多重多时标网络的可控性
Controllability of multiplex, multi-timescale networks
摘要：
分层网络的范例被用来描述许多现实世界的系统-从生物网络到社会组织和运输系统。近年来，人们对多层网络的一般性质的理解有了很大的进展，但对如何控制这类系统的理解仍然有限。使这一努力具有挑战性的一个方面是，每个层可以在不同的时间尺度上运行，因此我们不能直接应用单个网络的结构控制理论的标准思想。在这里，我们解决了控制多层和多时间尺度网络的问题，重点关注具有一对一层间耦合的两层复用网络。我们调查的情况下，当控制信号被施加到一层的节点。我们开发了一个理论的基础上不相交的路径覆盖，以确定完全控制所需的最小数量的输入（$N_\T i$）。我们表明，如果这两层在相同的时间尺度上运行，那么这两层的网络结构同样影响可控性。在时间尺度分离的情况下，如果控制器与较快层交互，则可控性增强：随着时间尺度差异增加到临界时间尺度差异，$N_\T i$减小，超过临界时间尺度差异，$N_\T i$保持恒定并且完全由较快层确定。相反，如果控制器与在较慢时标上操作的层交互，则控制变得越来越困难，并且增加时标分离导致增加的$N_\T i$，再次达到临界值，高于该临界值，$N_\T i$仍然取决于两个层的结构。通过识别一个简化模型的时间尺度差异和可控性连接的基本机制，我们提供了深入了解如何解开我们的能力，控制真正的相互作用的复杂系统是由各种来源的复杂性。
英文摘要：
The paradigm of layered networks is used to describe many real-world systems -- from biological networks, to social organizations and transportation systems. Recently there has been much progress in understanding the general properties of multilayer networks, our understanding of how to control such systems remains limited. One aspect that makes this endeavor challenging is that each layer can operate at a different timescale, thus we cannot directly apply standard ideas from structural control theory of individual networks. Here we address the problem of controlling multilayer and multi-timescale networks focusing on two-layer multiplex networks with one-to-one interlayer coupling. We investigate the case when the control signal is applied to the nodes of one layer. We develop a theory based on disjoint path covers to determine the minimum number of inputs ($N_\T i$) necessary for full control. We show that if both layers operate on the same timescale then the network structure of both layers equally affect controllability. In the presence of timescale separation, controllability is enhanced if the controller interacts with the faster layer: $N_\T i$ decreases as the timescale difference increases up to a critical timescale difference, above which $N_\T i$ remains constant and is completely determined by the faster layer. In contrast, control becomes increasingly difficult if the controller interacts with the layer operating on the slower timescale and increasing timescale separation leads to increased $N_\T i$, again up to a critical value, above which $N_\T i$ still depends on the structure of both layers. By identifying the underlying mechanisms that connect timescale difference and controllability for a simplified model, we provide insight into disentangling how our ability to control real interacting complex systems is affected by a variety of sources of complexity.
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