Spectra of adjacency and Laplacian matrices of Erd\H{o}s-R\'{e}nyi hypergraphs

TL;DR

This paper analyzes the spectral properties of adjacency and Laplacian matrices of Erdős-Rényi hypergraphs, revealing convergence to semi-circle law, eigenvalue phase transitions, and distributional limits for large hypergraphs with varying uniformity.

Contribution

It provides the first detailed spectral analysis of hypergraph adjacency matrices, including phase transitions and eigenvalue distributions, extending classical graph results to hypergraphs.

Findings

Expected spectral distribution converges to semi-circle law.

Eigenvalues exhibit phase transitions at specific hyperedge sizes.

Largest and smallest eigenvalues converge to predictable limits.

Abstract

We study adjacency and Laplacian matrices of Erd\H{o}s-R\'{e}nyi $r$-uniform hypergraphs on $n$ vertices with hyperedge inclusion probability $p$, in the setting where $r$ can vary with $n$ such that $r / n \to c \in [0, 1)$. Adjacency matrices of hypergraphs are contractions of adjacency tensors and their entries exhibit long range correlations. We show that under the Erd\H{o}s-R\'{e}nyi model, the expected empirical spectral distribution of an appropriately normalised hypergraph adjacency matrix converges weakly to the semi-circle law with variance $(1 - c)^2$ as long as $\frac{d_{\avg}}{r^7} \to \infty$, where $d_{\avg} = \binom{n-1}{r-1} p$. In contrast with the Erd\H{o}s-R\'{e}nyi random graph ($r = 2$), two eigenvalues stick out of the bulk of the spectrum. When $r$ is fixed and $d_{\avg} \gg n^{r - 2} \log^4 n$, we uncover an interesting Baik-Ben Arous-P\'{e}ch\'{e} (BBP) phase transition at the value $r = 3$. For $r \in \{2, 3\}$, an appropriately scaled largest (resp. smallest) eigenvalue converges in probability to $2$ (resp. $-2$), the right (resp. left) end point of the support of the standard semi-circle law, and when $r \ge 4$, it converges to $\sqrt{r - 2} + \frac{1}{\sqrt{r - 2}}$ (resp. $-\sqrt{r - 2} - \frac{1}{\sqrt{r - 2}}$). Further, in a Gaussian version of the model we show that an appropriately scaled largest (resp. smallest) eigenvalue converges in distribution to $\frac{c}{2} \zeta + \big[\frac{c^2}{4}\zeta^2 + c(1 - c)\big]^{1/2}$ (resp. $\frac{c}{2} \zeta - \big[\frac{c^2}{4}\zeta^2 + c(1 - c)\big]^{1/2}$), where $\zeta$ is a standard Gaussian. We also establish analogous results for the bulk and edge eigenvalues of the associated Laplacian matrices.