Percolation transitions in interdependent networks with reinforced dependency links

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.