Discontinuous percolation transitions in cluster merging processes

The cluster merging process was regarded as the central kinetics of a sol-gel transition and was solved analytically by Ziff. Since then, it has been applied to diverse phenomena, such as the evolution of social networks and spread of epidemic diseases. The sol-gel transition is applied to the robustness of complex networks with regard to the percolation transition. Percolation transition is regarded as a robust continuous transition; however, in complex systems, diseases or rumors can spread rapidly. Hence, it has been challenging to modify percolation models such that they exhibit a discontinuous transition that explains abruptly changing phenomena. Recently, researchers argued that a discontinuous percolation transition can occur when a network evolves under a rule with global information. In this study, we review earlier studies on percolation models that exhibit discontinuous transitions, focusing particularly on models with cluster-merging kinetics.