As the monoplex network has its limitations in describing the real world, a new framework called the multiplex network is put forward and has received much attention in recent years. This paper focuses on synchronization of the multiplex network with multiple delays and stochastic perturbations. Due to the complexity, pinning control of the multiplex network is of particular interest. Based on the LaSalle-type invariance principle for stochastic differential delay equations and the Lyapunov stability theory, some control schemes and synchronization criteria are obtained. It is concluded that under some mild conditions, one can determine which nodes should be pinned in a multiplex network. In addition, it is found that the number of pinned nodes increases with the varying interval of noise and time delay, and decreases with the varying interval of intralayer coupling strength. Some two-layer and three-layer networks are employed to validate the effectiveness of the proposed algorithm.

中文翻译：

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具有随机扰动的多重延迟网络的钉扎同步


由于单工网络在描述现实世界方面有其局限性，一种称为多路复用网络的新框架被提出，并近年来受到广泛关注。本文重点研究具有多重延迟和随机扰动的多路网络的同步。由于其复杂性，多路复用网络的固定控制特别令人感兴趣。基于随机微分时滞方程的LaSalle型不变性原理和Lyapunov稳定性理论，得到了一些控制方案和同步判据。结论是，在一些温和的条件下，我们可以确定哪些节点应该固定在多路复用网络中。此外，还发现，钉扎节点的数量随着噪声和时延间隔的变化而增加，随着层内耦合强度间隔的变化而减少。采用一些两层和三层网络来验证所提算法的有效性。