Spectral Analysis of the Adjacency Matrix of Random Geometric Graphs

TL;DR

This paper investigates the eigenvalue distribution of adjacency matrices in random geometric graphs, showing convergence to a deterministic limit and providing eigenvalue approximation methods in the connectivity regime.

Contribution

It establishes the limiting eigenvalue distribution of RGGs in the connectivity regime and introduces a method to approximate eigenvalues using a deterministic geometric graph model.

Findings

Eigenvalue distribution converges to that of a deterministic geometric graph.

Provided bounds for eigenvalue approximation errors.

Applicable in the connectivity regime where average degree scales with log(n).

Abstract

In this article, we analyze the limiting eigenvalue distribution (LED) of random geometric graphs (RGGs). The RGG is constructed by uniformly distributing $n$ nodes on the $d$-dimensional torus $\mathbb{T}^d \equiv [0, 1]^d$ and connecting two nodes if their $\ell_{p}$-distance, $p \in [1, \infty]$ is at most $r_{n}$. In particular, we study the LED of the adjacency matrix of RGGs in the connectivity regime, in which the average vertex degree scales as $\log\left( n\right)$ or faster, i.e., $\Omega \left(\log(n) \right)$. In the connectivity regime and under some conditions on the radius $r_{n}$, we show that the LED of the adjacency matrix of RGGs converges to the LED of the adjacency matrix of a deterministic geometric graph (DGG) with nodes in a grid as $n$ goes to infinity. Then, for $n$ finite, we use the structure of the DGG to approximate the eigenvalues of the adjacency matrix of the RGG and provide an upper bound for the approximation error.