Counterpart synchronization of duplex networks with delayed nodes and noise perturbation

Abstract

Abstract. 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.

Synchronizability of Multilayer Networks With K-nearest-neighbor Topologies

“…In 2014, Aguirre et al revealed that the connector nodes between layers play an important role in the synchronizability of interconnected networks [19]. Further, intra-layer synchronization, inter-layer synchronization, counterpart synchronization, and generalized synchronization in multiplex networks have been investigated [20][21][22][23][24]. Tang et al proposed three necessary regions to describe the different types of coherent behaviors (such as complete synchronization, intra-layer synchronization, and inter-layer synchronization) in multiplex networks based on the master stability function method [25].…”

Mean-square heterogeneous synchronization of interdependent networks with stochastic disturbances

“…At present, many scholars have explored the synchronization of complex networks [16][17][18]. Due to the disturbances of networks, there have been many works in the field of synchronization for complex networks with stochastic disturbances [19][20][21][22][23][24][25]. In Ref.…”

“…In Ref. [24], the synchronization between each node in the drive layer and its counterpart in the response layer of this sort of duplex networks with delayed nodes and noise perturbation is explored. Though there is increasing research relating to the synchronization of interdependent networks [8,9,26], few people have explored the synchronization problem of interdependent networks with stochastic disturbances.…”

Adaptive Finite-Time Mixed Interlayer Synchronization of Two-Layer Complex Networks with Time-Varying Coupling Delay

“…Interlayer synchronization is also called counterpart synchronization [11] which describes how the nodes in one network behave coherently with the corresponding ones in other connected networks. This concept can be regarded as the development of synchronization for coupled driveresponse systems [12].…”

“…This kind of interlayer synchronization can be also understood as outer synchronization [13] between two coupled networks. In the past decade, it has gained considerable attention-see [11,[13][14][15][16][17][18][19][20] and the references therein. Some control schemes have been proposed to realize all kinds of outer synchronization such as complete outer synchronization [13][14][15][16][17], inverse outer synchronization [18], generalized outer synchronization [19], and finite-time outer synchronization [21].…”