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In the real world, many complex systems are represented not by singlenetworks but rather by sets of interdependent ones. In these specific networks,nodes in one network mutually interact with nodes in other networks. This paperfocuses on a simple representative case of two-layer networks (the so-calledduplex networks) with unidirectional inter-layer couplings. That is, each nodein one network depends on a counterpart in the other network. Accordingly, theformer network is called the response layer and the latter network is the drivelayer. Specifically, synchronization between each node in the drive layer andits counterpart in the response layer (counterpart synchronization, or CS) ofthis sort of duplex networks with delayed nodes and noise perturbation isinvestigated. Based on the LaSalle-type invariance principle, a controltechnique is proposed and a sufficient condition is developed for realizingcounterpart synchronization of duplex networks. Furthermore, two corollariesare derived as special cases. In addition, node dynamics within each layer canbe various and topologies of the two layers are not necessarily identical.Therefore, the proposed synchronization method can be applied to a wide rangeof multiplex networks. Numerical examples are provided to illustrate thefeasibility and effectiveness of the results.

Counterpart synchronization of duplex networks with delayed nodes and noise perturbation