  We consider the problem of learning latent community structure from multiple
correlated networks. We study edge-correlated stochastic block models with two
balanced communities, focusing on the regime where the average degree is
logarithmic in the number of vertices. Our main result derives the precise
information-theoretic threshold for exact community recovery using multiple
correlated graphs. This threshold captures the interplay between the community
recovery and graph matching tasks. In particular, we uncover and characterize a
region of the parameter space where exact community recovery is possible using
multiple correlated graphs, even though (1) this is information-theoretically
impossible using a single graph and (2) exact graph matching is also
information-theoretically impossible. In this regime, we develop a novel
algorithm that carefully synthesizes algorithms from the community recovery and
graph matching literatures.
