Dependence can highly increase the vulnerability of interdependent networks under cascading failure. Recent studies have shown that a constant density of reinforced nodes can prevent catastrophic network collapses. However, the effect of reinforcing dependency links in interdependent networks has rarely been addressed. Here, we develop a percolation model for studying interdependent networks by introducing a fraction of reinforced dependency links. We find that there is a minimum fraction of dependency links that need to be reinforced to prevent the network from abrupt transition, and it can serve as the boundary value to distinguish between the first- and second-order phase transitions of the network. We give both analytical and numerical solutions to the minimum fraction of reinforced dependency links for random and scale-free networks. Interestingly, it is found that the upper bound of this fraction is a constant 0.088 01 for two interdependent random networks regardless of the average degree. In particular, we find that the proposed method has higher reinforcement efficiency compared to the node-reinforced method, and its superiority in scale-free networks becomes more obvious as the coupling strength increases. Moreover, the heterogeneity of the network structure profoundly affects the reinforcement efficiency. These findings may provide several useful suggestions for designing more resilient interdependent networks.

Failure and recovery underlie many complex systems ranging from critical infrastructures to organisms. In many real complex systems, the reliability of repaired components is improved due to the exceeding recovery mechanism, and such systems typically have enhanced failure resistance. The main motivation of this study lies in developing an exceeding recovery model to capture the exceeding recovery mechanics of complex network systems. In the proposed model, cascading failure and exceeding recovery perform concomitantly. The network resilience analysis is performed in the Barabá si-Albert and Erdős-Ré nyi networks by focusing on the exceeding recovery process from random and targeted attacks. The results show that for a given initial failure size, there is a critical value of the exceeding recovery coefficient above which the network can restore to the normal state, but below this value, the network abruptly collapses. The proposed model is compared with the conventional recovery model. The comparison indicates that the proposed model can recover to a significantly high level in a short recovery time and at a low recovery cost. The exceeding recovery mechanism strongly affects the failure-recovery property, which is expressed as reduced risk of a secondary failure at the micro level and enhanced heterogeneity of the load distribution at the macro level. These findings provide a guideline to address the exceeding recovery problem of a network and can help to design networks with better resilience against cascading failures.

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.