复杂网络的控制与观测谱
Spectrum of Controlling and Observing Complex Networks
摘要：
观察和控制复杂网络对于理解复杂的物理、生物和技术系统具有重要意义。最近的研究在识别传感器或驱动器节点方面取得了重要进展，通过这些节点我们可以观察或控制一个复杂系统。然而，由测量噪声引起的观测不确定性和控制所需的能量仍然是实际应用中的重大挑战。在这里，我们表明，控制能量和观测不确定性的状态空间的不同方向的变化强烈依赖于驱动节点的数量。特别是，我们发现，如果所有的节点直接驱动，控制是积极可行的，因为最大能量随系统大小呈次线性增加。然而，如果我们的目标是通过单个节点控制系统，在某些方向上的控制是积极禁止的，随着系统的大小呈指数增长。对于介于两者之间的情况，当驱动节点的数量增加时，最大能量呈指数衰减。我们在几个模型和真实网络中验证了我们的研究结果，得出了一系列描述控制能量的基本定律，这些定律共同加深了我们对复杂系统的理解。
英文摘要：
Observing and controlling complex networks are of paramount interest for understanding complex physical, biological and technological systems. Recent studies have made important advances in identifying sensor or driver nodes, through which we can observe or control a complex system. Yet, the observational uncertainty induced by measurement noise and the energy required for control continue to be significant challenges in practical applications. Here we show that the variability of control energy and observational uncertainty for different directions of the state space depend strongly on the number of driver nodes. In particular, we find that if all nodes are directly driven, control is energetically feasible, as the maximum energy increases sublinearly with the system size. If, however, we aim to control a system through a single node, control in some directions is energetically prohibitive, increasing exponentially with the system size. For the cases in between, the maximum energy decays exponentially when the number of driver nodes increases. We validate our findings in several model and real networks, arriving to a series of fundamental laws to describe the control energy that together deepen our understanding of complex systems.
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