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

In this paper, we obtain new results on 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. In particular, 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, but still obtain strong consistency in the logarithmic degree regime. Both methods are shown to be weakly consistent in the divergent degree regime. These results fill in the gap in the theory of community detection with maximum likelihood and integrated conditional likelihood, solving open problems in the literature.