Random Matrix Analysis of Multiplex Networks

TL;DR

This paper explores how the spectral properties of multiplex networks change with varying multiplexing strength and randomness, revealing transitions from Poisson to GOE statistics that have implications for controlling complex systems.

Contribution

It demonstrates how multiplexing and randomness influence spectral statistics, showing that a small-world transition in one layer can induce GOE behavior in the entire multiplex network.

Findings

Spectra transition from superposition of two GOEs to a single GOE with increasing multiplexing strength.

Random rewiring in one layer can induce a Poisson to GOE transition in the entire multiplex.

Small-world transition in one layer suffices to produce GOE statistics for the whole multiplex.

Abstract

We investigate the spectra of adjacency matrices of multiplex networks under random matrix theory (RMT) framework. Through extensive numerical experiments, we demonstrate that upon multiplexing two random networks, the spectra of the combined multiplex network exhibit superposition of two Gaussian orthogonal ensemble (GOE)s for very small multiplexing strength followed by a smooth transition to the GOE statistics with an increase in the multiplexing strength. Interestingly, randomness in the connection architecture, introduced by random rewiring to 1D lattice, of at least one layer may govern nearest neighbor spacing distribution (NNSD) of the entire multiplex network, and in fact, can drive to a transition from the Poisson to the GOE statistics or vice versa. Notably, this transition transpires for a very small number of the random rewiring corresponding to the small-world transition. Ergo, only one layer being represented by the small-world network is enough to yield GOE statistics for the entire multiplex network. Spectra of adjacency matrices of underlying interaction networks have been contemplated to be related with dynamical behaviour of the corresponding complex systems, the investigations presented here have implications in achieving better structural and dynamical control to the systems represented by multiplex networks against structural perturbation in only one of the layers.