Phase Transition in Recovery Process of Complex Networks

The dynamic characteristic of complex network failure and recovery is one of the main research topics in complex networks. Real world systems such as traffic jams and Internet recovery could be described by the complex network theory. We propose a model to study the recovery process in complex networks. Two different recovery mechanisms are considered in three kinds of networks： external recovery and internal recovery. By simulating the process of the nodes recovery in networks, it is found that the system exhibits the feature of first-order phase transition only when the external recovery is considered. Internal recovery cannot induce such a kind of transitions. As external recovery and internal recovery coexist on networks, the systems will retain the most efficient part of external recovery and internal recovery. Meanwhile, a hysteresis could be observed when increasing or decreasing the failure probability. Finally, a largest degree node protection strategy is proposed for improving the robustness of networks.