加权复杂网络的重分形分析和谱分析及其应用

发布时间：2018-03-28 11:59

本文选题：复杂网络　切入点：沙箱算法　出处：《湘潭大学》2017年博士论文

【摘要】：近几十年来,复杂网络在许多领域已经引起了越来越多的关注,复杂网络己然成为了科学研究热点之一。复杂网络中,小世界性、无标度性及自相似性是最常见也是最重要的三大基本特征。本文首先讨论加权复杂网络的自相似性,提出了用于研究加权复杂网络的分形及重分形特性的SBw算法。作为应用,我们通过构建加权的复杂网络来研究分数布朗运动性质,主要讨论了两种不同的构建方法得到的加权复杂网络的基本拓扑性质,一是分数布朗运动通过水平可视化构建加权网络,二是分数布朗运动通过相空间重构建立加权递归网络。主要有以下几点:1、提出适用于加权复杂网络的重分形分析方法。已有的对复杂网络重分形分析的方法主要都是针对无权的网络,其中也有我们课题组最近提出的沙箱算法。但这些已有的方法都不再适用对加权的复杂网络进行重分形分析。本论文提出了改进的沙箱算法(我们称它为SBw算法)以适用于对加权的复杂网络进行重分形分析,首先我们利用SBw算法通过对构造的“Sierpinski”加权分形网络家族和“Cantor dust”加权分形网络家族进行了分形及重分形的分析,我们也讨论了分形维数和广义分形维数随着加权分形网络的权值变化而变化的规律。通过比较加权分形网络的理论分形维数与用SBw算法得到的数值结果,表明SBw算法针对加权网络的分形及重分形分析是可行的也是有效的。然后,我们应用加权的沙箱算法研究几类实际加权科学家合作网络的多重分形性质。发现多重分形存在于这些加权网络,并受他们边的权重影响。2、分数布朗运动通过水平可视图方法构建加权水平可视网络,并研究了这些加权网络的基本拓扑性质。对于不同的Hurst指数H的分数布朗运动所构建的加权水平可视网络,本文数值研究了它们的度分布,强度分布,聚集系数以及经典的Laplace算子和改进的Laplace算子的次小特征值和最大特征值与Hurst指数H的关系,研究了Hurst指数H对加权水平可视网络拓扑性质的影响规律。数值分析了所构建网络的分形及重分形性质,分析比较不同的Hurst指数H/对网络分形及重分形特性的影响。通过比较已有的不加权的水平可视网络的基本特征,探究网络的权值对整个时间序列的影响。3、基于相空间重构的方法由分数布朗运动构建加权递归网络,并研究了这些加权网络的基本拓扑性质。与用水平可视化构建加权的网络类似,本文数值研究了度分布,强度分布,聚集系数与Hurst指数H关系;从几何角度,本文研究了加权递归网络的分形及重分形性质;从代数角度,本文对加权递归网络的谱进行了分析。所得结果与无权递归网络比较,探究权值对这些统计量的影响;与水平可视图方法构建的加权网络比较,探讨两种不同的方法对这些统计特征影响。这两种不同的构建加权网络的方法,都是对原始分数布朗运动更精细的刻画模型,为研究时间序列提供新的参考方法。
[Abstract]:In recent years, the complex network has attracted more and more attention in many fields, the complex network has become a hot topic of scientific research. In the complex networks, small world and scale-free and self similarity is the three most common basic characteristics is the most important. This paper first discusses the self similarity weighted complex network, SBw algorithm for Fractal Study on weighted complex network and fractal characteristics is proposed. As an application, we construct a weighted complex network to study the fractional Brown motion properties, mainly discuss the basic topological properties of two kinds of different construction method of the weighted complex network, one is the fractional Brown motion to construct a weighted network through the level of visualization, two is fractional Brown motion through phase space reconstruction based weighted recursive network. The following main points: 1, multifractal is proposed for weighted complex network Analysis method. The existing methods of network multifractal analysis complex are mainly for the unweighted network, which also has a sandbox algorithm we recently proposed. But the existing methods are no longer applicable to complex networks are weighted multifractal analysis. This thesis proposes a sandbox algorithm (we call it the SBw algorithm is applicable to the complex network) on the weighted multifractal analysis, we use the SBw algorithm was analyzed by fractal and multifractal structure of "Sierpinski" and "Cantor family weighted fractal network dust weighted fractal network family, we also discuss the generalized fractal dimension and fractal dimension varies with the weight the change law of the weighted fractal network. By comparing the weighted fractal dimension theory of fractal networks and numerical results obtained by the SBw algorithm, show that the SBw algorithm for weighted Fractal and multifractal analysis network is feasible and effective. Then, we study the multifractal properties of several kinds of sandbox algorithm of weighted real weighted network. Scientists found that multifractal exist in the weighted network, and by weight affect their side.2, the fractional Brown movement through the level of constructing weighted level visual network view method, and studied the basic topological properties of these weighted networks. The weighted level of visual network constructed for the Hurst index H of different fractional Brown motion, we numerically study their degree distribution, strength distribution, the relationship between Hurst index and H aggregation coefficient and classical Laplace operator Laplace operator and improved. The second smallest eigenvalue and maximum eigenvalue, studied the influence of Hurst H on the topological properties of weighted index level of visual network. Numerical analysis of the construction of network Fractal and multifractal properties, analysis of the influence of Hurst H/ index comparison of different fractal and multifractal characteristics of the network. The basic characteristics of the existing unweighted level visual network, explore the weights of the network impact on the entire time series.3, phase space reconstruction method based on the fractional Brown motion to construct weighted recursive network. And study the basic topological properties of these weighted networks. With the level of construction of weighted network visualization, we numerically study the degree distribution, strength distribution, aggregation coefficient and Hurst index H; from the geometric angle, in this paper the fractal weighted recursive network and multifractal properties; from the view of algebra, the weighted the recursive network spectrum was analyzed. Results compared with no recurrent network, inquiry weight effect on these statistics; weighted network and method of constructing the view level than A, to investigate the effects of two different methods for these statistical features. The two different construction method of weighted network, is the original fractional Brown motion more sophisticated models, provide new reference methods for research on time series.

【学位授予单位】：湘潭大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O157.5

【参考文献】