Partial recovery bounds for the sparse stochastic block model

In this paper, we study the information-theoretic limits of community detection in the symmetric two-community stochastic block model, with intra-community and inter-community edge probabilities a/n and b/n respectively. We consider the sparse setting, in which a and b do not scale with n, and provide upper and lower bounds on the proportion of community labels recovered on average. We provide a numerical example for which the bounds are near-matching for moderate values of a - b, and matching in the limit as a - b grows large.