Effect of primary cores in continuous and discontinuous percolation transition of random graph

Summary Percolation describes the transition of clusters in a graph into extensive cluster upon the addition of links. The initial state of network may change macroscopic linkage in networks explosively where links are added competitively according to certain rules. In a previous study, authors used the product of sizes of clusters to create network and concluded that percolation transition is discontinuous. Recently, vast studies have picked that a few models present a discontinuous transition in merging clusters. Dissimilar to the continuous transitions, comprehending the essence of discontinuous transitions needs a deeper study of the system, which has not been performed yet. In this paper, we consider the cluster size during the links addition in the order parameter of transition models and find that discontinuous transitions are impelled by primary cores of networks. Moreover, the nature of transition can be determined by the main characteristic of whether the primary cores in network are homogeneous. We also find the necessary conditions for discontinuous transition, which can be used effectively in the explosive percolation model. We find that discontinuous transitions are impelled by primary cores of networks. The nature of transition can be determined by the main characteristic of whether the primary cores in network are homogeneous. We also find the necessary conditions for discontinuous transition, which can be used effectively in the explosive percolation model.