Skip to main content
eScholarship
Open Access Publications from the University of California

UC Irvine

UC Irvine Previously Published Works bannerUC Irvine

Concentration and regularization of random graphs

Published Web Location

https://doi.org/10.1002/rsa.20713Creative Commons 'BY' version 4.0 license
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 Erdös-Rényi random graphs on n vertices, where edges form independently and possibly with different probabilities pij. 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 npij. Then we show that the resulting adjacency matrix A’ concentrates with the optimal rate: ||A’−EA||= O(√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(EA’)|| = O(1/√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. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 538–561, 2017.

Many UC-authored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.

Main Content
For improved accessibility of PDF content, download the file to your device.
Current View