复杂网络系统涵盖了从自然界的生物系统到人工制造的基础设施系统等诸多领域，对故障具有自发恢复能力。例如，大脑在癫痫发作后可能会自发恢复正常，交通堵塞后也可能再次变得顺畅。以往动态网络自发恢复的研究主要局限在无向网络上。然而，在现实世界中，大多数网络是有向的。为填补这一空白，本研究建立了一种节点故障与自发恢复交替发生的有向动态网络模型，同时开发了一套理论工具来分析网络的恢复特性。该工具可准确预测活跃节点的最终比例，预测准确性随网络中双向边比例的增加而降低，这凸显了网络方向性的重要性。根据不同的初始状态，有向动态网络在相同的控制参数下可能表现出不同的稳定状态并呈现出滞后行为。此外，对于小规模网络，活跃节点的比例可能在高低状态之间震荡，这种现象可以模拟重复的故障恢复过程。这些发现有助于阐明系统恢复机制，更好地指导高韧性网络系统的设计。

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.