Semicircle laws with combined variance for non-uniform Erd\H{o}s-R\'enyi hypergraphs

TL;DR

This paper analyzes the spectral distribution of adjacency matrices of non-uniform, inhomogeneous Erdős-Rényi hypergraphs, establishing semicircle laws with explicit variance formulas under certain conditions.

Contribution

It characterizes the limiting spectral distribution for complex hypergraph models, extending semicircle law results to inhomogeneous, non-uniform cases with explicit variance expressions.

Findings

The expected spectral distribution converges to a semicircle law under non-sparse conditions.

A Pastur-type condition allows Gaussianization of the adjacency matrix.

Explicit variance formulas are derived as convex combinations of uniform cases.

Abstract

We consider Erd\H{o}s-R\'enyi-type random hypergraphs that are non-uniform, in the sense that hyperedges of different sizes may coexist, and inhomogeneous, in that connection probabilities may depend on the hyperedge size. All parameters are allowed to scale with the hypergraph size. We study the random adjacency matrix whose $(u,v)$-entry counts the number of hyperedges containing both vertices $u$ and $v$, and characterize its expected limiting spectral distribution in terms of the connection probabilities and the hyperedge sizes. We provide a Pastur-type condition, in the sense of Chatterjee (2005), under which the matrix can be Gaussianized, as well as a more restrictive but simpler sufficient condition in terms of the generalized average degree of the model. As a second main result, based on such a Gaussianization, we characterize the limiting spectral distributions under non-sparse conditions as semicircle laws with an explicit parametric variance. The latter can be expressed as a convex combination of the variances arising in the uniform cases, with coefficients determined by the trade-off between the different sources of inhomogeneity.