Spectral statistics in Stochastic Block Models

Abstract

In this paper, we considered the spectral statistics in stochastic block models (SBMs), which used for community detection problems and other many applications. We proved results corresponding to the local semicircle law, an important analysis tool used in random matrix theory, and use it to prove the phase transition phenomenon of the largest eigenvalue and the central limit theorem applied to the entireeigenvalues. We further proposed an algorithm that could use the theoretical results to determine the number of blocks (communities) where it is unknown.