Concentration and regularization of random graphs
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 ErdösRényi 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 GrothendieckPietsch 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2015
 arXiv:
 arXiv:1506.00669
 Bibcode:
 2015arXiv150600669L
 Keywords:

 Mathematics  Probability;
 Computer Science  Social and Information Networks;
 Mathematics  Statistics Theory;
 05C80;
 60B20;
 05C85
 EPrint:
 21 pages. Elizaveta Levina is added as a coauthor. Application to community detection of networks is expanded