Exact recovery threshold in the binary censored block model

Given a background graph with n vertices, the binary censored block model assumes that vertices are partitioned into two clusters, and every edge is labeled independently at random with labels drawn from Bern(1 - ε) if two endpoints are in the same cluster, or from Bern(ε) otherwise, where ε E [0, 1/2] is a fixed constant. For Erdós-Rényi graphs with edge probability p = a log n/n and fixed a, we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold a(√1 - ε - √ε)² > 1 for exactly recovering the partition from the labeled graph with probability tending to one as n oo. For random regular graphs with degree scaling as a log n, we show that the semidefinite programming relaxation also achieves the optimal recovery threshold aD(Bern(1/2)IIBern(ε)) > 1, where D denotes the Kullback-Leibler divergence.