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