Eigenvalues Outside the Bulk of Inhomogeneous Erdos-Renyi Random Graphs

In this article, an inhomogeneous Erdos-Renyi random graph on {1,..., N} is considered, where an edge is placed between vertices i and j with probability epsilon(N) f (i/N, j/N), for i <= j, the choice being made independently for each pair. The integral operator I-f 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 Erdos-Renyi random graph under the assumption that N-epsilon N -> infinity sufficiently fast. Although the bulk of the spectrum of the adjacency matrix, scaled by root N epsilon(N), 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 I f has multiplicity 1. If I-f 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].