Characteristics of phase transitions via intervention in random networks

Abstract

We present a percolation process in which the classical Erdös—Rényi (ER) random evolutionary network is intervened by the product rule (PR) from some moment t0. The parameter t0 is continuously tunable over the real interval [0, 1]. This model becomes the random network under the Achlioptas process at t0 = 0 and the ER network at t0 = 1. For the percolation process at t0 ≤ 1, we introduce a relatively slow-growing point, after which the largest cluster begins growing faster than that in the ER model. A weakly discontinuous transition is generated in the percolation process at t0 ≤ 0.5. We take the relatively slow-growing point as the lower pseudotransition point and the maximum gap point of the order parameter as the upper pseudotransition point. The critical point can be approximately predicted by each fitting function of the two points about t0. This contributes to understanding the rapid mergence of the large clusters at the critical point. The numerical simulations indicate that the lower pseudotransition point and the upper pseudotransition point are equal in the thermodynamic limit. When t0 > 0.5, the percolation processes generate a continuous transition. The scaling analyses of several quantities are presented, including the relatively slow-growing point, the duration of the relatively slow-growing process, as well as the relatively maximum strength between the percolation percolation at t0 < 1 and the ER network about different t0. The presented mechanism can be viewed as a two-stage percolation process that has many potential applications in the growth processes of real networks.