Large deviation principle for the maximal eigenvalue of inhomogeneous ErdH{o}s-Renyi random graphs

We consider an inhomogeneous ErdH{o}s-Renyi random graph $G_N$ with vertex set $[N] = {1,dots,N}$ for which the pair of vertices $i,j in [N]$, $i eq j$, is connected by an edge with probability $r(tfrac{i}{N},tfrac{j}{N})$, independently of other pairs of vertices. Here, $rcolon,[0,1]^2 to (0,1)$ 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.