Recovery and Rigidity in a Regular Stochastic Block Model

TL;DR

This paper introduces the regular stochastic block model (RSBM), demonstrating that exact community recovery is achievable with high probability and spectral methods, and that partial recovery can be extended to full recovery.

Contribution

It presents the RSBM variant and proves rigidity, showing exact community recovery is possible and can be efficiently achieved using spectral methods, extending partial to full recovery.

Findings

Exact community recovery is possible with high probability.

Spectral methods can efficiently recover communities in certain regimes.

Partial recovery can be bootstrapped to full recovery.

Abstract

The stochastic block model is a natural model for studying community detection in random networks. Its clustering properties have been extensively studied in the statistics, physics and computer science literature. Recently this area has experienced major mathematical breakthroughs, particularly for the binary (two-community) version, see Mossel, Neeman, Sly (2012, 2013) and Massoulie (2013). In this paper, we introduce a variant of the binary model which we call the regular stochastic block model (RSBM). We prove rigidity by showing that with high probability an exact recovery of the community structure is possible. Spectral methods exhibit a regime where this can be done efficiently. Moreover we also prove that, in this setting, any suitably good partial recovery can be bootstrapped to obtain a full recovery of the communities.