Markov chains theory for scale-free networks

This paper proposes a Markov chain method to predict the growth dynamics of the individual nodes in scale-free networks, and uses this to calculate numerically the degree distribution. We first find that the degree evolution of a node in the BA model is a nonhomogeneous Markov chain. An efficient algorithm to calculate the degree distribution is developed by the theory of Markov chains. The numerical results for the BA model are consistent with those of the analytical approach. A directed network with the logarithmic growth is introduced. The algorithm is applied to calculate the degree distribution for the model. The numerical results show that the system self-organizes into a scale-free network.