March 2024 Mutual information for the sparse stochastic block model
Tomas Dominguez, Jean-Christophe Mourrat
Author Affiliations +
Ann. Probab. 52(2): 434-501 (March 2024). DOI: 10.1214/23-AOP1665


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.


We would like to warmly thank Dmitry Panchenko and Jean Barbier for sharing their notes [13] on the free energy in the disassortative sparse stochastic block model with us, which provided us with a very useful starting point and helped us with many of the computations in Section 2.


Download Citation

Tomas Dominguez. Jean-Christophe Mourrat. "Mutual information for the sparse stochastic block model." Ann. Probab. 52 (2) 434 - 501, March 2024.


Received: 1 October 2022; Revised: 1 August 2023; Published: March 2024
First available in Project Euclid: 4 March 2024

MathSciNet: MR4718399
Digital Object Identifier: 10.1214/23-AOP1665

Primary: 60K35 , 82B44
Secondary: 35D40 , 35R15

Keywords: Free energy , Hamilton–Jacobi equation , mutual information , Stochastic block model

Rights: Copyright © 2024 Institute of Mathematical Statistics


This article is only available to subscribers.
It is not available for individual sale.

Vol.52 • No. 2 • March 2024
Back to Top