Largest eigenvalues of sparse inhomogeneous Erd\H{o}s-R\'enyi graphs

TL;DR

This paper analyzes the asymptotic behavior of the largest eigenvalues of inhomogeneous Erdős-Rényi graphs with low average degree, revealing their dependence on degree distribution and expectation matrix eigenvalues, and identifying a crossover in behavior around the log n threshold.

Contribution

It characterizes the extreme eigenvalues in the sparse regime where mean degree is much less than log n, showing their dependence on degrees and expectation matrix eigenvalues, and introduces new tail estimates for Poisson approximations.

Findings

Extreme eigenvalues are governed by largest degrees and expectation matrix eigenvalues.

Eigenvalues do not converge to a nondegenerate point process in this regime.

A crossover in eigenvalue behavior occurs around the log n degree threshold.

Abstract

We consider inhomogeneous Erd\H{o}s-R\'enyi graphs. We suppose that the maximal mean degree $d$ satisfies $d \ll \log n$. We characterize the asymptotic behavior of the $n^{1 - o(1)}$ largest eigenvalues of the adjacency matrix and its centred version. We prove that these extreme eigenvalues are governed at first order by the largest degrees and, for the adjacency matrix, by the nonzero eigenvalues of the expectation matrix. Our results show that the extreme eigenvalues exhibit a novel behaviour which in particular rules out their convergence to a nondegenerate point process. Together with the companion paper [3], where we analyse the extreme eigenvalues in the complementary regime $d \gg \log n$, this establishes a crossover in the behaviour of the extreme eigenvalues around $d \sim \log n$. Our proof relies on a new tail estimate for the Poisson approximation of an inhomogeneous sum of independent Bernoulli random variables, as well as on an estimate on the operator norm of a pruned graph due to Le, Levina, and Vershynin.