Eigenvalue Distribution of Large Weighted Random Sparse Uniform $q$-Hypergraphs

TL;DR

This paper analyzes the eigenvalue distribution of adjacency matrices of large weighted random uniform hypergraphs, deriving recursive relations for the moments of the limiting spectral measure under finite moment assumptions.

Contribution

It introduces a method to determine the eigenvalue distribution of large weighted hypergraphs using recursive relations for moments.

Findings

Derived recursive formulas for spectral moments

Established convergence to a limiting eigenvalue distribution

Applicable to hypergraphs with finite-moment weight distributions

Abstract

We study eigenvalue distribution of the adjacency matrix $A^{(N,p,q)}$ of weighted random uniform $q$-hypergraphs $\Gamma= \Gamma_{N,p,q}$. We assume that the graphs have $N$ vertices and the average number of hyperedges attached to one vertex is $(q-1)!\cdot p$. To each edge of the graph $e_{ij}$ we assign a weight given by a random variable $a_{ij}$ with all moments finite. We consider the moments of normalized eigenvalue counting function $\sigma_{N,p,q}$ of $A^{(N,p,q)}$. Assuming all moments of $a$ finite, we obtain recurrent relations that determine the moments of the limiting measure $\sigma_{p,q} = \lim_{N\to\infty} \sigma_{N,p,q}$.