Mutual Information in Community Detection with Covariate Information and Correlated Networks

We study the problem of community detection when there is covariate information about the node labels and one observes multiple correlated networks. We provide an asymptotic upper bound on the per-node mutual information as well as a heuristic analysis of a multivariate performance measure called the MMSE matrix. These results show that the combined effects of seemingly very different types of information can be characterized explicitly in terms of formulas involving low-dimensional estimation problems in additive Gaussian noise. Our analysis is supported by numerical simulations.