In this paper we introduce an estimator for the number of communities in the Stochastic Block Model (SBM), based on the maximization of a penalized version of the so-called Krichevsky-Trofimov mixture distribution. We prove its eventual almost sure convergence to the underlying number of communities, without assuming a known upper bound on that quantity. Our results apply to both the dense and the sparse regimes. To our knowledge this is the first consistency result for the estimation of the number of communities in the SBM in the unbounded case, that is when the number of communities is allowed to grow with the same size.

翻译：在本文中,我们引入了斯托切斯特区块模型(SBM)中社区数量估计符号,其依据是最大限度地增加一个惩罚性版本的所谓克里切夫斯基-特罗菲莫夫混合配方。我们证明它最终几乎肯定地与基本社区数量趋同,而没有假定这一数量具有已知的上限。我们的结果既适用于密集政权,也适用于稀疏政权。据我们所知,这是在无约束的情况下,即允许社区数量以同样规模增长时,对标准媒体中社区数量进行估计的第一个一致性结果。