Consistent model selection for the Degree Corrected Stochastic Blockmodel

TL;DR

This paper proves that a penalized likelihood method consistently estimates the number of communities in dense or semi-sparse Degree Corrected Stochastic Block Models, even when the number of communities grows with the network size.

Contribution

It establishes the strong consistency of a penalized marginal likelihood estimator for community number estimation in DCSBM, applicable to large and complex networks.

Findings

Estimator is strongly consistent for community detection.

Applicable to dense and semi-sparse networks.

Number of communities can grow as large as the number of nodes.

Abstract

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.