Optimal Recovery of Block Models with $q$ Communities

TL;DR

This paper generalizes the optimal community recovery algorithm from 2 communities to q communities in the sparse stochastic block model, providing a provably optimal reconstruction method based on belief propagation.

Contribution

It extends the analysis and algorithm for optimal community detection to any number of communities, building on prior work for the 2-community case.

Findings

Proves the reconstruction accuracy matches the theoretical limit for q communities.

Develops a belief propagation-based algorithm with provable optimal recovery.

Generalizes key lemmas to multi-community settings.

Abstract

This paper is motivated by the reconstruction problem on the sparse stochastic block model. The paper "Belief Propagation, robust reconstruction and optimal recovery of block models" by Mossel, Neeman, and Sly provided and proved a reconstruction algorithm that recovers an optimal fraction of the communities in the 2 community case. The main step in their proof was to show that when the signal to noise ratio is sufficiently large, in particular $\theta^2d > C$, the reconstruction accuracy on a regular tree with or without noise on the leaves is the same. This paper will generalize their results, including the main step, to any number of communities, providing an algorithm related to Belief Propagation that recovers a provably optimal fraction of community labels.