Eigenvalues Outside the Bulk of Inhomogeneous Erdős–Rényi Random Graphs

Abstract

In this article, an inhomogeneous Erdős–Renyi random graph on $$\{1,\ldots , N\}$$ is considered, where an edge is placed between vertices i and j with probability $$\varepsilon _N f(i/N,j/N)$$ , for $$i\le j$$ , 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 Erdős–Renyi random graph under the assumption that $$N\varepsilon _N\rightarrow \infty $$ sufficiently fast. Although the bulk of the spectrum of the adjacency matrix, scaled by $$\sqrt{N\varepsilon _N}$$ , 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].