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

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