Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges

TL;DR

This paper proves that in random graphs with independent edges and sufficient expected degree, the adjacency matrix and Laplacian concentrate around their expected counterparts, extending known results and introducing a new matrix martingale concentration inequality.

Contribution

It generalizes matrix concentration results to a broad class of random graphs and introduces a novel matrix martingale inequality for analyzing such concentration.

Findings

Adjacency matrix and Laplacian concentrate around expected matrices in independent-edge random graphs.

Improved spectral gap bounds for bond percolation models.

Approximation of adjacency matrices by integral operators in inhomogeneous random graphs.

Abstract

Consider any random graph model where potential edges appear independently, with possibly different probabilities, and assume that the minimum expected degree is omega(ln n). We prove that the adjacency matrix and the Laplacian of that random graph are concentrated around the corresponding matrices of the weighted graph whose edge weights are the probabilities in the random model. While this may seem surprising, we will see that this matrix concentration phenomenon is a generalization of known results about the Er\"{o}s-R\'{e}nyi model. In particular, we will argue that matrix concentration is implicit the theory of quasi-random graph properties. We present two main applications of the main result. In bond percolation over a graph G, we show that the Laplacian of the random subgraph is typically very close to the Laplacian of G. As a corollary, we improve upon a bound for the spectral gap due to Chung and Horn that was derived via much more complicated methods. In inhomogeneous random graphs, there are points X_1,...,X_n uniformly distributed on the interval [0,1] and each pair is connected with probability p kappa(X_i,X_j). We show that if \ln n/n<< p<< 1 and kappa is bounded, then the adjacency matrix of the random graph is close to an integral operator defined in terms of kappa. Our main proof tool is a new concentration inequality for matrix martingales that generalizes Freedman's inequality for the standard scalar setting.