Compared Analysis With Non-Equilibrium Statistic Theory For The Degree Distribution Of Complex Dynamic Scale-Free Networks

Statistic physics methods play an important role in the research of the degree distribution of complex dynamic scale-free networks. However, the majority of really scale-free networks do not evolve in the equilibrium state, but in the non-equilibrium state. As a result, non-equilibrium statistic physics theory has more capability to analytically calculate the degree distribution of really scale-free networks more accurately. We utilize non-equilibrium statistic physics theory to analyze and calculate the degree distribution of scale-free networks. This paper can be divided into two parts: analyzing a newly established scale-free network and calculating an improved BA model.In part one, basing on BA model, a new model is proposed, which not only contains the exterior growth, but also includes the interior growth and annihilation. The original master-equation has been modified, and the modified master-equation is consisted of the preferential attachment and the growing mechanism which is lost in the original master-equation. We calculate the degree distribution of this model with mean-field theory and the modified master-equation, at the same time, compare their difference and excellence. Being a discrete method, the modified master-equation can calculate the degree distribution of scale-free network accurately. Mean-field theory over-valuates the probability density of degree distribution in the initial stage, but it is very accurate when the degree becomes large.The simply improved BA model can cover the shortage that the degree distribution of the original vertices is equal to zero. The new statistic definition of degree distribution probability density makes the significance of degree distribution more perfect. This new statistic definition involves the factor of time and the degree of the original vertices. Markov chains explicitly display the preferential attachment and growing mechanism of scale-free network. We do the solution for the probability equation of the degree distribution with the differential method and achieve the universal expression by solving the differential equation.