The different localisation properties of the eigenmodes of the Laplacian and adjacency matrix of 2D random geometric graphs

TL;DR

This paper compares the spectral and localization properties of eigenmodes of Laplacian and adjacency matrices in 2D random geometric graphs, revealing fundamental differences in their localization behaviors.

Contribution

It provides a detailed analysis of how eigenmodes of these matrices differ in localization, highlighting the impact of system size, connectivity, and network motifs.

Findings

All adjacency eigenmodes are localized for large systems.

Laplacian eigenmodes include a small proportion of system-spanning modes.

Power-law tails observed in participation ratio distributions.

Abstract

We compare the spectrum and the localisation properties of the eigenmodes of the Laplacian and the adjacency matrix of 2D random geometric graphs, using numerical diagonalization of these matrices for different system sizes and connectivities. For sufficiently large ensembles of systems, we evaluate the spectrum, the probability distribution of the participation ratio and the relation between participation ratios and eigenvalues. While all eigenmodes of the adjacency matrix are localised for sufficiently large system sizes, the Laplacian matrix always leads to a small proportion of system-spanning modes due to a conservation law, and therefore to power-law tails in the probability distribution of the participation ratio and its relation to the eigenvalues. By disentangling the effects of finite system size, of mean degree, of component size distribution, and of network motifs, we provide a thorough understanding of the data.