Research On Fractal Analysis Algorithm Of Complex Networks

Complex networks have drawn considerable attention of researchers in different disciplines owing to they are capable of describing the structure feature of many complex systems.Studies on the topology structure of complex networks are of great significance for understanding the network structure and network behavior,so the topology structure of complex networks is the basic issue for complex networks research.In 2005,the self-similarity and fractal properties of complex networks were revealed by Song et al.The fractal property is considered as the third fundamental topology property of complex networks,and thus have been reviewed and investigated by many scholars.In this paper,we focus on the fractal analysis algorithm of complex networks.The main work is shown as follows:(1)The classical algorithms for calculating the fractal dimension of complex networks proposed at home and abroad are reviewed,and some of them are described in detail,including box-covering method,information dimension method and volume dimension method.(2)A generalized information dimension of complex networks is proposed.The existing information dimension method can well reveal the fractal property of complex networks,but all nodes are undifferentiated in that method.Based on the existing definition of information dimension of complex networks,we consider further the difference between network nodes,and define firstly the probability of information containing the box as the ratio of the sum of nodes degree in that box to the sum of nodes degree in the network.Then we propose a new information dimension of complex networks according to the probability of information containing the box.The proposed method was applied to calculate the fractal dimensions of five real-world complex networks.The simulation results demonstrate that the proposed method can dealing with the fractal dimension problem of complex networks effectively.(3)A fractal analysis method of weighted networks based on information dimension is proposed.The existing information dimension method is mainly used to analyze the fractal property of unweighted networks,and it is not fully applicable to analyze the fractal property of weighted networks.Motivated by the idea of box-covering algorithm for weighted complex networks,a fractal analysis method of weighted networks based on information dimension is proposed.We apply the proposed method to study the fractal property of a family of constructed “Sierpinski” weighted fractal network and three real-world weighted networks,the results demonstrate that the proposed method is effective for the fractal scaling analysis of weighted complex networks,and the information dimension is affected by the edge-weights.(4)A volume dimension of weighted complex networks is proposed.Although the box-covering method and the modified information dimension are effective for the fractal scaling analysis of real-world weighted complex networks,but this two methods can only achieve an approximate solution since how to identify the fewest boxes needed to cover the entire network is an NP-hard problem.Here,a new volume dimension measure based on the node strength of weighted networks is proposed.We first apply the proposed method to study the fractal property of two families of weighted fractal networks with regular fractal structures: Sierpinski weighted fractal networks and Cantor dust weighted fractal networks.The result shows that the numerical fractal dimensions obtained by our method are very close to the theoretical similarity dimension of the network.Then,we use the proposed method to study the fractal property of four real-world weighted networks and the results have shown the effectiveness of the proposed method.