Analysis on the evolution process of BFW-like model with discontinuous percolation of multiple giant components

Recently, the modified BFW model on random graphs [W. Chen, R.M. D’Souza, Phys. Rev. Lett. 106 (2011) 115701], which shows a discontinuous percolation transition with multiple giant components, has attracted much attention from physicists and statisticians. In this paper, by establishing the evolution equations on the modified BFW model, the evolution process and steady-states on both random graphs and finite-dimensional lattices are analyzed. On a random graph, by varying the edge accepted rate α, the system stabilizes in a steady-state with different numbers of giant components. Moreover, a close correspondence is built between the values of α and the number of giant components in steady-states, the efficiency of which is verified by the numerical simulations. Then, the sizes of giant components for different evolution strategies can be obtained by solving some constraints derived from the evolution equations. Meanwhile, a similar analysis is expanded to finite-dimensional lattices, and we find the BFW (α) model on a finite-dimensional lattice has different steady-states from those on a random graph, but they have the same evolution mechanism. The analysis of the evolution process and steady-state is of great help to explain the properties of discontinuous percolation and the role of nonlocality.