No abstract

Eigenvalues statistics of various many-body systems have been widely
studied using the nearest neighbor spacing distribution under the random matrix
theory framework. Here, we numerically analyze eigenvalue ratio statistics of multiplex
networks consisting of directed Erd ̋os-R ́enyi random networks layers represented as,
first, weighted non-Hermitian random matrices and then weighted Hermitian random
matrices. We report that the multiplexing strength rules the behaviour of average
spacing ratio statistics for multiplexing networks represented by the non-Hermitian
and Hermitian matrices, respectively. Additionally, for both these representations
of the directed multiplex networks, the multiplexing strength appears as a guiding
parameter for the eigenvector delocalization of the entire system. These results could be important for driving dynamical processes in
several real-world multilayer systems, particularly, understanding significance of the
multiplexing in comprehending network properties.

The distribution of the ratios of consecutive eigenvalue spacings of random matrices has emerged as an important tool to study spectral properties of many-body systems. This article numerically investigates the eigenvalue ratios distribution of various model networks, namely, small-world, Erdős-Rényi random, and (dis)assortative random having a diagonal disorder in the corresponding adjacency matrices. Without any diagonal disorder, the eigenvalues ratio distribution of these model networks depict Gaussian orthogonal ensemble (GOE) statistics. Upon adding diagonal disorder, there exists a gradual transition from the GOE to Poisson statistics depending upon the strength of the disorder. The critical disorder (wc) required to procure the Poisson statistics increases with the randomness in the network architecture. We relate wc with the time taken by maximum entropy random walker to reach the steady-state. These analyses will be helpful to understand the role of eigenvalues other than the principal one for various network dynamics such as transient behaviour.

Eigenvalues statistics of various many-body systems have been widely studied using the nearest neighbor spacing distribution under the random matrix theory framework. Here, we numerically analyze eigenvalue ratio statistics of multiplex networks consisting of directed Erdős-Rényi random networks layers represented as, first, weighted non-Hermitian random matrices and then weighted Hermitian random matrices. We report that the multiplexing strength rules the behavior of average spacing ratio statistics for multiplexing networks represented by the non-Hermitian and Hermitian matrices, respectively. Additionally, for both these representations of the directed multiplex networks, the multiplexing strength appears as a guiding parameter for the eigenvector delocalization of the entire system. These results could be important for driving dynamical processes in several real-world multilayer systems, particularly, understanding the significance of multiplexing in comprehending network properties.