A vertex-number-evolving Markov chain of networks

We have introduced a vector Markov chain of the vertex number with degree k in network evolving process as a framework of theoretical analysis and proved the stability of the BA-1 model and the LCD-1 model. In this paper, we use the vertex-number-evolving Markov chain to prove rigorously the existence of the steady-state degree distribution P(k) for a special case of the initial attraction model allowing multiple edges. The application of our approach to the LCD-m model, the result shows that it is more simpler than Bollobás’ method.