  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.
