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

TL;DR

This paper provides a theoretical analysis of the modified BFW model's explosive percolation process, establishing equations that explain the emergence of multiple giant components and matching well with simulations.

Contribution

It develops a theoretical framework for the modified BFW model, linking parameters to the number of giant components and explaining the explosive percolation phenomenon.

Findings

Theoretical expressions match numerical simulations within 4% error.

Parameter ndorresponds to the number of giant components in steady state.

Analysis explains the explosive nature of the percolation transition.

Abstract

Recently, the modified BFW model on random graph [Phys. Rev. Lett., 106, 115701 (2011)], which shows a strongly discontinuous percolation transition with multiple giant components, has attracted much attention from physicists, statisticians and materials scientists. In this paper, by establishing the theoretical expression of evolution equations on the modified BFW model, the steady-state and evolution process are analyzed and a close correspondence is built between the values of parameter \alpha and the number of giant components in steady-states, which fits very well with the numerical simulations. In fact, with the value of \alpha decreasing to 0.25, the error between theoretical and numerical results is smaller than 4% and trends to 0 rapidly. Furthermore, the sizes of giant components for different evolution strategies can also be obtained by solving some constraints derived from the evolution equations. The analysis of the steady-state and evolution process is of great help to explain why the percolation of modified BFW model is explosive and how explosive it is.