Partial Recovery Bounds for the Sparse Stochastic Block Model

TL;DR

This paper investigates the fundamental limits of detecting communities in a sparse stochastic block model, providing bounds on the accuracy of label recovery and demonstrating near-optimality in certain regimes.

Contribution

It establishes upper and lower bounds on community detection accuracy in the sparse regime, advancing understanding of the information-theoretic limits.

Findings

Bounds are near-matching for moderate $a-b$

Bounds become tight as $a-b$ grows large

Provides numerical examples illustrating theoretical limits

Abstract

In this paper, we study the information-theoretic limits of community detection in the symmetric two-community stochastic block model, with intra-community and inter-community edge probabilities $\frac{a}{n}$ and $\frac{b}{n}$ respectively. We consider the sparse setting, in which $a$ and $b$ do not scale with $n$, and provide upper and lower bounds on the proportion of community labels recovered on average. We provide a numerical example for which the bounds are near-matching for moderate values of $a - b$, and matching in the limit as $a-b$ grows large.