The transition from erds-rényi percolation to explosive percolation under the partial product rule

摘要

Explosive percolation proposed by Achlioptas et 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 Erds-Rényi (ER) percolation to Achlioptas explosive percolation. In our model, only a fraction of the adding edges will obey the product rule. With the probability p, the edge is selected under the Achlioptas's product rule; with 1 - p the 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 AP_(-t)_c ER_(p+t)_c ER, where t_c ~(ER) and t_c ~(AP) are the critical points of ER percolation and Achlioptas one. We present the largest gap s _(max) and the critical region as a function of p and N. The giant component s in terms of the edge density t above t_c is also analyzed. We find that all the critical values have a linear relationship against p.