  The information-theoretic limits of community detection have been studied
extensively for network models with high levels of symmetry or homogeneity. The
contribution of this paper is to study a broader class of network models that
allow for variability in the sizes and behaviors of the different communities,
and thus better reflect the behaviors observed in real-world networks. Our
results show that the ability to detect communities can be described succinctly
in terms of a matrix of effective signal-to-noise ratios that provides a
geometrical representation of the relationships between the different
communities. This characterization follows from a matrix version of the I-MMSE
relationship and generalizes the concept of an effective scalar signal-to-noise
ratio introduced in previous work. We provide explicit formulas for the
asymptotic per-node mutual information and upper bounds on the minimum
mean-squared error. The theoretical results are supported by numerical
simulations.
