Eigenvalues and Spectral Dimension of Random Geometric Graphs in Thermodynamic Regime

TL;DR

This paper analytically determines the spectral dimension of random geometric graphs in the thermodynamic regime, revealing it approximates the space dimension through eigenvalue distribution analysis.

Contribution

It provides an analytical approximation for the eigenvalues of RGGs' Laplacian and links the spectral dimension to the space dimension in the thermodynamic limit.

Findings

Spectral dimension approximates the space dimension d

Eigenvalue distribution near minimum follows a power-law tail

Smallest non-zero eigenvalue converges to zero as graph size increases

Abstract

Network geometries are typically characterized by having a finite spectral dimension (SD), $d_{s}$ that characterizes the return time distribution of a random walk on a graph. The main purpose of this work is to determine the SD of a variety of random graphs called random geometric graphs (RGGs) in the thermodynamic regime, in which the average vertex degree is constant. The spectral dimension depends on the eigenvalue density (ED) of the RGG normalized Laplacian in the neighborhood of the minimum eigenvalues. In fact, the behavior of the ED in such a neighborhood characterizes the random walk. Therefore, we first provide an analytical approximation for the eigenvalues of the regularized normalized Laplacian matrix of RGGs in the thermodynamic regime. Then, we show that the smallest non zero eigenvalue converges to zero in the large graph limit. Based on the analytical expression of the eigenvalues, we show that the eigenvalue distribution in a neighborhood of the minimum value follows a power-law tail. Using this result, we find that the SD of RGGs is approximated by the space dimension $d$ in the thermodynamic regime.