The Model Based On Markov Process Possessing The General Characteristic Of Real Network

Complex networks describe a wide range of systems in nature and society. In the empirical study of complex network, scientists find that almost all the real networks possess the small-world property, the high clustering property and the scale-free property. Recent work on the networks has been driven largely by observations of the properties of real networks and attempts to model them. But up to date on the one hand, almost all the existing models can not satisfy the three real network properties; on the other hand, the existing models are lack of strict mathematical description and the results are simulative. When the network becomes large, whether the limit of the sequence of the degree distribution is unique? Whether the limit of the clustering coefficient exists and what are the expression of the limiting degree distribution and the clustering coefficient? This paper proposes a conjecture which causes the network possess the high clustering property: the aggregative connection; and gives a new model which is based on this conjecture and the growth and preferential attachment which supposed by Barabasi and Albert. By the Markov chain and some other mathematical methods, the uniqueness and existence of the limit of a sequence of degree distributions of the network is proved, the limiting degree distribution and a lower bound of the limiting clustering coefficient and a upper bound of the limiting average length of the nodes of the model are obtained as well. Thus, the model satisfies all the three mainly properties of the real network has been proved. When the model lacks of the aggregative connection mechanism, the high clustering property lacks too.