Eigenvalue distribution of bipartite large weighted random graphs. Resolvent approach

TL;DR

This paper analyzes the eigenvalue distribution of large weighted bipartite random graphs using resolvent methods, deriving equations for the limiting spectral measure and proving convergence of eigenvalue distributions.

Contribution

It introduces a resolvent approach to determine the eigenvalue distribution of weighted bipartite graphs and establishes the existence and convergence of the limiting spectral measure.

Findings

Derived a closed system of equations for the limiting functions

Proved the existence of the limiting eigenvalue measure

Established weak convergence of eigenvalue distributions

Abstract

We study eigenvalue distribution of the adjacency matrix $A^{(N,p, \alpha)}$ of weighted random bipartite graphs $\Gamma= \Gamma_{N,p}$. We assume that the graphs have $N$ vertices, the ratio of parts is $\frac{\alpha}{1-\alpha}$ and the average number of edges attached to one vertex is $\alpha\cdot p$ or $(1-\alpha)\cdot p$. To each edge of the graph $e_{ij}$ we assign a weight given by a random variable $a_{ij}$ with the finite second moment. We consider the resolvents $G^{(N,p, \alpha)}(z)$ of $A^{(N,p, \alpha)}$ and study the functions $f_{1,N}(u,z)=\frac{1}{[\alpha N]}\sum_{k=1}^{[\alpha N]}e^{-ua_k^2G_{kk}^{(N,p,\alpha)}(z)}$ and $f_{2,N}(u,z)=\frac{1}{N-[\alpha N]}\sum_{k=[\alpha N]+1}^Ne^{-ua_k^2G_{kk}^{(N,p,\alpha)}(z)}$ in the limit $N\to \infty$. We derive closed system of equations that uniquely determine the limiting functions $f_{1}(u,z)$ and $f_{2}(u,z)$. This system of equations allow us to prove the existence of the limiting measure $\sigma_{p, \alpha}$ . The weak convergence in probability of normalized eigenvalue counting measures is proved.