Recent Progress on Hybrid Percolation Transitions

Percolation describes the formation of a giant cluster once the average degree of a network exceeds a critical value. A hybrid percolation transition (HPT) denotes a phenomenon in which a discontinuous jump of the order parameter and the critical behavior, a basic pattern of a continuous transition, appear together at the same threshold. Such HPTs have been reported in many different systems. In this review, we present several representative examples of HPTs and classify them into two categories: global suppression-induced HPTs and cascading failure-induced HPTs. In the former class, critical behavior manifests itself in the distribution of cluster sizes, whereas in the latter it emerges in the distribution of avalanche sizes. We further outline the universal scaling relations shared by both types.