The degree distribution and degree correlations of the network with both preferential and random attachments

In this paper, the degree distribution and the two vertex degree correlations of the network with both preferential and random attachments are studied. Based on the concepts and techniques of Markov chain theory, a rigorous proof for the existence of the steady-state degree distribution is provided, and the exact analytic formula of the degree distribution and the scaling exponent are mathematically derived. By using the rate equation approach, we obtain an asymptotic expression for two vertex degree correlations.