Eigenvalues outside the bulk of inhomogeneous Erd\H{o}s-R\"enyi random graphs

TL;DR

This paper analyzes the spectral properties of inhomogeneous Erdős-Rényi random graphs, showing that certain eigenvalues diverge and follow Gaussian distributions, extending previous homogeneous graph results.

Contribution

It provides a detailed analysis of eigenvalues outside the bulk for inhomogeneous Erdős-Rényi graphs, including their asymptotic distributions and eigenvector behavior.

Findings

Largest eigenvalues diverge and are Gaussian distributed.

Joint distribution of top eigenvalues converges to a multivariate Gaussian.

Eigenvector behavior is characterized in the asymptotic regime.

Abstract

The article considers an inhomogeneous Erd\H{o}s-R\"enyi random graph on $\{1,\ldots, N\}$, 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 independent for each pair. The function $f$ is assumed to be non-negative definite, symmetric, bounded and of finite rank $k$. We study the edge of the spectrum of the adjacency matrix of such an inhomogeneous Erd\H{o}s-R\'enyi random graph under the assumption that $N\varepsilon_N\to \infty$ sufficiently fast. Although the bulk of the spectrum of the adjacency matrix, scaled by $\sqrt{N\varepsilon_N}$, is compactly supported, the $k$-th largest eigenvalue goes to infinity. It turns out that the largest eigenvalue after appropriate scaling and centering converge to a Gaussian law, if the largest eigenvalue of $f$ has multiplicity $1$. If $f$ 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 by-product of the above results. The results complement the homogeneous case derived by Erd\H{o}s et al.(2013).