Multifractal Analysis And Laplace Spectrum Analysis Of Weighted Networks And Some Applications

In recent decades, complex networks have attracted growing attention in many fields,and have become one of the hotspots of scientific research. Small world?scale-free and self similarity are three most common and most importan-t characteristics of complex networks. In this thesis we discuss the self simi?larity of weighted complex networks and propose the SBw algorithm for fractal and multifractal analysis of weighted complex network. As an application, we construct weighted complex networks from the fractional Brown motion, mainly discuss the basic topological properties and fractal analysis of weighted network constructed from fractional Brownian motions: horizontal visibility weighted net-works constructed from fractional Brownian motions , recurrence weighted network constructed from fractional Brownian motions. The main points are follows:1. Multifractal analysis of weighted networks is proposed. Some algorithms for multifractal analysis( MFA ) of unweighted complex networks have been proposed in the past a few years, including the sandbox (SB) algorithm recently employed by our group. In this thesis, a modified SB algorithm (we call it SBw algorithm)is proposed for MFA of weighted networks. First, we use the SBw algorithm to study the multifractal property of two families of weighted fractal networks (WFN-s): "Sierpinski" WFNs and “Cantor dust" WFNs. We also discuss how the fractal dimension and generalized fractal dimensions change with the edge-weights of the WFN. From the comparison between the theoretical and numerical fractal dimen-sions of these networks, we can find that the proposed SBw algorithm is efficient and feasible for MFA of weighted networks. Then, we apply the SBw algorithm to study multifractal properties of some real weighted networks -collaboration networks. It is found that the multifractality exists in these weighted networks,and is affected by their edge-weights.2. Weighted horizontal visibility networks have constructed from fractional Brownian motions, and the basic topological properties of these weighted networks are studied. For different exponents H, We construct weighted horizontal visibility networks from fractional Brown motion, and we study their degree distribution,vertex cumulative strength distribution, weighted clustering coefficient, we also numerical calculate the second-smallest eigenvalue, the third-smallest eigenvalue,and the maximum eigenvalue of the the general and normalized Laplace operator of the weighted horizontal visibility networks. We study the influence of the Hurst exponent H on the topological properties of the weighted horizontal visibility net-works. The fractal and multifractal properties of the constructed weighted network are analyzed numerically. The effects of different Hurst exponent H on fractal and multifractal characteristics of the weighted networks are analyzed and com-pared. Based on the comparison of the basic characteristics of horizontal visibility networks, we study the influence of network weight in the whole time series.3. Based on the phase space reconstruction method, a class of weighted re-currence networks are constructed from fractional Brownian motion, and we study their basic topological properties. Similar to the weighted networks constructed from horizontal visualization method, the relationships between degree distribu-tion, intensity distribution, clustering coefficient and the Hurst exponent H are numerically studied. Besides that, from the geometric point of view, we study the fractal and multifractal properties of the weighted recurrence networks; and from the algebraic point of view we analyse their spectrums. The results are com-pared with thoses of the unweighted recursive network to explore the influence of weights on these statistics. These two kinds of methods to construct the weighted network are more detailed description of the original fractional Brownian motion,and provide a new reference for the study of time series.