We consider the problem of recovering the community structure in the stochastic block model. We aim to describe the mutual information between the observed network and the actual community structure as the number of nodes diverges while the average degree of a given node remains bounded. Our main contribution is a representation of the limit of this quantity, assuming it exists, as an explicit functional evaluated at a critical point of that functional. While we mostly focus on the two-community setting for clarity, we expect our method to be robust to model generalizations. We also present an example involving four communities where we show the invalidity of a plausible candidate variational formula for this limit.

翻译：我们研究随机块模型中的社区结构恢复问题。当节点数量发散而给定节点的平均度保持有界时，我们旨在描述观测网络与实际社区结构之间的互信息。我们的主要贡献是将该极限（假设其存在）表示为一个显式泛函在其临界点处的取值。为清晰起见，我们主要聚焦于双社区场景，但我们预期该方法对模型泛化具有鲁棒性。我们还给出了一个四社区示例，通过该示例我们证明了关于该极限的一个看似合理的候选变分公式不成立。