demonstrating the robustness of spectral methods to geometric perturbations in the stochastic block model, with explicit conditions for successful community recovery.

Findings

Spectral methods remain effective under certain geometric perturbations.

Explicit regimes where eigenvectors correlate with true communities.

Spectrum analysis of the latent geometric graph enhances understanding.

Abstract

We consider the stochastic block model where connection between vertices is perturbed by some latent (and unobserved) random geometric graph. The objective is to prove that spectral methods are robust to this type of noise, even if they are agnostic to the presence (or not) of the random graph. We provide explicit regimes where the second eigenvector of the adjacency matrix is highly correlated to the true community vector (and therefore when weak/exact recovery is possible). This is possible thanks to a detailed analysis of the spectrum of the latent random graph, of its own interest.