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

Per Arne Rikvold

Home/Archive/arXiv

arXiv · 2604.26599

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

Per Arne Rikvold

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cond-mat.dis-nn

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Per Arne Rikvold. 2026-04-29. Effective length scales, dispersion relations, and discrete densities of states for Laplacian eigenvectors on complex networks. https://arxiv.org/abs/2604.26599

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

R-transforms for non-Hermitian matrices: a spherical integral approach

In this paper, we establish a connection between the formalism of $\mathcal{R}$-transforms for non-Hermitian random matrices and the framework of spherical integrals, using the replica method. This connection was previously proved in the Hermitian setting and in the case of bi-invariant random matrices. We show that the $\mathcal{R}$-transforms used in the non-Hermitian context in fact originate from a single scalar function of two variables. This provides a new and transparent way to compute $\mathcal{R}$-transforms, which until now had been known only in restricted cases such as bi-invariant, Hermitian, or elliptic ensembles.

cond-mat.dis-nn↗

arXiv2026

Spectral boundaries of deterministic matrices deformed by rotationally invariant random non-Hermitian ensembles

One of the great miracles of random matrix theory is that, in the $N \to \infty$ limit, many otherwise intractable matrix problems with horrendously complicated finite-$N$ expressions admit remarkably simple and elegant asymptotic solutions. In this paper, we illustrate this phenomenon in the context of spectral boundaries (or spectral edges) for deformed random matrices. Specifically, we consider matrices of the form $\mathbf{A} + \mathbf{B}$, where $\mathbf{A}$ is a deterministic $N\times N$ matrix (not necessarily Hermitian) and $\mathbf{B}$ is a rotationally invariant random matrix. In the large-$N$ limit, we show that the complex eigenvalue distribution of $\mathbf{A} + \mathbf{B}$ satisfies remarkably simple boundary equations that depend on the $\mathcal{R}_1$ and $\mathcal{R}_2$ transforms of $\mathbf{B}$. We illustrate our results on several explicit random matrix ensembles and support them with numerical simulations.

cond-mat.dis-nn↗

arXiv2026

Electrical conductivity of crack-template-based transparent conducting films: mean-field approximation, effective-medium theory, and simulation

In this work, crack-template-based transparent conducting films were modeled as networks corresponding to the edges of a two-dimensional Poisson--Voronoi diagram. Two types of networks were considered: the original one, in which the conductance of each edge was inversely proportional to its length, and the effective one, in which all edges had the same conductance obtained from the effective-medium theory. The mean-field approximation was used for analytical evaluation of the electrical conductivity. Direct numerical calculations for the Poisson--Voronoi diagram showed that the mean-field approximation overestimated the effective conductivity of the original network by approximately 13\%, and of the effective network by 79\%. In addition, a honeycomb network with an edge conductance distribution corresponding to the Poisson--Voronoi diagram was studied: for it, the predictions of the effective-medium theory turned out to be more accurate than for the Poisson--Voronoi diagram, which was explained by the greater structural homogeneity of the periodic honeycomb lattice. The results indicate that, when modeling crack-template-based transparent conducting films, the application of the mean-field approximation may lead to significant errors if the resistance of individual conductors is not simply proportional to their length. This possibility is discussed as a motivation for future studies of hierarchical cracks with variable width, which are not directly investigated here.

cond-mat.dis-nn↗