Bad local minima exist in the stochastic block model

TL;DR

This paper demonstrates that in the disassortative stochastic block model, there are local minima in the posterior distribution that hinder MAP inference, despite the existence of efficient algorithms for community detection.

Contribution

It provides evidence of the existence of poor local minima in the posterior landscape of the stochastic block model, challenging assumptions about inference optimality.

Findings

MAP inference can fail due to local minima.

Efficient algorithms can still succeed despite MAP failure.

Posterior modes may have low agreement with ground truth.

Abstract

We study the disassortative stochastic block model with three communities, a well-studied model of graph partitioning and Bayesian inference for which detailed predictions based on the cavity method exist [Decelle et al. (2011)]. We provide strong evidence that for a part of the phase where efficient algorithms exist that approximately reconstruct the communities, inference based on maximum a posteriori (MAP) fails. In other words, we show that there exist modes of the posterior distribution that have a vanishing agreement with the ground truth. The proof is based on the analysis of a graph colouring algorithm from [Achlioptas and Moore (2003)].