Spectral Properties and Weak Detection in Stochastic Block Models

TL;DR

This paper analyzes the spectral properties of stochastic block models in sparse and dense regimes, identifying phase transitions at the Kesten--Stigum threshold and proposing a hypothesis test for community detection.

Contribution

It establishes phase transition points for eigenvalues and proves a central limit theorem for spectral statistics in both regimes, introducing a new community detection method.

Findings

Eigenvalue phase transition at Kesten--Stigum threshold

Central limit theorem for spectral statistics

Effective hypothesis test for community detection

Abstract

We consider the spectral properties of balanced stochastic block models of which the average degree grows slower than the number of nodes (sparse regime) or proportional to it (dense regime). For both regimes, we prove a phase transition of the extreme eigenvalues of SBM at the Kesten--Stigum threshold. We also prove the central limit theorem for the linear spectral statistics for both regimes. We propose a hypothesis test for determining the presence of communities of the graph, based on the central limit theorem for the linear spectral statistics.