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Synchronization phenomena are of broad interest across disciplines andincreasingly of interest in a multiplex network setting. Here we show how theMaster Stability Function, a celebrated framework for analyzing synchronizationon a single network, can be extended to certain classes of multiplex networkswith different intra-layer and inter-layer coupling functions. We derive threemaster stability equations that determine respectively the necessary regions ofcomplete synchronization, intra-layer synchronization and inter-layersynchronization. We calculate these three regions explicitly for the case of atwo-layer network of R{\"o}ssler oscillators and show that the overlap of theregions determines the type of synchronization achieved. In particular, if theinter- or intra-layer coupling function is such that the inter-layer orintra-layer synchronization region is empty, complete synchronization cannot beachieved regardless of the coupling strength. Furthermore, for any given nodaldynamics and network structure, the occurrence of intra-layer and inter-layersynchronization depend mainly on the coupling functions of nodes within a layerand across layers, respectively. Our mathematical analysis requires that theintra- and inter-layer supra-Laplacians commute. But we show this is only asufficient, and not necessary, condition and that the results can be appliedmore generally.

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