本文選題：復雜網(wǎng)絡(luò)　切入點：沙箱算法　出處：《湘潭大學》2017年博士論文

【摘要】：近幾十年來,復雜網(wǎng)絡(luò)在許多領(lǐng)域已經(jīng)引起了越來越多的關(guān)注,復雜網(wǎng)絡(luò)己然成為了科學研究熱點之一。復雜網(wǎng)絡(luò)中,小世界性、無標度性及自相似性是最常見也是最重要的三大基本特征。本文首先討論加權(quán)復雜網(wǎng)絡(luò)的自相似性,提出了用于研究加權(quán)復雜網(wǎng)絡(luò)的分形及重分形特性的SBw算法。作為應(yīng)用,我們通過構(gòu)建加權(quán)的復雜網(wǎng)絡(luò)來研究分數(shù)布朗運動性質(zhì),主要討論了兩種不同的構(gòu)建方法得到的加權(quán)復雜網(wǎng)絡(luò)的基本拓撲性質(zhì),一是分數(shù)布朗運動通過水平可視化構(gòu)建加權(quán)網(wǎng)絡(luò),二是分數(shù)布朗運動通過相空間重構(gòu)建立加權(quán)遞歸網(wǎng)絡(luò)。主要有以下幾點:1、提出適用于加權(quán)復雜網(wǎng)絡(luò)的重分形分析方法。已有的對復雜網(wǎng)絡(luò)重分形分析的方法主要都是針對無權(quán)的網(wǎng)絡(luò),其中也有我們課題組最近提出的沙箱算法。但這些已有的方法都不再適用對加權(quán)的復雜網(wǎng)絡(luò)進行重分形分析。本論文提出了改進的沙箱算法(我們稱它為SBw算法)以適用于對加權(quán)的復雜網(wǎng)絡(luò)進行重分形分析,首先我們利用SBw算法通過對構(gòu)造的“Sierpinski”加權(quán)分形網(wǎng)絡(luò)家族和“Cantor dust”加權(quán)分形網(wǎng)絡(luò)家族進行了分形及重分形的分析,我們也討論了分形維數(shù)和廣義分形維數(shù)隨著加權(quán)分形網(wǎng)絡(luò)的權(quán)值變化而變化的規(guī)律。通過比較加權(quán)分形網(wǎng)絡(luò)的理論分形維數(shù)與用SBw算法得到的數(shù)值結(jié)果,表明SBw算法針對加權(quán)網(wǎng)絡(luò)的分形及重分形分析是可行的也是有效的。然后,我們應(yīng)用加權(quán)的沙箱算法研究幾類實際加權(quán)科學家合作網(wǎng)絡(luò)的多重分形性質(zhì)。發(fā)現(xiàn)多重分形存在于這些加權(quán)網(wǎng)絡(luò),并受他們邊的權(quán)重影響。2、分數(shù)布朗運動通過水平可視圖方法構(gòu)建加權(quán)水平可視網(wǎng)絡(luò),并研究了這些加權(quán)網(wǎng)絡(luò)的基本拓撲性質(zhì)。對于不同的Hurst指數(shù)H的分數(shù)布朗運動所構(gòu)建的加權(quán)水平可視網(wǎng)絡(luò),本文數(shù)值研究了它們的度分布,強度分布,聚集系數(shù)以及經(jīng)典的Laplace算子和改進的Laplace算子的次小特征值和最大特征值與Hurst指數(shù)H的關(guān)系,研究了Hurst指數(shù)H對加權(quán)水平可視網(wǎng)絡(luò)拓撲性質(zhì)的影響規(guī)律。數(shù)值分析了所構(gòu)建網(wǎng)絡(luò)的分形及重分形性質(zhì),分析比較不同的Hurst指數(shù)H/對網(wǎng)絡(luò)分形及重分形特性的影響。通過比較已有的不加權(quán)的水平可視網(wǎng)絡(luò)的基本特征,探究網(wǎng)絡(luò)的權(quán)值對整個時間序列的影響。3、基于相空間重構(gòu)的方法由分數(shù)布朗運動構(gòu)建加權(quán)遞歸網(wǎng)絡(luò),并研究了這些加權(quán)網(wǎng)絡(luò)的基本拓撲性質(zhì)。與用水平可視化構(gòu)建加權(quán)的網(wǎng)絡(luò)類似,本文數(shù)值研究了度分布,強度分布,聚集系數(shù)與Hurst指數(shù)H關(guān)系;從幾何角度,本文研究了加權(quán)遞歸網(wǎng)絡(luò)的分形及重分形性質(zhì);從代數(shù)角度,本文對加權(quán)遞歸網(wǎng)絡(luò)的譜進行了分析。所得結(jié)果與無權(quán)遞歸網(wǎng)絡(luò)比較,探究權(quán)值對這些統(tǒng)計量的影響;與水平可視圖方法構(gòu)建的加權(quán)網(wǎng)絡(luò)比較,探討兩種不同的方法對這些統(tǒng)計特征影響。這兩種不同的構(gòu)建加權(quán)網(wǎng)絡(luò)的方法,都是對原始分數(shù)布朗運動更精細的刻畫模型,為研究時間序列提供新的參考方法。
[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.

【學位授予單位】：湘潭大學
【學位級別】：博士
【學位授予年份】：2017
【分類號】：O157.5

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本文編號：1676236