Achieving Exact Cluster Recovery Threshold via Semidefinite Programming: Extensions

Recently it has been shown in \cite{HajekWuXuSDP14} that the semidefinite programming (SDP) relaxation of the maximum likelihood estimator achieves the sharp threshold for exactly recovering the community strucuture under the binary stochastic block model of two equal-sized clusters. Extending the techniques in \cite{HajekWuXuSDP14}, in this paper we show that SDP relaxations also achieve the sharp recovery threshold in the following cases: (1) Binary stochastic block model with two clusters of sizes proportional to $n$ but not necessarily equal; (2) Stochastic block model with a fixed number of equal-sized clusters; (3) Binary censored block model with the background graph being Erd\H{o}s-R\'enyi.