Recent network research has focused on the cascading failures in a system of interdependent networks and the necessary preconditions for system collapse. An important question that has not been addressed is how to repair a failing system before it suffers total breakdown. Here we introduce a recovery strategy for nodes and develop an analytic and numerical framework for studying the concurrent failure and recovery of a system of interdependent networks based on an efficient and practically reasonable strategy. Our strategy consists of repairing a fraction of failed nodes, with probability of recovery γ, that are neighbors of the largest connected component of each constituent network. We find that, for a given initial failure of a fraction 1 - p of nodes, there is a critical probability of recovery above which the cascade is halted and the system fully restores to its initial state and below which the system abruptly collapses. As a consequence we find in the plane γ - p of the phase diagram three distinct phases. A phase in which the system never collapses without being restored, another phase in which the recovery strategy avoids the breakdown, and a phase in which even the repairing process cannot prevent system collapse.
Bibliographical noteFunding Information:
HES thanks the NSF PHY-1505000, ONR Grant N00014-14-1-0738, and DTRA Grant HDTRA1-14-1-0017 for financial support. SH thanks the Defense Threat Reduction Agency (DTRA), the Office of Naval Research (ONR) (N62909-14-1-N019), the United States-Israel Binational Science Foundation, the LINC (Grant No. 289447) and the Multiplex (Grant No. 317532) European projects, the Italy-Israel binational support and the Israel Science Foundation for support. MAD, CEL and LAB wish to thank to UNMdP, FONCyT and CONICET (Pict 0429/2013, Pict 1407/2014 and PIP 00443/2014) for financial support. We thank L. D. Valdez for useful discussions and support throughout this research.