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

We consider an inhomogeneous Erdős-Rényi random graph G N with vertex set [ N ] = { 1 , ⋯ , N } for which the pair of vertices i , j ∈ [ N ] , i ≠ j , is connected by an edge with probability r ( i N , j N ) , independently of other pairs of vertices. Here, r : [ 0 , 1 ] 2 → ( 0 , 1 ) is a symmetric function that plays the role of a reference graphon. Let λ N be the maximal eigenvalue of the adjacency matrix of G N . It is known that λ N / N satisfies a large deviation principle as N → ∞ . The associated rate function ψ 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 ψ r , specially when the reference graphon is of rank 1.