Non-Reconstructability in the Stochastic Block Model

TL;DR

This paper investigates the limits of clustering in the stochastic block model, especially for multiple and unbalanced clusters, establishing conditions for impossibility and exploring the relationship between reconstructability and distinguishability.

Contribution

It extends the understanding of non-reconstructability to more general stochastic block models, connecting it with model distinguishability and providing new sufficient conditions.

Findings

Impossibility conditions for clustering in general SBMs

Connection between non-reconstructability and non-distinguishability

Reconstruction possible even below the Kesten-Stigum threshold in some cases

Abstract

We consider the problem of clustering (or reconstruction) in the stochastic block model, in the regime where the average degree is constant. For the case of two clusters with equal sizes, recent results by Mossel, Neeman and Sly, and by Massoulie, show that reconstructability undergoes a phase transition at the Kesten-Stigum bound of $\lambda_2^2 d = 1$, where $\lambda_2$ is the second largest eigenvalue of a related stochastic matrix and $d$ is the average degree. In this paper, we address the general case of more than two clusters and/or unbalanced cluster sizes. Our main result is a sufficient condition for clustering to be impossible, which matches the existing result for two clusters of equal sizes. A key ingredient in our result is a new connection between non-reconstructability and non-distinguishability of the block model from an Erd\H{o}s-R\'enyi model with the same average degree. We also show that it is some times possible to reconstruct even when $\lambda_2^2 d < 1$. Our results provide evidence supporting a series of conjectures made by Decelle, Krzkala, Moore and Zdeborov\'a regarding reconstructability and distinguishability of stochastic block models (but do not settle them).