Some Researches About The Theory And Application Of Directed Network

After finding the small-world effect and scale-free property of many real complex networks by empirical studies, complex networks have gradually attracted particular attention from all fields of science, the research about complex networks is becoming a hot issue over recent years. Fortunately, People have carried through kinds of comprehensive researches in topology structure and model of undirected networks and have attained many exciting results. In fact, there are many real directed networks besides undirected networks in nature and society which exist in a wide range of technology, information, biological, social and other fields. For examples, the world wide web(web pages as nodes, the hyperlinks between different pages as edges, the edges are obviously directional); the citation network(papers as nodes, the citation relationships between different papers as directed edges); the food web in ecosystem(each species is represented as a node of the network, a directed edge is placed between two species whenever one of them feeds the other); the goods web in economy field(the articles as nodes, the production relationships between articles as directed edges). However, it is a pity that we still can't find a paper or a book which has a systematic introduction to the basic theory of directed networks. Few of researches focused on the theory and application of directed networks, and most of them treated directed networks as a naive extension of undirected networks. They simplified the problems belonging to directed networks as those of undirected networks, which makes us unable to truly grasp the problems. Using the complexity science, nonlinear dynamics, statistics, differential calculus, cybernetics, matrix theory and the computer which is a power analysis, calculation and simulation tool, we do a summary and broaden research to the theory and application of directed networks from the following several aspects:1. We do a summary and broaden research to the theory of directed networks, including the primary definitions, the primary topology characteristics, the basic structure and matrix theory of directed networks. Especially, we do a more in-depth study to the adjacent matrix and Correlation matrix of directed networks.2. Base on some real networks with hierarchical structure, group structure and statistical self-similarity, we build a directed network model with deterministic degree distribution, analyze some of whose topology property and discuss some of whose potential applications in some real directed networks.3. According to the characteristics of some real networks such as the internet and the goods distribution network, we build an evolving directed network model on the base of the BA model. In particular, we study its distribution and discuss some of whose potential applications in some real directed networks.The main innovation points of this paper are as follows:1. Using deterministic iterative method, we build a hierarchical directed network model which can approximately simulate some real systems such as the food web and analyze some of its topology properties such as degree distribution. The model overcomes the defect that the stochastic model is difficult to analyze, which has great theoretical significance.2. We build an evolving directed network model on the base of the BA model which obeys power-law distribution and its index is tunable in [2,∞). The model can approximately simulate some real systems such as the Internet.