A Sufficient Condition For One Class Of Graph-valued Markov Processes Not Possessing High Clustering Property

In recent years, complex networks have set off a wave of domestic and foreign research, and have been strongly concerned by the researchers in all areas from the science and engineering. Many real world systems can be described by complex networks, such as social networks, the World Wide Web, the Internet and so on.. From the mathematical point of view, any network can be abstracted as a graph consisting of node-set V and edge-set E, where nodes represent the elements of the network and edges represent the relationship between elements. Many evolution network models can be viewed as graph-valued Markov processes. Different from the classical random graph process, graph-valued stochastic process discusses the graph process where nodes and edges vary randomly. Here, add and remove nodes randomly over the time, when a node is removed, the edges linking to it are also removed. The scale-free behavior for the high clustering property of the graph-valued process are the main property that people are interested in, especially the sufficient condition for graph-valued process possessing the property really catches researchers’attention. To find out features of the formation mechanism of complex networks, researches have propose many models of evolution networks. However, up to date, the features researches of graph-valued process of evolution models have been remained mostly in numerical simulation or the mean-field theory and the main equation and so on, which are the methods of statistical physics, and some models even lack rigorly mathematically. In this paper, we study the graph-valued Markov process with new nodes and edges, from the perspectiov of stochastic analysis, we get the sufficient condition for the Markov processes which do not express the high clustering property.