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 rho 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 rho 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 rho min of reinforced nodes (or the minimum group size S-min) to prevent abrupt collapse for Erdos-Renyi, scale-free, and regular random interdependent networks. Furthermore, we present the validity of the analytic solutions for a triple point rho(c) * (or S-c *), the corresponding phase transition point p( c )*, and second-order phase transition points p(c)( pi) in interdependent networks. These findings might yield a broad perspective for designing more resilient interdependent infrastructure networks. Published under an exclusive license by AIP Publishing.