Abstract: Based on the classic scale-free network model, we set up the partial differential equation satisfied a more general network dynamic system, and then we not only find another important topological property of scale-free network, but also discuss the real background meaning of every function. Meanwhile, we extend the BA-network-model growth principle, degree-preferential attachment mechanism. Starting from a more general situation, we establish a network model containing degree-preferential attachment probability and special-graph-preferential attachment probability. By analysis, this model is scale-free. Finally, we distinguish the connect between the scope of the power law parameter γ of scale-free network and the phenomena all kinds of preferential attachment probabilities co-existing. According to the contribution level from the vertex to the whole network, we come up with a preferential attachment probability.

Key words: scale-free network model, dynamical evolution, partial differential equation, degree preferential attachment, graph preferential attachment