Concentration and regularization of random graphs

TL;DR

This paper investigates how random graphs concentrate around their expected structure in spectral norm, identifying issues with sparse graphs and proposing regularization techniques to restore concentration, with applications to community detection.

Contribution

It introduces regularization methods for inhomogeneous Erdős-Rényi graphs to achieve optimal spectral concentration rates, addressing sparsity-related issues.

Findings

Regularization restores spectral concentration in sparse graphs.

Reweighting or removing edges bounds degrees and improves concentration.

Application demonstrated in community detection analysis.

Abstract

This paper studies how close random graphs are typically to their expectations. We interpret this question through the concentration of the adjacency and Laplacian matrices in the spectral norm. We study inhomogeneous Erd\"os-R\'enyi random graphs on $n$ vertices, where edges form independently and possibly with different probabilities $p_{ij}$. Sparse random graphs whose expected degrees are $o(\log n)$ fail to concentrate; the obstruction is caused by vertices with abnormally high and low degrees. We show that concentration can be restored if we regularize the degrees of such vertices, and one can do this in various ways. As an example, let us reweight or remove enough edges to make all degrees bounded above by $O(d)$ where $d=\max np_{ij}$. Then we show that the resulting adjacency matrix $A'$ concentrates with the optimal rate: $\|A' - \mathbb{E} A\| = O(\sqrt{d})$. Similarly, if we make all degrees bounded below by $d$ by adding weight $d/n$ to all edges, then the resulting Laplacian concentrates with the optimal rate: $\|L(A') - L(\mathbb{E} A')\| = O(1/\sqrt{d})$. Our approach is based on Grothendieck-Pietsch factorization, using which we construct a new decomposition of random graphs. We illustrate the concentration results with an application to the community detection problem in the analysis of networks.