Critical point representation of the mutual information in the sparse stochastic block model

TL;DR

This paper investigates the mutual information in the sparse stochastic block model, providing a critical point representation of its limit as the network size grows, with implications for community detection.

Contribution

It introduces a novel representation of the mutual information limit as an explicit functional evaluated at a critical point, applicable to the two-community case and beyond.

Findings

Explicit functional representation of mutual information limit

Validation of the critical point approach in community detection

Counterexample showing limitations of a variational formula

Abstract

We consider the problem of recovering the community structure in the stochastic block model. We aim to describe the mutual information between the observed network and the actual community structure as the number of nodes diverges while the average degree of a given node remains bounded. Our main contribution is a representation of the limit of this quantity, assuming it exists, as an explicit functional evaluated at a critical point of that functional. While we mostly focus on the two-community setting for clarity, we expect our method to be robust to model generalizations. We also present an example involving four communities where we show the invalidity of a plausible candidate variational formula for this limit.