本文選題：復(fù)雜網(wǎng)絡(luò) + 二分網(wǎng)絡(luò)�。� 參考：《湘潭大學(xué)》2017年博士論文

【摘要】：復(fù)雜網(wǎng)絡(luò)已經(jīng)吸引了來(lái)自科學(xué)技術(shù)不同領(lǐng)域研究者的大量關(guān)注。繼復(fù)雜網(wǎng)絡(luò)中的小世界特性與無(wú)標(biāo)度性質(zhì)之后,自相似性已經(jīng)成為復(fù)雜網(wǎng)絡(luò)的第三大基本特征并且在近年來(lái)得到了廣泛的研究。分形分析可以有效地揭示一些分形對(duì)象的自相似性。然而,重分形分析是一個(gè)用來(lái)系統(tǒng)性地刻畫(huà)理論和實(shí)驗(yàn)復(fù)雜分形對(duì)象空間異質(zhì)性的更加強(qiáng)大而有效的工具。盡管已有一些研究復(fù)雜網(wǎng)絡(luò)自相似性的分形和重分形分析算法,但是這些算法的效率并不高而且已有的算法并不是對(duì)所有類(lèi)型的復(fù)雜網(wǎng)絡(luò)總是有效的。因此,研發(fā)高效、可行、精確的分形和重分形分析算法以及針對(duì)特殊類(lèi)型的網(wǎng)絡(luò)提出特殊的算法顯得尤為重要。在本文中,我們改進(jìn)分形幾何中已有的沙箱算法(sandbox algorithm)去研究復(fù)雜網(wǎng)絡(luò)的重分形性質(zhì)。首先我們通過(guò)計(jì)算一些確定性模型網(wǎng)絡(luò)的質(zhì)量指數(shù)τ(q)去將改進(jìn)的沙箱算法與已有的兩個(gè)重分形分析算法進(jìn)行了比較,這兩個(gè)算法分別是改進(jìn)的緊盒子燃燒算法(compact-box-burning algorithm)和改進(jìn)的計(jì)盒算法(box-counting algorithm)。我們對(duì)這些確定性模型網(wǎng)絡(luò)的質(zhì)量指數(shù)τ(q)的理論解和由這三個(gè)算法計(jì)算得到的數(shù)值結(jié)果做了詳細(xì)的比較。比較結(jié)果顯示改進(jìn)的沙箱算法對(duì)于計(jì)算網(wǎng)絡(luò)的質(zhì)量指數(shù)τ(q)并進(jìn)一步研究復(fù)雜網(wǎng)絡(luò)的重分形性質(zhì)是最有效、最可行的。然后,我們使用改進(jìn)的沙箱算法去研究了其他一些經(jīng)典模型網(wǎng)絡(luò)的重分形性質(zhì),包括無(wú)標(biāo)度網(wǎng)絡(luò)、小世界網(wǎng)絡(luò)與隨機(jī)網(wǎng)絡(luò)。數(shù)值結(jié)果表明在無(wú)標(biāo)度網(wǎng)絡(luò)中存在重分形性質(zhì),重分形性質(zhì)在小世界網(wǎng)絡(luò)中并不明顯,而隨機(jī)網(wǎng)絡(luò)則幾乎不具有重分形性質(zhì)。最近,二分網(wǎng)絡(luò)(bipartite network)也已經(jīng)引起了不同領(lǐng)域研究者的大量興趣。單分網(wǎng)絡(luò)(unipartite network)或者經(jīng)典網(wǎng)絡(luò)(classical network)的分形和重分形性質(zhì)在近年來(lái)已經(jīng)得到了研究,但是目前還沒(méi)有工作去研究二分網(wǎng)絡(luò)的這些性質(zhì)。在本文中,我們通過(guò)對(duì)大量真實(shí)二分網(wǎng)絡(luò)數(shù)據(jù)集和由一些二分網(wǎng)絡(luò)模型生成的二分網(wǎng)絡(luò)進(jìn)行分形和重分形分析試圖去揭示二分網(wǎng)絡(luò)的自相似結(jié)構(gòu)。首先,我們發(fā)現(xiàn)在一些二分網(wǎng)絡(luò)中存在分形性質(zhì),包括CiteULike、Netflix、MoieLens(ml-20m)、Delicious真實(shí)二分網(wǎng)絡(luò)數(shù)據(jù)集和由(u,v)-flower模型所生成的第7代(2, 2)-flower二分網(wǎng)絡(luò)。同時(shí),我們也發(fā)現(xiàn)并不是所有二分網(wǎng)絡(luò)都具有明顯的自相似性,即在其他幾個(gè)二分網(wǎng)絡(luò)中觀察到了偏移冪律和指數(shù)行為。我們接下來(lái)研究了二分網(wǎng)絡(luò)的重分形性質(zhì)。結(jié)果表明上面具有分形性質(zhì)的二分網(wǎng)絡(luò)還具有重分形性質(zhì)。為了捕捉二分網(wǎng)絡(luò)包含兩類(lèi)不同節(jié)點(diǎn)的本質(zhì)特征,我們受到推薦系統(tǒng)中基于網(wǎng)絡(luò)資源配置動(dòng)力學(xué)的啟發(fā)給二分網(wǎng)絡(luò)中不同類(lèi)的節(jié)點(diǎn)賦予了不同的權(quán)重。重分形分析的結(jié)果顯示這些節(jié)點(diǎn)加權(quán)二分網(wǎng)絡(luò)存在重分形性質(zhì)。另外,對(duì)于帶評(píng)分的二分網(wǎng)絡(luò)數(shù)據(jù)集,我們構(gòu)建了對(duì)應(yīng)的邊加權(quán)二分網(wǎng)絡(luò)。相應(yīng)地,已有的兩個(gè)用來(lái)研究邊加權(quán)單分網(wǎng)絡(luò)分形和重分形性質(zhì)的算法被修改以使得它們能夠用于去研究這些邊加權(quán)二分網(wǎng)絡(luò)的自相似性。研究結(jié)果表明這些修改的算法是可行的,能夠有效地揭示這些邊加權(quán)二分網(wǎng)絡(luò)和相應(yīng)的節(jié)點(diǎn)加權(quán)二分網(wǎng)絡(luò)的自相似結(jié)構(gòu)。最后,我們把修改的算法用于去研究幾個(gè)已經(jīng)常被其他研究人員所研究的無(wú)權(quán)單分網(wǎng)絡(luò)的分形和重分形性質(zhì)。我們的分形結(jié)果和之前的研究結(jié)果是一致的。因此,這些修改的算法也能有效地用來(lái)研究無(wú)權(quán)單分網(wǎng)絡(luò)的自相似性。
[Abstract]:Complex networks have attracted a lot of attention from various fields of science and technology. The following small world properties in complex networks and scale-free properties, self similarity has become the third basic characteristics of complex networks and has been widely studied in recent years. The fractal analysis can effectively reveal the self similarity of fractal object. However, multifractal analysis is used to systematically describe the theory and experiment of complex fractal object spatial heterogeneity is more powerful and effective tool. Although there have been some research on Fractal and multifractal analysis algorithm of self similar complex networks, but the efficiency of these algorithms is not high and the existing algorithm is not all types of complex networks is effective. Therefore, developing an efficient, feasible, and put forward the algorithm for a special type of network analysis accurate fractal and multifractal A special algorithm is particularly important. In this paper, we improved the existing algorithm