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.

中文翻译：

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稀疏随机块模型的互信息

我们考虑在具有两个社区的随机块模型中恢复社区结构的问题。我们旨在描述在稀疏状态下观察到的网络与实际社区结构之间的互信息，其中节点总数发散，而给定节点的平均度数保持有界。我们的主要贡献是对这个量的极限的猜想，我们用在概率度量空间上的 Hamilton-Jacobi 方程来表达，并证明这个猜想的极限为渐近互信息提供了一个下限。在我们的配套论文中获得了 Hamilton-Jacobi 方程的适定性。如果跨社区的链接比社区内的链接更有可能，已知渐近互信息由变分公式给出。我们还表明，在这种情况下，我们推测的极限与该公式一致。