Sparse random graphs: regularization and concentration of the Laplacian

TL;DR

This paper proves that regularizing the Laplacian of sparse random graphs by adding a small constant to the adjacency matrix entries ensures concentration, validating a fast community detection method.

Contribution

It provides a rigorous proof that simple regularization guarantees Laplacian concentration in sparse graphs, supporting spectral clustering in network analysis.

Findings

Regularization ensures Laplacian concentration in sparse graphs.

Validates spectral clustering for community detection under the stochastic block model.

Uses Grothendieck's inequality and paving arguments in the proof.

Abstract

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.