Localized and delocalized modes on random geometric graphs in 1D

TL;DR

This paper investigates the localization properties of eigenmodes in 1D random geometric graphs, analyzing how various factors influence eigenvector localization and comparing results to ordered graphs and tight-binding models.

Contribution

It provides a comprehensive analysis of eigenmode localization in 1D random geometric graphs, considering multiple influencing factors and comparisons to related models.

Findings

Eigenmode localization depends on system size and graph structure.

Participation ratio distribution varies with eigenvalue.

Comparison reveals similarities and differences with ordered graphs.

Abstract

We perform an extensive investigation of the localization properties of the eigenmodes of the Laplace and adjacency matrix for one-dimensional random geometric graphs. We evaluate the density of states, the probability distribution of the participation ratio and its relation to the eigenvalue. By disentangling the influence of system size, graph component size distribution, mean degree of nodes, network motifs, and degeneracy, we provide a comprehensive understanding of this system. We compare our findings to ordered graphs with the same mean degree and to one-dimensional tight-binding models.