Eigenvalues Outside the Bulk of Inhomogeneous Erdős–Rényi Random Graphs

Abstract

In this article, an inhomogeneous Erdős–Rényi random graph on {1,…,N} is considered, where an edge is placed between vertices i and j with probability εNf(i/N,j/N), for i≤j, the choice being made independently for each pair. The integral operator If associated with the bounded function f is assumed to be symmetric, non-negative definite, and of finite rank k. We study the edge of the spectrum of the adjacency matrix of such an inhomogeneous Erdős–Rényi random graph under the assumption that NεN→∞ sufficiently fast. Although the bulk of the spectrum of the adjacency matrix, scaled by NεN−−−−√, is compactly supported, the kth largest eigenvalue goes to infinity. It turns out that the largest eigenvalue after appropriate scaling and centering converges to a Gaussian law, if the largest eigenvalue of If has multiplicity 1. If If has k distinct non-zero eigenvalues, then the joint distribution of the k largest eigenvalues converge jointly to a multivariate Gaussian law. The first order behaviour of the eigenvectors is derived as a byproduct of the above results. The results complement the homogeneous case derived by [18].