Multilayer directed random networks: Scaling of spectral properties

TL;DR

This study analyzes the spectral properties of multilayer directed random networks using random matrix theory, revealing a universal scaling law for eigenfunction localization and related spectral measures across different network configurations.

Contribution

It introduces a universal scaling law for eigenfunction localization in multilayer directed networks, linking spectral properties to network parameters within a random matrix theory framework.

Findings

Eigenfunction localization length follows a simple scaling law.

Spectral measures scale with the effective bandwidth and network size.

Universal behavior observed across different multilayer network setups.

Abstract

Motivated by the wide presence of multilayer networks in both natural and human-made systems, within a random matrix theory (RMT) approach, in this study we compute eigenfunction and spectral properties of multilayer directed random networks (MDRNs) in two setups composed by $M$ layers of size $N$: A line and a complete graph (node-aligned multiplex network). First, we numerically demonstrate that the normalized localization length $\beta$ of the eigenfunctions of MDRNs follows a simple scaling law given by $\beta=x^*/(1+x^*)$, with $x^*\propto (b_{\rm eff}^2/L)^\delta$, $\delta\sim 1$ and $b_{\rm eff}$ being the effective bandwidth of the adjacency matrix of the network of size $L=M\times N$. Here, $b_{\rm eff}$ incorporates both intra- and inter-layer edges. Then, we show that other eigenfunction and spectral RMT measures (the inverse participation ratio of eigenfunctions, the ratio between nearest- and next-to-nearest-neighbor eigenvalue distances, and the ratio between consecutive singular-value spacings) of MDRNs also scale with $x^*$.