The ability to steer the state of a dynamical network towards a desired state within a time horizon is intrinsically dependent on the number of driven nodes considered, as well as the network’s topology. The trade-off between time-to-control and the minimum number of driven nodes is captured by the notion of the actuation spectrum (AS). We study the actuation spectra of a variety of artificial and real-world networked systems, modeled by fractional-order dynamics that are capable of capturing non-Markovian time properties with power-law dependencies. We find evidence that, in both types of networks, the actuation spectra are similar when the time-to-control is less or equal to about 1/5 of the size of the network. Nonetheless, for a time-to-control larger than the network size, the minimum number of driven nodes required to attain controllability in networks with fractional-order dynamics may still decrease in comparison with other networks with Markovian properties. These differences suggest that the minimum number of driven nodes can be used to determine the true dynamical nature of the network. Furthermore, such differences also suggest that new generative models are required to reproduce the actuation spectra of real fractional-order dynamical networks.

中文翻译：

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具有幂律时间依赖性的时空网络的驱动谱

在时间范围内将动态网络的状态引导至所需状态的能力本质上取决于所考虑的驱动节点的数量以及网络的拓扑结构。控制时间和驱动节点的最小数量之间的权衡由驱动谱 (AS) 的概念捕获。我们研究了各种人工和现实世界网络系统的驱动谱，这些系统由分数阶动力学建模，能够捕获具有幂律依赖性的非马尔可夫时间属性。我们发现证据表明，在这两种类型的网络中，当控制时间小于或等于网络大小的大约 1/5 时，驱动谱是相似的。尽管如此，对于大于网络规模的控制时间，与其他具有马尔可夫性质的网络相比，在具有分数阶动力学的网络中实现可控性所需的最小驱动节点数可能仍会减少。这些差异表明，驱动节点的最小数量可用于确定网络的真实动态特性。此外，这些差异还表明，需要新的生成模型来再现真实分数阶动态网络的驱动谱。