Concentration and regularization of random graphs

• Published in: Random structures & algorithms Vol. 51; no. 3; pp. 538 - 561
• Main Authors: Le, Can M., Levina, Elizaveta, Vershynin, Roman
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.10.2017
• Subjects: community detection; concentration; Economic models; graph Laplacian; Graph theory; Graphs; Random graphs; Regularization; sparse networks
• ISSN: 1042-9832; 1098-2418
• DOI: 10.1002/rsa.20713

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(logn) 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=maxnpij. 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