Complex networked systems, which range from biological systems in the natural world to infrastructure systems in the human-made world, can exhibit spontaneous recovery after a failure; for example, a brain may spontaneously return to normal after a seizure, and traffic flow can become smooth again after a jam. Previous studies on the spontaneous recovery of dynamical networks have been limited to undirected networks. However, most real-world networks are directed. To fill this gap, we build a model in which nodes may alternately fail and recover, and we develop a theoretical tool to analyze the recovery properties of directed dynamical networks. We find that the tool can accurately predict the final fraction of active nodes, and the prediction accuracy decreases as the fraction of bidirectional links in the network increases, which emphasizes the importance of directionality in network dynamics. Due to different initial states, directed dynamical networks may show alternative stable states under the same control parameter, exhibiting hysteresis behavior. In addition, for networks with finite sizes, the fraction of active nodes may jump back and forth between high and low states, mimicking repetitive failure-recovery processes. These findings could help clarify the system recovery mechanism and enable better design of networked systems with high resilience.

中文翻译：

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有向动态网络中的自发恢复

复杂的网络系统，从自然界的生物系统到人造世界的基础设施系统，都可以在发生故障后表现出自发恢复；例如，大脑在癫痫发作后可能会自发恢复正常，交通在拥堵后可能会再次变得顺畅。先前关于动态网络自发恢复的研究仅限于无向网络。然而，大多数现实世界的网络都是有向的。为了填补这一空白，我们建立了一个模型，其中节点可能会交替失败和恢复，并开发了一种理论工具来分析有向动态网络的恢复特性。我们发现该工具可以准确地预测活动节点的最终比例，并且预测精度随着网络中双向链路比例的增加而降低，这强调了方向性在网络动力学中的重要性。由于初始状态不同，有向动力网络在相同的控制参数下可能会表现出不同的稳定状态，表现出滞后行为。此外，对于大小有限的网络，活动节点的一部分可能会在高状态和低状态之间来回跳跃，模仿重复的故障恢复过程。这些发现有助于阐明系统恢复机制，并能够更好地设计具有高弹性的网络系统。