Research On General Complex Networks And The Dynamic Model And Stability Of Economic Networks

Complex networks have been found many applications in a variety of fields including society, politics, medicine, economics and management, and attract more and more attentions from various fields of science and engineering. Evolving networks and epidemic dynamics are two important fields in complex networks. And, Economists and econophysicists have been starting to research on the network formation of economic systems from the rise of the complex networks. Economic networks describe the relationships and the influences among the economic agents precisely. The relationships can be the information changing among the economic agents, commodity transaction, dealing in securities, investment relation, credit relationship, membership and so on. It is neglected that the real economic systems always have a large and complex network formation in classic economic theory. It is supposed that the economic systems have the simple network formation like star network full connected network. It can partly explain why the economic theory can not resolve the economic phenomena sometimes. So, it is important to research on the economic network formation. It is a new field in economics and there are many fields need further study. But it shows the potentiality on economic networks through current progress report.The theories and methods which contain uncertainty theory, utility theory, Game, stochastic dominance theory and the mean-filed method are used to the research on network formation in this paper. And the research contains network formation, the characters and the dynamics. It can be used to explain the characters of real-world networks through the results that received from the research. And by studying the dynamic properties of complex networks, on the one hand, we can learn the mechanism of network formation and build better network structure by using the results; on the other hand, we can apply these theoretical results to influence the formation process of real-world networks and let it to change advantaged. So, the importance of the research on network formation and the properties of networks are clearly self-evident.The main contents and originalities in this paper can be summarized as follows: 1. The evolving network based on uncertainty theory is proposed.It contains various uncertainties in the process of network formation. First, an evolving network model by using proving rules is presented. The proving contains both adding the uncertainty variable to preferential probability and deleting the old links. Then the dynamical equation is established via using mean-filed approach. The expression equations of degree distribution and the degree exponent are deduced both by discussing two kinds of distributing of uncertainty variable and by the different relationship between the numbers of added and deleted links. And then, the degree of fit between the theoretical result and simulation is compared. But it will have a higher error if the degree is lower, it is analyzed by Markov-based numerical method. It is well fit by comparing the numerical result and simulation. It is showed that the uncertainty evolving network can evolve to scale-free network and it will be used generally through the degree exponent and the theoretical result.2. The influence of randomness on the network structure and the properties is studied.There is not only preferential attachment but also random attachment in the process of network formation. And the different probability of the two kinds of attachment will lead to different network structure. The model which includes both random attachment and preferential attachment is presented. The parameters are set to the probability of two kinds of attachment and the probability of deleting, and the change of the network formation and the properties following the change of parameters is analyzed. Firstly, the equation of degree distribution, cumulative degree distribution and degree exponent are deduced via mean field approach. The influence of randomness on the degree distribution is discussed, and network stucture is different because of the change of degree distribution and degree exponent. Then, the influence of randomness on the properties of network which include average path length and the clustering coefficient is discussed, and the regularity of the influence is analyzed.3. The epidemic dynamics and immunization in complex networks based on stochastic dominance theory is studied.The epidemic dynamics and immunization are the important dynamic behaviors. It is to help learn properties of the spread of epidemic and gossips through studying the epidemic dynamics of network. Network structure is the important factor to influence the properties of epidemic dynamics and immunization. So, it is important to learn the influence. Firstly, the equation of the positive steady state is built and the positive steady state is discussed. Then, the epidemic speed and ratio between different network structures can be made comparison based on the stochastic dominance theory in SIS model. The diffusion speed and ration can be learned from the comparison results, and the immunization which can influenced epidemic dynamics the can be carried on from the results, all these conclusions can be used to change the epidemic property of real-world networks like epidemic and gossip broadcasting. So, It is supplied a new method to these problems. Based on the conclusion, the randomness model which is presented above is discussed. The changes of speed and the ratio of epidemic dynamics following the change of the parameters are analyzed. And then, the property of immunization changing by the parameters is also analyzed.4. The dynamic economic models and the stability are analyzed.The endogenous model which is one kind of economic networks is described the relationship between the agents in economic systems precisely. The stability and efficiency studied in the model can assist to explain some economic phenomenon. But, it is some difference from the static state of the model to the dynamic process of real-world economic systems and let the theoretical results show the shortage in some analysis. The dynamic model that contains new nodes adding is presented in this paper in order to solve this problem. The definition of the stability and efficiency which give in static model is not suit here, so, firstly, the new meaning is defined based on the dynamic formation process. Then the dynamic stability and the dynamic efficiency network formation are showed based on the rules of endogenous model and the hypotheses that the agent is selfish and myopic. And the process of the linking is decided by the benefit and the cost. The dynamic stability and the dynamic efficiency network formations are proved through utility theory. In order to expand the applied range of the results, the hypotheses is changed to that the agent is heterogeneous (the cost of agent is different). The dynamic stability network formation is showed in this kind of hypotheses and is proved through utility theory.