Research On Evolving Model, Stability And Application Of Complex Network

Small-world property and Scale-free property have been discovered from many real-world networks such as social networks, Internet networks, collaboration networks, biological networks and so on, which makes research on complex networks become a focus. In order to theoretically explain these characteristics, researchers have proposed many models. The small-world network explains "six-degree separations" phenomenon in social networks, the BA model reflects the mechanism that formed the power-law. Based on these two models, many other improved models emerge one after another incessantly, such as the growth network model, the evolution network model, the evolution of local world network model and so on. They have done lots of work from the evolution mechanism and algorithm of network model. But many models did not consider the fact that, personal energy and resources are limited in the social network. So nodes have limited bearing capacity when the network evolves and can not unlimitedly connect to others. The router ports are limited in Internet network, so the data of each router connected to the line can not be infinite. These phenomenons can be found everywhere. So we must consider the cost of nodes in the evolution of network. It will be very meaningful to research the structure character of actual network.The main content and innovation can be summarized as follows.1. The introduction of mathematical method on the complex network.A most important feature of complex network is scaling-free degree, that is, its degree distribution is power-law. First of all, this paper introduces some properties of pow-law distribution and does a little derivation from the mathematical point of view on these properties. However, research of complex network often focused on the calculation method of model. At present there are relatively common methods of mean field, master equation and rate equation. The master equation is a mathematics method and others are statistical mechanics methods. Why master equation can solve such evolution problem? Therefore, this paper introduces how come to the master equation from the Markov process and analyzes its physical meaning and the application of evolution network model.2. On the Model of Complex NetworkBecause of the cost of nodes that can not be overlooked, it is necessary to build a evolution model that consider the cost of nodes. Because the research on other mechanisms of evolution model have been proved useful, in order to simplify study, we build LBA model which add limited degree of nodes on the base of BA model. We find that when M < 2m(M is the cost of nodes,m is number of edges of new node), network evolution will stop after the limited steps and shows homogeneity network. Numerical simulation shows that the conclusions are accurate, and the evolution network have a high concentration and a short average shortest Path. The attenuation rate of cluster coefficient with network size increasing slower than BA model. Compared to the BA model, this model is more suitable for practical network, such as Internet network, Social Network, etc.3 On the stability of the complex network model study.The nodes which have limited degree may be more in LBA network. How about its stability when attack this nodes? The researchers found that there are different robustness between random network and scale-free network with various attack strategies. In this paper, I find that the robustness of LBA network with random attacking is higher. The average shortest path doesn't change drastically when delete some nodes randomly. But the connectivity of network was damaged when delete less than 30% large degree nodes. So the robustness of LBA network is very low when attacks important nodes deliberate.4 On the application of complex networks.As a typical application of LBA model, friendship network also has a number of other characteristics, such as friendship as time past, transplant, death and other reasons will decay. There is high transitivity in friendship network, just as my friend's friend may be also my friend. So I build a evolution model of friendship network and find that the scale-free degree distribution and small-world phenomena appear together by using master equation. But the scale-free index is smaller than BA model. Moreover, not only friendship network have highly concentrated community structure and observably small world, but also there are isolated nodes in network.