Large Deviation Principle for the Maximal Eigenvalue of Inhomogeneous Erdős-Rényi Random Graphs

Abstract

We consider an inhomogeneous Erdős-Rényi random graph \(G_N\) with vertex set \([N] = \{1,\dots ,N\}\) for which the pair of vertices \(i,j \in [N]\), \(i\ne j\), is connected by an edge with probability \(r(\tfrac{i}{N},\tfrac{j}{N})\), independently of other pairs of vertices. Here, \(r:\,[0,1]^2 \rightarrow (0,1)\) 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 \(N \rightarrow \infty \). 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.