The Transition From Erdős–Rényi Percolation To Explosive Percolation Under The Partial Product Rule

Explosive percolation proposed by Achlioptaset al.has attracted much attention recently. To understand how the product rule delays the phase transition, in this paper, we will concentrate our efforts on the influence of product rule in the transition from Erdős–Rényi (ER) percolation to Achlioptas explosive percolation. In our model, only a fraction of the adding edges will obey the product rule. With the probabilityp, the edge is selected under the Achlioptas's product rule; with1 - pthe edge is added between two randomly chosen nodes, that is, the ER rule. When the product rule plays a dominant role in the evolution,p > 0.5, the percolation transition delays more abruptly and instantaneously. The location of the critical points is independent of the network size and satisfies$t_{c}=(t_{c}^{\rm {AP}}-t_{c}^{\rm {ER}})p+t_{c}^{\rm {ER}}$, where$t_{c}^{\rm {ER}}$and$t_{c}^{\rm {AP}}$are the critical points of ER percolation and Achlioptas one. We present the largest gapΔsmaxand the critical region Δ as a function ofpandN. The giant componentsin terms of the edge densitytabovetcis also analyzed. We find that all the critical values have a linear relationship againstp.