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. Much effort has been devoted to describe the properties of HPTs of individual systems. Yet the fundamental question about the possible universal mechanism underlying those HPTs has not been investigated at a microscopic level. Here, we find that the discontinuity in the order parameter in such HPTs results from two steps: a durable critical branching (CB) and an explosive, supercritical (SC) process. 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. Those vulnerable nodes are activated then in the short SC process. This crossover mechanism and scaling behavior are universal for different HPT systems. 
Deokjae Lee, Wonjun Choi, J. Kert\'esz, B. Kahng
physics.soc-ph cond-mat.stat-mech