Eigenvalues Outside the Bulk of Inhomogeneous Erdős–Rényi Random Graphs

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