Spectral bounds of the regularized normalized Laplacian for random geometric graphs

TL;DR

This paper investigates the spectral properties of the regularized normalized Laplacian in random geometric graphs, providing convergence results and explicit eigenvalue approximations in different connectivity regimes.

Contribution

It establishes the limiting eigenvalue distribution for RGGs and offers an explicit approximation in the thermodynamic regime, linking RGG spectra to deterministic geometric graphs.

Findings

LED converges to a Dirac measure in the connectivity regime

An explicit eigenvalue approximation is provided for the thermodynamic regime

Bound on Levy distance quantifies the approximation accuracy

Abstract

In this work, we study the spectrum of the regularized normalized Laplacian for random geometric graphs (RGGs) in both the connectivity and thermodynamic regimes. We prove that the limiting eigenvalue distribution (LED) of the normalized Laplacian matrix for an RGG converges to the Dirac measure in one in the full range of the connectivity regime. In the thermodynamic regime, we propose an approximation for the LED and we provide a bound on the Levy distance between the approximation and the actual distribution. In particular, we show that the LED of the regularized normalized Laplacian matrix for an RGG can be approximated by the LED of the regularized normalized Laplacian for a deterministic geometric graph with nodes in a grid (DGG). Thereby, we obtain an explicit approximation of the eigenvalues in the thermodynamic regime.