Hypergraph-based modeling of cascading failures with probabilistic node-to-group interactions

Higher-order interactions are widespread in real-world complex systems and play a crucial role in the functionality and robustness of these systems. In this paper, we introduce a cascading failure model with interactions between individual nodes and groups by using hypergraphs to represent the systems with higher-order interactions, where the failure of one individual within a group results in group-wide failure in a probability β representing the interaction strength. Through extensive simulations and theoretical analysis, we find that lower interaction strength leads to a gradual decrease in the final fraction of viable nodes. Conversely, higher interaction strength results in a sudden drop in the final fraction of viable nodes at an infinitesimal fraction of initiators. For both cases, the size of the giant component in the viable nodes undergoes a continuous decline to zero at the critical fraction of initiators. Additionally, we identify two critical values for interaction strength when the initiator fraction is infinitesimal. Failures rarely propagate if the interaction strength is less than the first critical point. Beyond the first, failures propagate significantly, while surpassing the second results in a completely fragmented state, impairing overall system functionality. This study provides theoretical insights into the cascading behavior of systems with higher-order interactions, contributing to the construction of more robust complex systems and the effective management of potential risk in complex systems with higher-order interactions.