We study the problem of learning latent community structure from multiple correlated networks, focusing on edge-correlated stochastic block models with two balanced communities. Recent work of Gaudio, R\'acz, and Sridhar (COLT 2022) determined the precise information-theoretic threshold for exact community recovery using two correlated graphs; in particular, this showcased the subtle interplay between community recovery and graph matching. Here we study the natural setting of more than two graphs. The main challenge lies in understanding how to aggregate information across several graphs when none of the pairwise latent vertex correspondences can be exactly recovered. Our main result derives the precise information-theoretic threshold for exact community recovery using any constant number of correlated graphs, answering a question of Gaudio, R\'acz, and Sridhar (COLT 2022). In particular, for every $K \geq 3$ we uncover and characterize a region of the parameter space where exact community recovery is possible using $K$ correlated graphs, even though (1) this is information-theoretically impossible using any $K-1$ of them and (2) none of the latent matchings can be exactly recovered.

翻译：我们研究从多个相关网络中学习潜在社区结构的问题，重点关注具有两个平衡社区的边相关随机块模型。Gaudio、R\'acz 和 Sridhar（COLT 2022）的最新工作确定了使用两个相关图实现精确社区恢复的精确信息论阈值；这尤其揭示了社区恢复与图匹配之间微妙的相互作用。本文研究超过两个图的自然场景。主要挑战在于理解当任何一对潜在顶点对应关系都无法被精确恢复时，如何跨多个图聚合信息。我们的主要成果推导出使用任意常数个相关图实现精确社区恢复的精确信息论阈值，回答了 Gaudio、R\'acz 和 Sridhar（COLT 2022）提出的问题。具体而言，对于每个 $K \geq 3$，我们发现并刻画了参数空间的一个区域：在该区域内，使用 $K$ 个相关图可以实现精确社区恢复，尽管（1）使用其中任意 $K-1$ 个图在信息论上均不可能实现，且（2）所有潜在匹配均无法被精确恢复。