MUTUAL INFORMATION FOR THE SPARSE STOCHASTIC BLOCK MODEL

• 成果类型: Article
• 署名作者: Dominguez, Tomas; Mourrat, Jean-christophe
• 署名单位: University of Toronto; Ecole Normale Superieure de Lyon (ENS de LYON)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/23-AOP1665
• 发表日期: 2024
• 页码: 434-501
• 关键词: hamilton-jacobi equations; reconstruction; blockmodels; RECOVERY; LIMITS

摘要

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.