Spectral Analysis of the Adjacency Matrix of Random Geometric Graphs

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.