of fractal geometry in the sandbox (sandbox algorithm) to study the multifractal properties of complex networks. We calculate the quality index of some deterministic model of network tau (q) to two multifractal algorithm with the existing sand box improved analysis algorithm comparing the two algorithms are tight box improved burning algorithm (compact-box-burning algorithm) and the improved box counting algorithm (box-counting algorithm). Our quality index of these networks is a deterministic model (q) theory solution and the numerical calculated by the three algorithm results made a detailed comparison. Comparison results show that the improved algorithm for computing network sandbox quality index (q) the multifractal properties and further the study of complex networks is the most effective, the most feasible. Then, we use the The improved algorithm to study the sandbox multifractal properties of some other classical models including network, scale-free network, small world network and the random network. The numerical results show that the existence of multifractal properties in scale-free networks, multifractal properties in the small world network and the random network is not obvious, little has multifractal nature. Recently, two network (bipartite network) has attracted a lot of interest of researchers in different fields. A single network (unipartite network) or classical network (classical network) the fractal and multifractal properties have been studied in recent years, but there is no work to study these properties of two networks. In this paper, we based on a large number of real data sets and two network generated by some two network models of two sub networks of fractal and multifractal analysis to reveal the two network Self similar network structure. Firstly, we found that there are some two points in the fractal property of the network including CiteULike, Netflix, MoieLens, Delicious (ml-20m) two real network data set and by (U, V) -flower model generated by the seventh generation (2, 2) -flower two network. At the same time, we also found that not all two networks have obvious self similarity, i.e. the observed power-law index and migration behavior in several other two points in the network. Then we study the multifractal properties of two sub networks. The results show that the two points above the network with fractal properties has multifractal properties in order to capture. Two different types of nodes of the network includes two essential features, we are inspired by the cyber source to different types of dynamic configuration of two nodes in the network gives different weights based on recommendation system. The results of multifractal analysis show that these day Two point weighted network has multifractal properties. In addition, the two network data with the score set, we constructed two points corresponding to the edge weighted network. Accordingly, the two has been used to study the single edge weighted network fractal and multifractal properties of the algorithm are modified to make them can be used for self similar to study these two edge weighted network. The results show that the modified algorithm is feasible and can effectively reveal the self similar structure of these two edge weighted network and the corresponding node weighted two points of the network. Finally, we have modified the fractal method is applied to study several other researchers have often been the right to a single network and multifractal properties. The results of our research on Fractal and previous results are consistent. Therefore, the modified algorithm can be effectively used to study a single network from unauthorized Similarity.

【學(xué)位授予單位】：湘潭大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O157.5

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本文編號(hào)：1741101