The multiplex decomposition: An analytic framework for multilayer dynamical networks

TL;DR

This paper introduces an analytic framework for multiplex networks with simultaneously triagonalizable adjacency matrices, linking their spectra to individual layers and applying it to stability analysis of synchronized states.

Contribution

It develops a spectral relation for a special class of multiplex networks and applies it to stability analysis, including explicit conditions for linear diffusive systems.

Findings

Interlayer interactions can stabilize or destabilize synchronization.

Spectral relations simplify stability analysis in multiplex networks.

Explicit stability conditions are derived for linear diffusive systems.

Abstract

Multiplex networks are networks composed of multiple layers such that the number of nodes in all layers is the same and the adjacency matrices between the layers are diagonal. We consider the special class of multiplex networks where the adjacency matrices for each layer are simultaneously triagonalizable. For such networks, we derive the relation between the spectrum of the multiplex network and the eigenvalues of the individual layers. As an application, we propose a generalized master stability approach that allows for a simplified, low-dimensional description of the stability of synchronized solutions in multiplex networks. We illustrate our result with a duplex network of FitzHugh-Nagumo oscillators. In particular, we show how interlayer interaction can lead to stabilization or destabilization of the synchronous state. Finally, we give explicit conditions for the stability of synchronous solutions in duplex networks of linear diffusive systems.