Sparse random graphs: regularization and concentration of the Laplacian

We study random graphs with possibly different edge probabilities in the challenging sparse regime of bounded expected degrees. Unlike in the dense case, neither the graph adjacency matrix nor its Laplacian concentrate around their expectations due to the highly irregular distribution of node degrees. It has been empirically observed that simply adding a constant of order $1/n$ to each entry of the adjacency matrix substantially improves the behavior of Laplacian. Here we prove that this regularization indeed forces Laplacian to concentrate even in sparse graphs. As an immediate consequence in network analysis, we establish the validity of one of the simplest and fastest approaches to community detection -- regularized spectral clustering, under the stochastic block model. Our proof of concentration of regularized Laplacian is based on Grothendieck's inequality and factorization, combined with paving arguments.