Stochastic process rule-based Markov chain method for degree correlation of evolving networks

There is yet to be a unified theoretical framework for defining and solving degree correlation in evolving networks, which limits applied research in evolving networks. To address this problem, we proposed a stochastic process-based Markov chain method. The transition rules of network nodes and edges designed in this method ensure that the network topology and statistical characteristics at any time are the same as those in natural evolution. Then, the Markov chain model constructed based on this rule gives the theoretical results of the steady-state joint degree distribution of directed pure growth networks and corresponding undirected networks. Finally, the accuracy of the solution was verified by Monte Carlo simulation, and the probability functions of the joint degree distribution under different parameters were given. This work not only provides a theoretical research framework for the steady-state degree correlation of evolving networks for the first time but is also applicable to the study of many complex network evolution mechanisms and high-order statistical characteristics. In addition, this method can also study the transient degree correlation of networks at any time, providing a new perspective for network dynamics control.