Multilayer directed random networks: Scaling of spectral properties

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 $\beta =x^{\ast }/(1+x^{\ast })$, where $x^{\ast }$ is a nontrivial function of $M$, $N$, and number of 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 $x^{\ast }$. We validate our results on real-world networks.