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.
Acknowledgments
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.
Citation
Tomas Dominguez. Jean-Christophe Mourrat. "Mutual information for the sparse stochastic block model." Ann. Probab. 52 (2) 434 - 501, March 2024. https://doi.org/10.1214/23-AOP1665
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