Multiplex networks have been intensively studied during the last few years as they offer a more realistic representation of many interdependent and multilevel complex networked systems. However, even if most real networks have some degree of directionality, the vast majority of the existent literature deals with multiplex networks where all layers are undirected. Here, we study the dynamics of diffusion processes acting on coupled multilayer networks where at least one layer consists of a directed graph; we call these directed multiplex networks. We reveal a new and unexpected signature of diffusion dynamics on directed multiplex networks, namely, that different from their undirected counterparts, they can exhibit a nonmonotonic rate of convergence to steady state as a function of the degree of coupling, resulting in a faster diffusion at an intermediate degree of coupling than when the two layers are fully coupled. We use synthetic multiplex examples and real-world topologies to illustrate the characteristics of the underlying dynamics that give rise to a regime in which an optimal coupling exists. We further provide analytical and numerical evidence that this new phenomenon is solely a property of directed multiplex, where at least one of the layers exhibits sufficient directionality quantified by a normalized metric of asymmetry in directional path lengths. Given the ubiquity of both directed and multilayer networks in nature, our results have important implications for studying the dynamics of multilevel complex systems.

中文翻译：

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有向层多重网络中的扩散动力学和最优耦合

在过去的几年中，对复用网络进行了深入的研究，因为它们提供了许多相互依存和多层次的复杂联网系统的更真实的表示。然而，即使大多数真实网络具有一定程度的方向性，现有文献中的绝大多数还是涉及所有层都不定向的多路复用网络。在这里，我们研究了作用于耦合多层网络的扩散过程的动力学，其中至少一层由有向图组成；我们称这些定向多路复用网络。我们揭示了有向多重网络上扩散动力学的一个新的和出乎意料的特征，即与无向对应网络不同，它们可以表现出非单调收敛到稳态的速率，该速率是耦合度的函数，从而导致扩散速度更快。与两层完全耦合时相比，耦合程度为中等。我们使用合成的多重示例和现实世界的拓扑来说明潜在动力学的特征，这些动力学导致了存在最佳耦合的机制。我们进一步提供分析和数值证据，表明这一新现象仅仅是定向多路复用的一种特性，其中至少一层具有足够的方向性通过定向路径长度中的归一化不对称度量来量化。考虑到自然界中定向网络和多层网络的普遍存在，我们的结果对研究多层复杂系统的动力学具有重要意义。