Large deviation principle for the norm of the Laplacian matrix of inhomogeneous Erd\H{o}s-R\'enyi random graphs

TL;DR

This paper establishes large deviation principles for the scaled maximum eigenvalue of the Laplacian matrix in inhomogeneous Erdős-Rényi graphs, linking the eigenvalue behavior to the underlying graphon structure.

Contribution

It introduces a large deviation framework for the eigenvalues of Laplacian matrices in inhomogeneous random graphs with convergence to a limiting graphon.

Findings

Downward LDP with rate inom{N}{2} for igenvalue/N

Upward LDP with rate N for igenvalue/N

Identification of rate functions  and igenvalue behaviors

Abstract

We consider an inhomogeneous Erd\H{o}s-R\'enyi random graph $G_N$ with vertex set $[N] = \{1,\dots,N\}$ for which the pair of vertices $i,j \in [N]$, $i\neq j$, is connected by an edge with probability $r_N(\tfrac{i}{N},\tfrac{j}{N})$, independently of other pairs of vertices. Here, $r_N\colon\,[0,1]^2 \to (0,1)$ is a symmetric function that plays the role of a reference graphon. Let $\lambda_N$ be the maximal eigenvalue of the Laplacian matrix of $G_N$. We show that if $\lim_{N\to\infty} \|r_N-r\|_\infty = 0$ for some limiting graphon $r\colon\,[0,1]^2 \to (0,1)$, then $\lambda_N/N$ satisfies a downward LDP with rate $\binom{N}{2}$ and an upward LDP with rate $N$. We identify the associated rate functions $\psi_r$ and $\widehat{\psi}_r$, and derive their basic properties.