Spectral community detection in heterogeneous large networks

In this article, we study spectral methods for community detection based on $ \alpha$-parametrized normalized modularity matrix hereafter called $ {\bf L}\alpha $ in heterogeneous graph models. We show, in a regime where community detection is not asymptotically trivial, that $ {\bf L}\alpha $ can be well approximated by a more tractable random matrix which falls in the family of spiked random matrices. The analysis of this equivalent spiked random matrix allows us to improve spectral methods for community detection and assess their performances in the regime under study. In particular, we prove the existence of an optimal value $ \alpha{\rm opt} $ of the parameter $ \alpha $ for which the detection of communities is best ensured and we provide an on-line estimation of $ \alpha{\rm opt} $ only based on the knowledge of the graph adjacency matrix. Unlike classical spectral methods for community detection where clustering is performed on the eigenvectors associated with extreme eigenvalues, we show through our theoretical analysis that a regularization should instead be performed on those eigenvectors prior to clustering in heterogeneous graphs. Finally, through a deeper study of the regularized eigenvectors used for clustering, we assess the performances of our new algorithm for community detection. Numerical simulations in the course of the article show that our methods outperform state-of-the-art spectral methods on dense heterogeneous graphs.