The Application Of Fractal Theory In The Study Of Complex Networks

The theory of complex networks has been studied extensively owing to their close relevance to many real networks. In the real world, these systems can be represented by complex networks, such as Internet, social networks, movie actor collaboration network, metabolic networks, and so on. In these complex networks, the nodes can represent individuals or organizations and edges mimic the interactions among them. And many empirical studies indicate that these various kinds of networks have three common characteristics: scale-free, fractal and small-world. These characteristics receive the researchers' great interest and considerable attention.This thesis takes MATLAB simulative method to propose a model of scale-free complex network with aging mechanism and fractal based on the model of BA scale-free network.The thesis uses renormalization and box-covering method of fractal theory to probe into origins of three features of complex networks and underlying mechanism of their relation and depict entire features of complex networks.The thesis takes iterative method to build a class of network graphs with fractality and small-world effect based on Apollonian Networks and Sierpinski Gasket. By calculating analytically the clustering coefficient, the average length of the graphs, and the diameter and calculating the box-counting dimension and Hausdoff dimension as a measure of their fractality, we prove that the graphs have small-world and fractal.Finally, the thesis takes box-covering method and regressive method of SPSS to calculate the value of the fractal dimension and exponent of degree distribution of faultage cranny on some area, depict the cranny in real world has scale-free and fractal characteristics.