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 β of the eigenfunctions of MDRNs follows a simple scaling law given by β=x∗/(1+x∗), where x∗ 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∗. We validate our results on real-world networks.