Weighted random--geometric and random--rectangular graphs: Spectral and eigenfunction properties of the adjacency matrix

TL;DR

This study investigates the spectral and eigenfunction properties of weighted random geometric and rectangular graphs using random matrix theory, revealing universal behaviors and scaling laws dependent on geometric and connection parameters.

Contribution

It introduces a detailed analysis of spectral properties of weighted RGGs and RRGs, identifying universal scaling laws and the influence of geometry and connection radius.

Findings

Spectral properties depend on the ratio r/N^γ with γ ≈ -1/2.

Universal spectral and eigenfunction behaviors emerge for large a when r/(a N^γ) is fixed.

Scaling laws connect geometric parameters to spectral characteristics.

Abstract

Within a random-matrix-theory approach, we use the nearest-neighbor energy level spacing distribution $P(s)$ and the entropic eigenfunction localization length $\ell$ to study spectral and eigenfunction properties (of adjacency matrices) of weighted random--geometric and random--rectangular graphs. A random--geometric graph (RGG) considers a set of vertices uniformly and independently distributed on the unit square, while for a random--rectangular graph (RRG) the embedding geometry is a rectangle. The RRG model depends on three parameters: The rectangle side lengths $a$ and $1/a$, the connection radius $r$, and the number of vertices $N$. We then study in detail the case $a=1$ which corresponds to weighted RGGs and explore weighted RRGs characterized by $a\sim 1$, i.e.~two-dimensional geometries, but also approach the limit of quasi-one-dimensional wires when $a\gg1$. In general we look for the scaling properties of $P(s)$ and $\ell$ as a function of $a$, $r$ and $N$. We find that the ratio $r/N^\gamma$, with $\gamma(a)\approx -1/2$, fixes the properties of both RGGs and RRGs. Moreover, when $a\ge 10$ we show that spectral and eigenfunction properties of weighted RRGs are universal for the fixed ratio $r/{\cal C}N^\gamma$, with ${\cal C}\approx a$.