Time Evolution Of The Degree Distribution Of Model A Of Random Attachment Growing Networks

Complex network is the powerful tool in the investigating to complex systems. Recently, the investigating to complex networks attracts more and more attention. Based on the fact that the degree distribution of many real complex networks displays power law format, Barabás and Albert proposed the revolutionary BA model of scale-free networks in 1999. As growing and preferential attachment mechanisms are incorporated in the evolution of BA network model, so the degree distribution of this model follows the power law format. In order to verify that preferential attachment mechanism is one of the absolutely necessary factors which cause to power law degree distribution, they also proposed the model A of network. Because model A keeps the growing mechanism and removes the preferential attachment mechanism, therefore the degree distribution of this model is the exponential format rather than power format. The degree distribution of network reflects the hints of topology of the whole network. As random attachment growing way of networks is one of the very important growing ways, so the investigation to random attachment growing networks is very important. In their paper, Barabás and Albert only give the degree distribution of model A at infinite time, but they don't give the degree distribution at arbitrary time in the evolution of this model. By far, the time evolution of the degree distribution of this model hasn't been given in any literature. In this paper, using nonequilibrium statistic method, we give the master equation satisfied by degree distribution of model A in differential format, and obtain its rigorous analytical solution. The obtained solution is composed of two terms. At given finite time, one term decays as exponential way, well the other reflects the size effect. At infinite time of the evolution of this model, the normalized degree distribution is well accord with Barabás and Albert'; but at finite time, has some deviation from exponential format. The deviation of degree distribution from exponential format is introduced by size effect. At given time (or given network size), as the increases of degree, size effect will bring more and more influence to the degree distribution, or even destroy the exponential format.