Abstract
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.
Highlights
Recent network research has focused on the cascading failures in a system of interdependent networks and the necessary preconditions for system collapse
We note that the dynamics of cascading failures when γ > 0 differ greatly from when γ > 0
The main difference is that when p > pc and γ = 035 the number of iteration steps (NOI) needed to reach the steady state decays sharply, but when γ > 0 it remains high
Summary
Recent network research has focused on the cascading failures in a system of interdependent networks and the necessary preconditions for system collapse. We develop a model for the competition between the cascading failures and the restoration strategy that repairs failed nodes in the boundary of the functional network and reconnects them to it (see Fig. 1) The reasoning behind this repairing strategy is based on the fact that (a) in many real systems it is easier to repair boundary nodes (for example, in a transportation system one needs to bring equipment to the damaged site and it is easier to bring it near using the existing transportation system) and (b) fixing a node that is not in the boundary will cause the node to fail in the step since it is not connected to the giant component and such a repair will be a wasted effort. We find that there is a critical probability γc that depends on p that separates a regime of full system fragmentation from a regime of complete system restoration
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