Fractal Property Of Complex Networks And Its Application

Many complex systems arising from natural and social science can be modeled as complex networks. Recently, the topology structure of complex networks is the basic issue for com-plex networks research. Fractal, small-world and scale-free properties were regarded as three foundation properties of complex networks. These properties of complex networks have been extensively studied by many researchers.In this thesis, we focus on fractal property of complex networks and its application. The results are shown as follows:1. The fractal property of complex networks is revealed from varies perspectives. Fractal theory is used for describing the complexity and irregularity of nature. Fractal dimension, as an important parameter describing the fractal structure, reflects the regularity degree of a structure. Box-covering algorithm of complex networks is the most common way to calculate fractal dimension of complex networks. The fractal property has been revealed, not only in the structure, but in the function, information and so on. In this thesis, the covering ability of box-covering algorithm is represented by the number of nodes covered in complex networks. This ability is defined as information entropy of complex networks. And then, fractal property of complex networks is considered as the information dimension. An algorithm of the information dimension is proposed, which is more suitable for real complex networks than the traditional method. Another perspective, volume dimension is proposed as a generalization method based on box-covering algorithm of complex networks. Some real complex networks such as power gird network, c.elegans network, yeast network, scientific collaboration network in computational geometry and e-mail network are verified by two methods. Our results show that the proposed method is efficient when dealing with the fractal dimension of complex networks.2. Analysis of fractal property of weighted networks is proposed. Box-covering algorithm is a widely used method to measure the fractal dimension of complex networks. Existing researches mainly deal with the fractal dimension of unweighted networks. Here, the classical box-covering algorithm is modified to deal with the fractal dimension of weighted networks. Box size length is obtained by accumulating the distance between two nodes connected di-rectly. The proposed method is applied to calculate the fractal dimensions of the Sierpinski weighted fractal networks, e.coli network, scientific collaboration network, c.elegans network and USAir97network. Our results show that the proposed method is efficient when dealing with the fractal dimension problem of complex networks. We find that the fractal proper-ty is influenced by the edge-weight in weighted networks. The possible variation of fractal dimension due to changes in edge-weights of weighted networks is also discussed.3. Multi-fractal of weighted networks is considered. Only fractal dimension is not ade-quate when complex networks may exhibit multi-fractal behaviors. Multi-fractal analysis is a useful method to systematically characterize the spatial heterogeneity of both theoretical and experimental fractal patterns. In this thesis, multi-fractal analysis method of weighted net-works is proposed based on box-covering algorithm for fractal dimension of weighted networks (BCANw). The proposed method is applied to calculate the generalized fractal dimensions of some real networks. Our numerical results indicate that these weighted networks have multi-fractal properties.4. Identifying influential nodes is proposed based on local information dimension. The design of an effective ranking method to identify influential nodes is an important problem in the study of complex networks. In this thesis, a definition of the local information dimension is proposed. In our method, the information content of one node is represented by ratio of node number using covering ball in box-covering algorithm. For a given node, the local information dimension is defined as the change rate of information content. It indicates that the smaller value of the local information dimension is, the more important the nodes are. The local information dimension is applied to identify influential nodes. The results of the local information dimension are compared with the results of betweenness centrality and closeness centrality for Sierpinski weighted fractal networks and Zachary’s Karate club network. Numerical results show that the proposed method is efficient when dealing with the identifying influential nodes.