Research On Fractal Properties Of Complex Networks Based On Correlation Dimension

Complex networks are an important tool for characterizing and studying complex systems.The research on the topology properties of complex networks is still the main challenge in the field.In recent decades,it has received extensive attention from different groups in different disciplines and fields and has broad application prospects.In 2005,the fractal properties and self-similarity of complex networks were revealed,which provided new ideas for studying the topology of complex networks.Fractal properties are considered to be the third-largest complex network fundamental topological property after small world and scale-free properties.And it has become an important research field of complex networks.This thesis focuses on the fractal dimension of complex networks and their applications.The main research work includes the following aspects:(1)Introduced and ordered out the current development of domestic and foreign fractal theory in recent years and made a detailed introduction to some fractal dimensions of complex networks,including box-covering dimension method,information dimension method,and correlation dimension method.(2)A fractal property analysis method of a weighted network based on the correlation dimension is proposed.There are two main existing methods of correlation dimension analysis.One is to study the fractal characteristics of unweighted networks,and the other is to embed the network into the Euclidean space before analyzing the fractal properties.These two methods are not completely suitable for analyzing the fractal properties of the weighted network.Therefore,in this thesis,the weighted network’s fractal theory system is used to propose the correlation dimension method for weighted networks.This method validates the effectiveness of Sierpinski and Cantor Dust weighted fractal networks with theoretical fractal dimensions and characterizes the fractal properties of six real-world weighted networks.The experimental results show that the method can describe the fractal properties of the weighted network well,and the consideration of edge-weight is very important for the analysis of the fractal properties of partially weighted networks.(3)A new method of correlation dimension analysis of complex network embedded space is proposed.The existing embedding space method is to embed the node position information into the Euclidean space,which cannot be directly used in the traditional metric space.Therefore,this thesis defines the correlation dimension of complex networks in non-Euclidean space.The original method traverses the random walk between nodes when embedding space to reconstruct a high-dimensional node vector.The random walk process only considers the neighbors to reconstruct the node vector to contain more similar node information,which is not conducive to restoring the dynamics of complex systems.Therefore,this thesis introduces the concept of the delay time of the embedded space in chaos theory into a complex network to expand the distance of the traveling objects and reduce the correlation between the nodes in the node vector.The analysis through three networks with certain fractal dimensions shows that the method is effective in analyzing the dimensions of the dynamic system,and the dimensions calculated by the original method tend to the true correlation dimension values.(4)A global efficiency estimation method for complex networks based on correlation dimensions is proposed.Global network efficiency was originally used to quantitatively analyze the small-world properties of the networks,and this feature can also be reflected through the correlation dimension.The relationship between the global efficiency of the network and the correlation dimension is analyzed,and the concept of the node correlation dimension is defined.Its intuitive expression is the difficulty of the node reaching other nodes for information exchange.The global network efficiency estimation method is proposed by considering the topological structure,fractal properties of a complex network.The efficiency of part of the calculation process is estimated by the correlation dimension.This method estimates the efficiency through two construction networks and three large road networks.The results show that the complex network with fractal properties can significantly shorten the time required to calculate the global efficiency.Further,this method can allow for parallelized operations.