The Statistical Description Of Compex Networks And The Researches On Establishing Network Models

At present the researches on the complex question and the complex networks are on the upsurge in the international. The massive complex systems, such as Internet, the computer network, the nervous system, the social relate network and so on, exist in the nature. These complex systems are all described through the complex networks. The aim of the studies on these networks is to investigate their mechanisms, understand their developing rules and find the relationship between their structure and the complex functioning processes on them, so as to accumulate the knowledge about the natural disciplinarians dominating the complex systems. This thesis starts from the establishment of network models, takes the statistical physics and random-graph theory as tools, researches the macroscopic rules of the statistical properties, the structure and the function of complex networks.The first chapter introduces the background and the developing process as well as the research status quo of complex networks. Most of our works concentrate on the following several chapters.The second chapter firstly elaborates several statistical targets of complex networks, such as degree distribution, average path length and clustering coefficient etc., then introduces several kinds of early network models,such as the ER model and the WS model etc.,introduces two kinds of the epidemic disease models as well, finally describes several real networks,such as the Internet, the living system network and the scientific research cooperation network and so on.The third chapter discusses the content of random-graph theory.Regarding the network researches, the earliest studies are started from the mathematician.Its basic theory is the graph theory which is a powerful tool of discussion on network general character. This chapter respectively analyzes the relative properties of complex networks, such as subgraphs, graph evolution,degree distribution,clustering coefficient, graph spectra and so on.The fourth chapter studies the percolation model and the relative theory. The percolation theory starts from the critical probability's angle which is similar but different to random-graph theory, analyzes the structure of complex networks and the changes of properties. This chapter emphatically introduces several important properties of percolation model and the relative theory, percolation on a Cayley and cluster structure.The fifth chapter analyzes and inspectes the formation mechanism and the basic characteristics of scale-free networks from the statistical mechanics angle, then introduces BA model which is most commonly used dynamics model in scale-free networks. We emphatically analyze the dynamic method (including Mean-field and Rate-equation) based on the Continuum theory and Master-equation method based on the probability theory in the scale-free networks, research their application and reciprocity of scale-free networks. Then we propose the formation mechanism of growth networks, introduce emphatically the BA model which is composed of the linear growth and preferential attachment, and attempt the value computational method based on Markov-chains theory. Moreover, we study the accelerating growth model which has the logarithm growth by the value computational method.