  In the real world, many complex systems are represented not by single
networks but rather by sets of interdependent ones. In these specific networks,
nodes in one network mutually interact with nodes in other networks. This paper
focuses on a simple representative case of two-layer networks (the so-called
duplex networks) with unidirectional inter-layer couplings. That is, each node
in one network depends on a counterpart in the other network. Accordingly, the
former network is called the response layer and the latter network is the drive
layer. Specifically, synchronization between each node in the drive layer and
its counterpart in the response layer (counterpart synchronization, or CS) of
this sort of duplex networks with delayed nodes and noise perturbation is
investigated. Based on the LaSalle-type invariance principle, a control
technique is proposed and a sufficient condition is developed for realizing
counterpart synchronization of duplex networks. Furthermore, two corollaries
are derived as special cases. In addition, node dynamics within each layer can
be various and topologies of the two layers are not necessarily identical.
Therefore, the proposed synchronization method can be applied to a wide range
of multiplex networks. Numerical examples are provided to illustrate the
feasibility and effectiveness of the results.
