Spontaneous recovery in dynamical networks

Networks that fail can sometimes recover spontaneously—think of traffic jams suddenly easing or people waking from a coma. A model for such recoveries reveals spontaneous ‘phase flipping’ between high-activity and low-activity modes, in analogy with first-order phase transitions near a critical point. Much research has been carried out to explore the structural properties1,2,3,4,5,6,7,8,9,10 and vulnerability11,12,13,14,15,16,17,18,19 of complex networks. Of particular interest are abrupt dynamic events that cause networks to irreversibly fail13,14,15,16,17. However, in many real-world phenomena, such as brain seizures in neuroscience or sudden market crashes in finance, after an inactive period of time a significant part of the damaged network is capable of spontaneously becoming active again. The process often occurs repeatedly. To model this marked network recovery, we examine the effect of local node recoveries and stochastic contiguous spreading, and find that they can lead to the spontaneous emergence of macroscopic ‘phase-flipping’ phenomena. As the network is of finite size and is stochastic, the fraction of active nodes z switches back and forth between the two network collective modes characterized by high network activity and low network activity. Furthermore, the system exhibits a strong hysteresis behaviour analogous to phase transitions near a critical point. We present real-world network data exhibiting phase switching behaviour in accord with the predictions of the model.