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 to. At to=0, the model becomes the classical ErdosRenyi (ER) random graph model where the order parameter undergoes a continuous phase transition. When to=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 to 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 to. For the percolation processes at different values of to, the pseudotransition point t(A) can be predicted by the fitting function t(N) about to. 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 to<0.888449. To clearly understand the behavior of the percolation processes at to<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.