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. 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 in the same regime. Both methods are shown to be weakly consistent in the divergent degree regime. These results confirm the optimality of maximum likelihood on the task of community detection, something that has remained as an open problem until now.

翻译：本文针对具有k个社区的随机块模型中的社区检测问题，获得了关于最大条件似然和积分条件似然估计量弱一致性和强一致性的新结果。具体而言，我们证明最大条件似然在对数度区域实现了精确恢复的已知最优阈值。对于积分条件似然，我们在相同区域获得了一个次优常数。两种方法在发散度区域均被证明具有弱一致性。这些结果证实了最大似然方法在社区检测任务中的最优性——这一问题此前一直悬而未决。