The Degree Corrected Stochastic Block Model (DCSBM) was introduced by \cite{karrer2011stochastic} as a generalization of the stochastic block model in which vertices of the same community are allowed to have distinct degree distributions. On the modelling side, this variability makes the DCSBM more suitable for real life complex networks. On the statistical side, it is more challenging due to the large number of parameters when dealing with community detection. In this paper we prove that the penalized marginal likelihood estimator is strongly consistent for the estimation of the number of communities. We consider \emph{dense} or \emph{semi-sparse} random networks, and our estimator is \emph{unbounded}, in the sense that the number of communities $k$ considered can be as big as $n$, the number of nodes in the network.

翻译：度修正随机块模型（DCSBM）由\cite{karrer2011stochastic}提出，作为随机块模型的一般化形式，允许同一社区内的顶点具有不同的度分布。在建模层面，这种变异性使得DCSBM更适用于现实世界的复杂网络。在统计层面，由于处理社区检测时涉及大量参数，该模型更具挑战性。本文证明，基于惩罚边际似然的估计量在估计社区数量时具有强一致性。我们考虑\emph{密集}或\emph{半稀疏}随机网络，且所提出的估计量是\emph{无界}的，即考虑的社区数量$k$可以大到与网络节点数$n$相同。