Cascading failures in multiple-to-multiple interdependent networks considering interdependent failure threshold

In recent years, the study of interdependent networks has emerged as a focus within the field of complex networks. In analyses of the robustness of networks with multiple support-dependent relationships, traditional interdependent failure rules often assume that a node fails when one or all of its interdependent nodes fail. Such assumptions often fail to capture the probabilistic nature of real-world network failures. More generally, interdependent networks in the real world typically exhibit failure dynamics in which a node fails if a certain proportion of its interdependent nodes fail. To address this fact, this paper proposes a model of multiple-to-multiple interdependent networks that introduces an interdependent failure threshold, defined as the proportion of failed interdependent nodes required to trigger the failure of a given node. The study investigates how the varying number of interdependent edges and the interdependent failure threshold influence the robustness of the network under random or intentional attacks. In intentional attacks, the robustness of the network decreases as the number of interdependent links increases. Additionally, assortatively coupled networks demonstrate greater robustness under random attacks, while randomly coupled networks are more robust under intentional attacks. Finally, we verify the correctness of our results in real-world settings and obtain conclusions that are basically consistent with the theoretical model. These findings provide valuable information on the resilience mechanisms of interdependent networks in the real world, which has implications for the design of more robust systems.