Strong consistency of Krichevsky-Trofimov estimator for the number of communities in the Stochastic Block Model

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.