A two-stage percolation process in random networks

We study a percolation model where the product rule (PR) is intervened by the randomly adding-edges rule from some moment t₀. At t₀=0, the model becomes the classical Erdös-Rényi (ER) random graph model where the order parameter undergoes a continuous phase transition. When t₀=1, the model becomes the PR model with two competitive edges, in which the order parameter exhibits an extremely abrupt transition with a significant jump in large but finite system. To study how the parameter t₀ affects the percolation behavior of the PR model, the numerical simulations investigate the pseudotransition point and the maximum gap of the order parameter for the percolation processes with different values of t₀. For the percolation processes at different values of t₀, the pseudotransition point t_c(N) can be predicted by the fitting function t_c(N) about t₀. As opposed to the weakly continuity of the order parameter in the PR model, it is found that the weakly continuity of the order parameter becomes weaker or even continuous in sufficiently large and finite system for the percolation processes at t₀<;0.888449. To clearly understand the behavior of the percolation processes at t₀<;0.888449, the numerical simulations investigate the cluster size distribution of the evolution. The characteristics of the phase transition in this model might provide reference for network intervention and control.