Complex Network Evolution Model And Transmission Dynamics

Many complex systems in the world can be described by networks made up of interactive individuals. Since the small-world effect and scale-free character of the real network were discovered, complex network has become one of the most rapidly developing areas. At present, the research of the complex network has already permeated through different fields such as mathematics, physics, engineering and life. The networks involved includes traffic network, electric network, Internet, financial network, protein function network, etc. We can say that complex network theory has already become an essential tool for people to study complex system.The study of complex networks, on one hand is helpful for people to find out about the structural characteristic of the real networks and their formation mechanism, on the other hand is useful for people to recognize the dynamical process in the complex networks, which is of great importance for people to optimize and control the dynamic process in the real networks.Over the past years, scholars from different fields have carried extensive research on the structure and the dynamical process of complex networks. However, there are still a lot of problems worthing further studying for the complexity of the real system. In this thesis, several closely linked works such as the evolving models of network, the spreading of virus and the cascading errors on complex networks were carried out.The main points and innovations of this thesis are as follows:(1) Proposed a random-scale free unified network model based on random evolving network and scale-free network model. The random evolving network and scale-free network model are unified by a parameter p. When p is regulated from 0 to 1, the structure of network is changed from BA scale-free network to random evolving network. The mean-field method and computer simulation are separately used to study the distribution of the degree. The results are more precise than the previous results. In addition, we extended this model to a unified network, in which the evolution can take place both inside and outside of the network. This researches discovered the evolution mechanism of the real network which is helpful for people to understand the growth procedure of real network.(2) Studied virus spreading process based on random-scale free unified network. The result shows that when the scale of the network is very large and the parameter p is positive, the threshold of the spreading efficiency will be a positive number. Only when the spreading efficiency is greater than the threshold, can viruses spread in the network continuously. When p is close to 0, the threshold will be close to 0, which means that a weak spreading source can induce the virus spread in the network continuously. This result proves that the structure of the network is an important factor influencing the virus to spread. The result is of great directive significance for people to design the network. During the process of network designing, we can change the structure of the network by regulating the parameter p, so as to prevent the spreading of virus.(3) Proposed the SIS (susceptible-infected-susceptible) and SIRS (susceptible -infected-recovered-susceptible) virus spreading models based on two dimensional lattices, the moving of individuals and the colony density is considered during the study procedure. The theory analysis and computer simulation shows that colony density, spreading efficiency and the moving of individuals are all important factors that influent the spreading process of virus. According to the result, we provided some schemes to control the spreading of virus, and these schemes keep the same with schemes which the government makes when SARS break out.(4) Proposed an adaptive network model, in which the evolution of the network's structure and the spreading process of the virus can go on at the same time. We investigated the network's structure and the virus' spreading effect in different conditions. The results show that when re-link probability is set to 0 and the iinfected probability, cure probability and recover probability are all above 0, the system will maintain a random network structure and the final infected rate will achieve a fixed value. When re-link probability is greater than 0 and the infected probability, cure probability and recover probability equal to 0, the system will be split into two independent random networks. When re-link probability, the infected probability, cure probability and recover probability are all above 0, the system will evolve to a broad-scale network. In this case the final infected rate will be oscillated, at the same time the bistable state is observed when the parameters are set at proper values. We also discovered that the re-link of network can not completely stop the spreading of virus, which shows that the individuals' spontaneous escaping acts can not stop virus spreading in the network.(5) Studied errors spreading process based on two dimensional regular network and BA scale-free network. During this processes energy dissipation and capability enlarging is allowed. The result shows that energy dissipation and capability enlarging are two important factors which lead the system develop toward SOC (self-organized criticality) status. This conclusion well explained the SOC phenomenon exists in electric network. In addition, the controlling strategies for avoiding cascading errors of networks are also studied. The result shows that as far as a few key nodes are protected in the network the large cascading errors will be avoided. This conclusion is of great importance to control the cascading errors of network.