Eigenvalue distribution of large weighted random graphs

Abstract

We study eigenvalue distribution of the adjacency matrix A(N,p) of weighted random graphs Γ=ΓN,p. We assume that the graphs have N vertices and the average number of edges attached to one vertex is p. To each edge of the graph eij we assign a weight given by a random variable aij with zero mathematical expectation and all moments finite. In the first part of the paper, we consider the moments of normalized eigenvalue counting function σN,p of A(N,p). Assuming all moments of a finite, we obtain recurrent relations that determine the moments of the limiting measure σp=limN→∞σN,p. The method developed is applied to the Laplace operator ΔΓ closely related with A(N,p). Using the recurrent relations, we analyze the form of σp for the both of random matrix families. In the second part of the paper we consider the resolvents G(A,Δ)(z) of A(N,p) and ΔΓ of ΓN,p and study the functions fN(A,Δ)(z,u)=(1/N)∑k=1N exp{−uGkk(A,Δ)(z)} in the limit N→∞. We derive closed equations that uniquely determine the limiting functions f(A,Δ)(z,u). These equations allow us to prove the existence of the limiting σp for adjacency matrix and the Laplace operator under a rather weak condition that only the fourth moment of aij is finite. Besides, equations for f(A,Δ)(z,u) give us the asymptotic expansions for the Stieltjes transform of the limiting σp with respect to z−k and pk.