Achieving exact cluster recovery threshold via semidefinite programming under the stochastic block model

Resolving a conjecture of Abbe, Bandeira and Hall, the authors have recently shown that the semidefinite programming (SDP) relaxation of the maximum likelihood estimator achieves the sharp threshold for exactly recovering the community structure under the binary stochastic block model of two equal-sized clusters. Extending the proof techniques, in this paper it is shown that SDP relaxations also achieve the sharp recovery threshold under the stochastic block model with a fixed number of equal-sized clusters.