Synchronization in a network of connected elements is essential to the proper functioning of both natural and engineered systems and is thus of increasing interest across disciplines. In many cases, synchronization phenomena involve not just actions within a single network in isolation, but the coordinated and coherent behaviors of several networks interacting with each other. The interactions between multiple systems play a crucial role in determining the emergent dynamics. One paradigm capable of representing real-world complex systems is that of multiplex networks, where the same set of nodes exists in multiple layers of the network. Recent studies have made significant progress in understanding synchronization in multiplex networks. In this review, we primarily focus on two key aspects: structural complexity and dynamical complexity. From the perspective of structural complexity, we present how the topological setting, such as the interlayer coupling pattern, affects the synchronizability of a multiplex network. The structural characteristics of a multiplex network, in particular, give rise to dynamical complexity, including the emergence of intralayer synchronization (within each layer) and interlayer synchronization (between layers). We also discuss the major methods for studying the stability of complete, intralayer, and interlayer synchronization, as well as synchronization control in multiplex networks. Additionally, we briefly introduce some relevant applications. Lastly, the review provides a comprehensive summary of the notable findings in the study of synchronization in multiplex networks, emphasizing the interplay between their structural and dynamical complexities, and identifies open problems that present opportunities for future research in this field.

中文翻译：

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多路复用网络中的同步

连接元素网络中的同步对于自然系统和工程系统的正常运行至关重要，因此跨学科的兴趣日益浓厚。在许多情况下，同步现象不仅涉及单个网络内孤立的行为，还涉及多个网络相互作用的协调一致的行为。多个系统之间的相互作用在确定新兴动态方面发挥着至关重要的作用。能够表示现实世界复杂系统的一种范例是多路复用网络，其中同一组节点存在于网络的多个层中。最近的研究在理解多路网络同步方面取得了重大进展。在这篇综述中，我们主要关注两个关键方面：结构复杂性和动态复杂性。从结构复杂性的角度来看，我们提出了拓扑设置（例如层间耦合模式）如何影响复用网络的同步性。特别是，多路复用网络的结构特征引起了动态复杂性，包括层内同步（每层内）和层间同步（层间）的出现。我们还讨论了研究完全同步、层内同步和层间同步稳定性以及复用网络中的同步控制的主要方法。此外，我们还简要介绍了一些相关应用。最后，该综述全面总结了多路网络同步研究的显着发现，强调其结构和动态复杂性之间的相互作用，并确定了为该领域未来研究提供机会的开放问题。