Cascading Failures in Interdependent Networks with Reinforced Crucial Nodes and Dependency Groups

Previous studies of group percolation models in interdependent networks with reinforced nodes have rarely addressed the effects of the degree of reinforced nodes and the heterogeneity of group size distribution. In this paper, a cascading failure model in interdependent networks with reinforced crucial nodes and dependency groups is investigated numerically and analytically. For each group, we assume that if all the nodes in a group fail on one network, a node on another network that depends on that group will fail. We find that rich percolation transitions can be classified into three types: discontinuous, continuous, and hybrid phase transitions, which depend on the density of reinforced crucial nodes, the group size, and the heterogeneity of group size distribution. Importantly, our proposed crucial reinforced method has higher reinforcement efficiency than the random reinforced method. More significantly, we develop a general theoretical framework to calculate the percolation transition points and the shift point of percolation types. Simulation results show that the robustness of interdependent networks can be improved by increasing the density of reinforced crucial nodes, the group size, and the heterogeneity of group size distribution. Our theoretical results can well agree with numerical simulations. These findings might develop a new perspective for designing more resilient interdependent infrastructure networks.