Scaling Properties of Multilayer Random Networks

TL;DR

This paper uncovers a simple scaling law for the localization length of eigenfunctions in multilayer random networks, aiding understanding of their critical properties and eigenfunction localization.

Contribution

It introduces a novel scaling law for eigenfunction localization length in multilayer random networks based on numerical analysis.

Findings

Eigenfunction localization length follows a universal scaling law

The scaling law relates localization length to network bandwidth and size

Results may improve understanding of critical phenomena in multilayer networks

Abstract

Multilayer networks are widespread in natural and manmade systems. Key properties of these networks are their spectral and eigenfunction characteristics, as they determine the critical properties of many dynamics occurring on top of them. In this paper, we numerically demonstrate that the normalized localization length $\beta$ of the eigenfunctions of multilayer random networks follows a simple scaling law given by $\beta=x^*/(1+x^*)$, with $x^*=\gamma(b_{\text{eff}}^2/L)^\delta$, $\gamma,\delta\sim 1$ and $b_{\text{eff}}$ being the effective bandwidth of the adjacency matrix of the network, whose size is $L=M\times N$. The reported scaling law for $\beta$ might help to better understand criticality in multilayer networks as well as to predict the eigenfunction localization properties of them.