Intralayer synchronization in evolving multiplex hypernetworks: analytical approach

Abstract

In this paper, we study intralayer synchronization of multiplex networks where nodes in each layer interact through diverse types of coupling functions associated with different time-varying network topologies, referred to as multiplex hypernetworks. Here, the intralayer connections are evolving with respect to time, and the interlayer connections are stagnant. In this context, an interesting and important problem is to analyze the stability of the intralayer synchronization in such temporal networks. We prove that if the dynamical multiplex hypernetwork for the time-average topology possesses intralayer synchronization, then each layer of the time-varying multiplex hypernetwork will also be synchronized for sufficiently fast switching. Then through master stability function formalism, we analytically derive necessary and sufficient stability conditions of intralayer synchronous states for such temporal architecture in terms of a time-average network. In this regard, we are able to decouple the transverse error component of the intralayer synchronization states for some special cases. Also, we extend our study for nonlinear intralayer coupling functions as well as multilayer hypernetwork architectures. Finally, the theoretical findings are verified numerically by taking the network of paradigmatic chaotic R\"ossler oscillators.