Eigenvalue distribution of large weighted multipartite random sparse graphs

TL;DR

This paper studies the eigenvalue distribution of large, weighted, multipartite random sparse graphs constructed from a fixed graph structure, revealing convergence properties and deriving moments of the limiting spectral measure.

Contribution

It introduces a new model for weighted multipartite random graphs and establishes the convergence of their eigenvalue distributions, including explicit moment calculations.

Findings

Eigenvalue distribution converges weakly to a deterministic measure.

Moments of the limiting measure are obtained via recurrence relations.

The model generalizes existing random graph spectral results.

Abstract

We investigate the distribution of eigenvalues of weighted adjacency matrices from a specific ensemble of random graphs. We distribute $N$ vertices across a fixed number $\kappa$ of components, with asymptotically $\alpha_j \dot N$ vertices in each component, where the vector $(\alpha_1,\alpha_2, \ldots, \alpha_{\kappa})$ is fixed. Consider a connected graph $\Gamma$ with $\kappa$ vertices. We construct a multipartite graph with $N$ vertices, in which all vertices in the $i$-th component are connected to all vertices in the $j$-th component if if $\Gamma_{ij}=1$. Conversely, if $\Gamma_{ij}=0$, no edge connects the $i$-th and $j$-th components. In the resulting graph, we independently retain each edge with a probability of $p/N$, where $p$ is a fixed parameter. To each remaining edge, we assign an independent weight with a fixed distribution, that possesses all finite moments. We establish the weak convergence in probability of a random counting measure to a non-random probability measure. Furthermore, the moments of the limiting measure can be derived from a system of recurrence relations.