Spectral Analysis of the Adjacency Matrix of Random Geometric Graphs

Abstract

In this article, we analyze the limiting eigen-value distribution (LED) of random geometric graphs (RGGs). The RGG is constructed by uniformly distributing n nodes on the d-dimensional torus $\Gamma^{d}\equiv [0, 1]^{d}$ 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 $\log(n)$ or faster, i.e., $\Omega(\log(n))$. 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. Index Terms--Random geometric graphs, adjacency matrix, limiting eigenvalue distribution, Levy distance.