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

We consider an inhomogeneous Erdős-Rényi random graph $$G_N$$ G N with vertex set $$[N] = \{1,\dots ,N\}$$ [ N ] = { 1 , ⋯ , N } for which the pair of vertices $$i,j \in [N]$$ i , j ∈ [ N ] , $$i\ne j$$ i ≠ j , is connected by an edge with probability $$r(\tfrac{i}{N},\tfrac{j}{N})$$ r ( i N , j N ) , independently of other pairs of vertices. Here, $$r:\,[0,1]^2 \rightarrow (0,1)$$ r : [ 0 , 1 ] 2 → ( 0 , 1 ) is a symmetric function that plays the role of a reference graphon. Let $$\lambda _N$$ λ N be the maximal eigenvalue of the adjacency matrix of $$G_N$$ G N . It is known that $$\lambda _N/N$$ λ N / N satisfies a large deviation principle as $$N \rightarrow \infty $$ N → ∞ . The associated rate function $$\psi _r$$ ψ r is given by a variational formula that involves the rate function $$I_r$$ I r of a large deviation principle on graphon space. We analyse this variational formula in order to identify the properties of $$\psi _r$$ ψ r , specially when the reference graphon is of rank 1.