Spectra of high-dimensional sparse random geometric graphs
Yifan Cao, Yizhe Zhu
TL;DR
This work analyzes the spectrum of high-dimensional, sparse random geometric graphs $\mathcal{G}(n,d,p)$ formed by thresholding inner products of random sphere points. It proves a universal semicircle law for the centered/adaptively scaled adjacency matrix in the regime $d=\omega(np\log^2(1/p))$, $np\to\infty$, and establishes a matching limiting spectrum $\nu_{\alpha}$ for $p=\alpha/n$ when $d=\omega(\log^2 n)$, showing universality with Erdős–Rényi graphs. The authors develop a novel moment method with a recursive decomposition based on block-cut trees and ear decompositions to control non-tree closed walks, enabling near-optimal second-eigenvalue bounds $\lambda(A)=O(\log^4 n\sqrt{np}+\tau np)$ and improving sparsity/dimensional thresholds for spontaneous synchronization of the Kuramoto model on these graphs. They also provide an application to synchronization, removing prior technical constraints and tightening parameter regimes. Overall, the paper advances understanding of how geometry-induced dependencies influence global spectral behavior and dynamic processes on random graphs in high dimensions.
Abstract
We analyze the spectral properties of the high-dimensional random geometric graph $\mathcal G(n, d, p)$, 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 $\langle v_i, v_j \rangle \geq τ$ so that $p=\mathbb P(\langle v_i,v_j\rangle \geq τ)$. This model defines a nonlinear random matrix ensemble with dependent entries. We show that if $d =ω( np\log^{2}(1/p))$ 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 $p=α/n$, we further show that if $d=ω(\log^2(n))$ the limiting spectral distribution of $A$ in $G(n,α/n)$ coincides with that of the Erdős-Rényi graph $\mathcal G(n,α/n)$. 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 $np=Ω(\log^4 (n))$, 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.