Spectral statistics of random geometric graphs

TL;DR

This paper applies random matrix theory to analyze the spectral properties of random geometric graphs, revealing a parameter-dependent transition in spectral statistics that aligns with universal behaviors observed in other random graph models.

Contribution

It introduces a spectral analysis framework for random geometric graphs using random matrix theory, identifying a transition between Poisson and GOE statistics.

Findings

Spectral statistics exhibit a transition from Poisson to GOE with parameter variation.

Level spacings reveal eigenvector localization and community structure.

Spectral behavior aligns with universality in other random graph models.

Abstract

We use random matrix theory to study the spectrum of random geometric graphs, a fundamental model of spatial networks. Considering ensembles of random geometric graphs we look at short range correlations in the level spacings of the spectrum via the nearest neighbour and next nearest neighbour spacing distribution and long range correlations via the spectral rigidity Delta_3 statistic. These correlations in the level spacings give information about localisation of eigenvectors, level of community structure and the level of randomness within the networks. We find a parameter dependent transition between Poisson and Gaussian orthogonal ensemble statistics. That is the spectral statistics of spatial random geometric graphs fits the universality of random matrix theory found in other models such as Erdos-Renyi, Barabasi-Albert and Watts-Strogatz random graph.