Effect of interlayer dependence strength on continuous to discontinuous transitions in multi-layer networks

Studies on multilayer networks have aroused great interest in recent years because they allow a deeper description of the complexity present in many real systems. In this study, we introduce a dependency strength controller into multilayer interdependent networks, refining the traditional models that often inadequately represent cascading failures in real-world systems like urban transportation networks. This refined model enables a more realistic depiction of failure dynamics, where a node’s failure leads to its dependent node’s failure with a probability λ or survival with 1−λ. We derive exact equations that describe this process in a multilayer network and its structural evolution, and solve them numerically. We found the existence of a critical transition from continuous to discontinuous percolation as the strength of the dependence between layers increases. The critical percolation thresholds for these transitions are numerically derived for two-layer Erdős-Rényi networks and two-layer scale-free networks. This enhanced understanding of complex systems, incorporating initial disturbances and varying dependency strengths, contributes to designing more resilient multilayer complex systems.