Mutual information for the sparse stochastic block model

TL;DR

This paper investigates the mutual information between observed networks and true community structures in sparse stochastic block models, proposing a conjecture expressed via a Hamilton-Jacobi equation and providing bounds and consistency results.

Contribution

It introduces a conjecture for the mutual information limit in sparse stochastic block models using a Hamilton-Jacobi framework and proves it bounds the true mutual information.

Findings

Conjectured the mutual information limit via Hamilton-Jacobi equation.

Proved the conjecture provides a lower bound for the mutual information.

Confirmed the conjecture matches known formulas in certain cases.

Abstract

We consider the problem of recovering the community structure in the stochastic block model with two communities. We aim to describe the mutual information between the observed network and the actual community structure in the sparse regime, where the total number of nodes diverges while the average degree of a given node remains bounded. Our main contributions are a conjecture for the limit of this quantity, which we express in terms of a Hamilton-Jacobi equation posed over a space of probability measures, and a proof that this conjectured limit provides a lower bound for the asymptotic mutual information. The well-posedness of the Hamilton-Jacobi equation is obtained in our companion paper. In the case when links across communities are more likely than links within communities, the asymptotic mutual information is known to be given by a variational formula. We also show that our conjectured limit coincides with this formula in this case.