Master stability functions for complete, intra-layer and inter-layer synchronization in multiplex networks

TL;DR

This paper extends the Master Stability Function framework to multiplex networks, deriving conditions for complete, intra-layer, and inter-layer synchronization, and analyzing their regions for Rössler oscillator networks.

Contribution

It introduces a generalized MSF approach for multiplex networks with different coupling functions, including explicit calculations and conditions for various synchronization types.

Findings

Regions for synchronization types are explicitly calculated.

Overlap of regions determines the achieved synchronization.

Synchronization depends mainly on coupling functions within and across layers.

Abstract

Synchronization phenomena are of broad interest across disciplines and increasingly of interest in a multiplex network setting. Here we show how the Master Stability Function, a celebrated framework for analyzing synchronization on a single network, can be extended to certain classes of multiplex networks with different intra-layer and inter-layer coupling functions. We derive three master stability equations that determine respectively the necessary regions of complete synchronization, intra-layer synchronization and inter-layer synchronization. We calculate these three regions explicitly for the case of a two-layer network of R{\"o}ssler oscillators and show that the overlap of the regions determines the type of synchronization achieved. In particular, if the inter- or intra-layer coupling function is such that the inter-layer or intra-layer synchronization region is empty, complete synchronization cannot be achieved regardless of the coupling strength. Furthermore, for any given nodal dynamics and network structure, the occurrence of intra-layer and inter-layer synchronization depend mainly on the coupling functions of nodes within a layer and across layers, respectively. Our mathematical analysis requires that the intra- and inter-layer supra-Laplacians commute. But we show this is only a sufficient, and not necessary, condition and that the results can be applied more generally.