Spectra of high-dimensional sparse random geometric graphs

TL;DR

This paper studies the spectral properties of high-dimensional sparse random geometric graphs, showing convergence to the semicircle law under certain conditions and improving bounds on eigenvalues relevant for synchronization phenomena.

Contribution

It introduces a novel recursive method to analyze spectral distributions of sparse high-dimensional geometric graphs, extending results to regimes previously unexplored.

Findings

Spectral distribution converges to the semicircle law in high dimensions.

Eigenvalue bounds are improved for sparse regimes, aiding synchronization analysis.

The method combines classical moment techniques with graph decomposition strategies.

Abstract

We analyze the spectral properties of the high-dimensional random geometric graph $G(n, d, p)$, formed by sampling $n$ i.i.d vectors $\{v_i\}_{i=1}^{n}$ uniformly on a $d$-dimensional unit sphere and connecting each pair $\{i,j\}$ whenever $\langle v_i, v_j \rangle \geq \tau$ so that $p=\mathbb P(\langle v_i,v_j\rangle \geq \tau)$. This model defines a nonlinear random matrix ensemble with dependent entries. We show that if $d =\omega( np\log^{2}(1/p))$ and $np\to\infty$, the limiting spectral distribution of the normalized adjacency matrix $\frac{A}{\sqrt{np(1-p)}}$ is the semicircle law. To our knowledge, this is the first such result for $G(n, d, p)$ in the sparse regime. In the constant sparsity case $p=\alpha/n$, we further show that if $d=\omega(\log^2(n))$ the limiting spectral distribution of $A$ in $G(n,\alpha/n)$ coincides with that of the Erd\H{o}s-R\'{e}nyi graph $G(n,\alpha/n)$. Our approach combines the classical moment method in random matrix theory with a novel recursive decomposition of closed-walk graphs, leveraging block-cut trees and ear decompositions, to control the moments of the empirical spectral distribution. A refined high trace analysis further yields a near-optimal bound on the second eigenvalue when $np=\Omega(\log^4 (n))$, removing technical conditions previously imposed in (Liu et al. 2023). As an application, we demonstrate that this improved eigenvalue bound sharpens the parameter requirements on $d$ and $p$ for spontaneous synchronization on random geometric graphs in (Abdalla et al. 2024) under the homogeneous Kuramoto model.