The Study On The Evolving Model Of Three Sierpinski Graphs And Their Fractality

In the real world, there are many systems, which can be represented by complex networks, such as Internet, movie actor collaboration network, metabolic networks, aviation network, and so on. Many empirical studies indicate that these various kinds of networks have some common characteristics: power-law distribution of degree, high clustering coefficient, and small average path length. These characteristics receive the researchers' great interest and considerable attention. This thesis takes mathematic analytical method, fractal, these methods into the complex network, and builds some improved models on the complex network to study on the character of the complex network.This thesis first introduces the background and the significance of the complex networks and the relevant conceptions are explained, too. Then according to the situation of complex studying, three new better models are proposed:1. On the basis of the Sierpinski tetra, we propose a kind of complex network in high dimensions with fractality and small-world effect. Using mathematical induction, we obtain that the clustering coefficient is 0.53, the diameter is 4, the average path length is 2.5, and the average degree is 7.5. The values of these properties testify that the network is small-world. Then we calculate the box-counting dimension and Similarity dimension as a measure of their fractality and the value of the dimension is about 2.2. On the basis of the Sierpinski carpet, we construct a kind of Sierpinski network. Using mathematical induction, we obtain that the clustering coefficient is 0.054, the diameter is 6, the average path length is less than 6, and the average degree is 4.61. The values of these properties testify that the network is small-world. Then we calculate the box-counting dimension and Similarity dimension as a measure of their fractality and the value of the dimension is about 1.8928. The similarity and elaborate structure proves the fractality of the network.3. A class of graphs with fractality and small-world effect is proposed. The clustering coefficient, the average length of the graphs, and the diameter are calculated analytically. Also we calculate the box-counting dimension and Hausdorff dimension as a measure of their fractality and the value of the dimension is 1.585. At last, the method of the constructing graphs is expanded, and the expressions of the relevant characteristics of the expanded graphs are given. The expanded graphs and the graphs proposed before are considered as a class of graphs with fractality and small-world effect.