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In this paper, we prove the weak and strong consistency of the maximum andintegrated conditional likelihood estimators for the community detectionproblem in the Stochastic Block Model with $k$ communities and unknownparameters. We show that maximum conditional likelihood achieves the optimalknown threshold for exact recovery in the logarithmic degree regime. For theintegrated conditional likelihood, we obtain a sub-optimal constant in the sameregime. Both methods are shown to be weakly consistent in the divergent degreeregime. The results also hold when the number of communities is allowed toincrease with the network size.

Optimal recovery by maximum and integrated conditional likelihood in the general Stochastic Block Model