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

TL;DR

This paper establishes a large deviation principle for the maximal eigenvalue of inhomogeneous Erdős-Rényi graphs, analyzing the associated rate function, especially for rank 1 graphons, to understand the eigenvalue fluctuations.

Contribution

It provides a detailed analysis of the large deviation rate function for the maximal eigenvalue in inhomogeneous Erdős-Rényi graphs, particularly for rank 1 reference graphons, extending previous results.

Findings

Large deviation principle for the maximal eigenvalue established.

Rate function characterized via a variational formula involving graphon space.

Special analysis conducted for rank 1 reference graphons.

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 $r(\tfrac{i}{N},\tfrac{j}{N})$, independently of other pairs of vertices. Here, $r\colon\,[0,1]^2 \to (0,1)$ is a symmetric function that plays the role of a reference graphon. Let $\lambda_N$ be the maximal eigenvalue of the adjacency matrix of $G_N$. It is known that $\lambda_N/N$ satisfies a large deviation principle as $N \to \infty$. The associated rate function $\psi_r$ is given by a variational formula that involves the rate function $I_r$ of a large deviation principle on graphon space. We analyse this variational formula in order to identify the properties of $\psi_r$, specially when the reference graphon is of rank 1.