Universal mechanism for hybrid percolation transitions.

Hybrid percolation transitions (HPTs) induced by cascading processes have been observed in diverse complex systems such as k-core percolation, breakdown on interdependent networks and cooperative epidemic spreading models. Here we present the microscopic universal mechanism underlying those HPTs. We show that the discontinuity in the order parameter results from two steps: a durable critical branching (CB) and an explosive, supercritical (SC) process, the latter resulting from large loops inevitably present in finite size samples. In a random network of N nodes at the transition the CB process persists for O(N (1/3)) time and the remaining nodes become vulnerable, which are then activated in the short SC process. This crossover mechanism and scaling behavior are universal for different HPT systems. Our result implies that the crossover time O(N (1/3)) is a golden time, during which one needs to take actions to control and prevent the formation of a macroscopic cascade, e.g., a pandemic outbreak.