Multifractal And Laplacian Spectrum Analyses Of The Horizontal Visibility Networks Constructed From ?-stable Lévy Motions

In recent years, researchers have proposed some methods to transform time series into complex networks. They studied the time series from different com-plex networks (most of them constructed from fractional Brownian motions) and obtained well results.In this thesis according to different ?,?, we generate a-stable Levy motions time series and map it to the horizontal visibility network by using the horizontal visibility graph, and study the multifractal properties, Laplacian spectrum and energy of this complex network. First, we employ the sandbox algorithm to com-pute the generalized fractal dimensions and mass exponents of complex network.We find that almost all of them have well multifractal properties. Secondly, we aim at the fractal dimension, information, dimension and correlation dimension of the generalized fractal dimensions to research the dependency relationship be-tween those dimensions and ?,? by choosing suitable model to fit, respectively.Meanwhile, we study S?,S motion (for the case ? = 0) and find the average frac-tal dimension is (D0(?,0)) = 2-1/?, which shows that the fractal dimension ofcomplex network is close to the fractal dimension of the graph of ?-stable Levy motions time series. Moreover, we study the dependency relationships between those dimensions and ?,?, which a is fixed and ? are different, find those figures are essentially symmetrical. We find those fitting equations are even function about ? when we choose surface fitting between ?,? and those results.Last, we further research the logarithm of second-samllest eigenvalue, loga-rithm of third-smallest eigenvalue, logarithm of largest eigenvalue and the energy for the Laplacian operator and the normalization Laplacian operator of the hor-izontal visibility network we construct, respectively. Similarly, we find that the relationships between ?,? and those numerical results for the two Laplacian oper-ators are similar to the conclusion for multifractal analysis. We consider that the numerical results result of corresponding -? is roughly as same as the numerical results of ? when we calculate a ? value.