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

In this paper we introduce the Krichevsky-Trofimov estimator for the number of communities in the Stochastic Block Model (SBM) and 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 as well as the sparse regimes. To our knowledge this is the first strong consistency result for the estimation of the number of communities in the SBM, even in the bounded case.