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.

(1) Let $\lambda_N$ be the maximal eigenvalue of the \emph{adjacency matrix} of $G_N$. It is known that $\lambda_N/N$ satisfies an LDP with rate $N$ 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 basic properties of $\psi_r$, especially when the reference graphon $r$ is of rank 1.

(2) Let $\hat\lambda_N$ be the maximal eigenvalue of the \emph{Laplacian matrix} of $G_N$. We show that $\hat\lambda_N/N$ satisfies a downward LDP with rate $\binom{N}{2}$ and an upward LDP with rate $N$. The associated rate functions $\hat\psi_r$ and $\hat\psi^*_r$ are those of the downward LDP and upward LDP for the maximal degree in the graph. We identify the basic properties of both $hat\psi_r$ and $\hat\psi^*_r$.