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

In this paper, we prove the weak and strong consistency of the maximum and integrated conditional likelihood estimators for the community detection problem in the Stochastic Block Model with $k$ communities and unknown parameters. We show that maximum conditional likelihood achieves the optimal known threshold for exact recovery in the logarithmic degree regime. For the integrated conditional likelihood, we obtain a sub-optimal constant in the same regime. Both methods are shown to be weakly consistent in the divergent degree regime. The results also hold when the number of communities is allowed to increase with the network size.