Effective length scales, dispersion relations, and discrete densities of states for Laplacian eigenvectors on complex networks

TL;DR

This paper introduces a method adapted from condensed-matter physics to estimate effective length scales of Laplacian eigenvectors on complex networks, aiding in understanding diffusion and oscillation processes.

Contribution

It applies a correlation length estimation method to various real and artificial networks, revealing localized and distributed eigenvectors and their dispersion relations.

Findings

Identified localized and distributed eigenvectors across different networks.

Provided graphical and tabular summaries of eigenvector properties and dispersion relations.

Demonstrated the method's applicability to diverse network types.

Abstract

To construct dispersion relations for diffusion or oscillation processes on random networks, it is necessary to obtain effective length scales for the eigenvectors of a graph Laplacian matrix, whose eigenvalues represent inverse time scales. For this purpose, we adapt a method originally introduced in condensed-matter physics to estimate correlation lengths for disordered materials as the ratio of volume to interface area [P. Debye, H.R. Anderson and H. Brumberger, J. Appl. Phys. 28, 679 (1957)]. In a graph setting of vertices connected by edges, we interpret this as the ratio of twice the total number of edges to the number of edges connecting vertices bearing values of different sign on the particular eigenvector. After describing the method and the necessary concepts in pedagogical detail, we apply it to nine different graphs representing natural and artificial networks, including two tree graphs without and with random shortcuts, the nervous system of a roundworm, a food web, a social network of dolphins, an electrical power grid, and a model porous material. The results identify both distributed and localized eigenvectors. They are given in graphical format showing example eigenvectors, dispersion relations, and discrete densities of states, as well as tables summarizing the main numerical results.