Group percolation in interdependent networks with reinforcement network layer.

In many real-world interdependent network systems, nodes often work together to form groups, which can enhance robustness to resist risks. However, previous group percolation models are always of a first-order phase transition, regardless of the group size distribution. This motivates us to investigate a generalized model for group percolation in interdependent networks with a reinforcement network layer to eliminate collapse. Some backup devices that are equipped for a density ρ of reinforced nodes constitute the reinforcement network layer. For each group, we assume that at least one node of the group can function in one network and a node in another network depends on the group to function. We find that increasing the density ρ of reinforcement nodes and the size S of the dependency group can significantly enhance the robustness of interdependent networks. Importantly, we find the existence of a hybrid phase transition behavior and propose a method for calculating the shift point of percolation types. The most interesting finding is the exact universal solution to the minimal density of reinforced nodes (or the minimum group size ) to prevent abrupt collapse for Erdős-Rényi, scale-free, and regular random interdependent networks. Furthermore, we present the validity of the analytic solutions for a triple point (or ), the corresponding phase transition point , and second-order phase transition points in interdependent networks. These findings might yield a broad perspective for designing more resilient interdependent infrastructure networks.