Multivariate recovery coupling in interdependent networks with cascading failure.

Interdependent networks are susceptible to catastrophic consequences due to the interdependence between the interacting subnetworks, making an effective recovery measure particularly crucial. Empirical evidence indicates that repairing the failed network component requires resources typically supplied by all subnetworks, which imposes the multivariate dependence on the recovery measures. In this paper, we develop a multivariate recovery coupling model for interdependent networks based on percolation theory. Considering the coupling structure and the failure-recovery relationship, we propose three recovery strategies for different scenarios based on the local stability of nodes. We find that the supporting network plays a more important role in improving network resilience than the network where the repaired component is located. This is because the recovery strategy based on the local stability of the supporting nodes is more likely to obtain direct benefits. In addition, the results show that the average degree and the degree exponent of the networks have little effect on the superior performance of the proposed recovery strategies. We also find a percolation phase transition from first to second order, which is strongly related to the dependence coefficient. This indicates that the more the recovery capacity of a system depends on the system itself, the more likely it is to undergo an abrupt transition under the multivariate recovery coupling. This paper provides a general theoretical frame to address the multivariate recovery coupling, which will enable us to design more resilient networks against cascading failures.