基于度分布的复杂网络拓扑结构建模研究
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摘要
复杂网络广泛地存在于自然界和人类社会中,从Internet, WWW到化学反应、生物食物链,再到人类社会的人际关系、人们之间的工作合作、科技引文,甚至人类性伙伴等都呈现出复杂网络的拓扑特性。近年来,复杂网络的研究得到了迅速地发展,己经遍及各个学科领域,如生物学、物理学,甚至社会科学。究其原因主要是由于计算能力的提高,使人们能够对包含数以千万计节点的各种现实网络进行研究。另外,人类迫切需要从整体上去认识各种复杂网络内部各部分之间的相互关系,以揭示出具有某些指导意义的规律。本文利用图论、统计学、计算机模拟等方法,获得了具有任意度分布的复杂网络拓扑结构。
主要研究内容如下:
(1)本文首先对复杂网络及其相关领域作了概述,给出了描述复杂网络结构特性的几个重要统计特征:度分布、最短路径和集团系数。同时介绍了ER模型、WS模型和BA模型等著名的网络拓扑模型。
(2)给出了度秩函数的概念,推导出了网络节点度分布和度秩函数的数学关系并给出了详细的理论证明,给出了具有代表性的无标度网络和指数网络的度秩函数的数学表达式并进行了仿真分析。利用度秩函数,分析研究了无标度网络在各种标度指数下网络节点的最大度和平均度并进行了仿真。
(3)建立了具有任意度分布的复杂网络拓扑结构模型并用无标度网络及指数网络进行了验证。
(4)构造了复杂网络拓扑结构生成器,用实例展现了拓扑生成器的功能及效果。
Complex networks describe a wide range of systems in nature and society. Frequently cited examples include Internet, WWW, a network of chemicals linked by chemical reactions, social relationship networks, citation networks, etc. In recent years, the research on complex network has been developing rapidly and is extended to many science fields, such as biology, physics and even social science. It boils down to the first reason is with the improvement of computing capability, people can do research in various realistic networks including multimillion nodes which could not be realized in the past, the second reason is they also need to recognize various networks urgently to find out instructional rules. Make use of graph theory, statistical theory and computer simulation etc, this thesis could get the network topology with arbitrary degree distribution. The main contents are outlined as follows:
(1) We first introduced the recent progress in the study of complex networks. The basic concepts, such as degree distribution, clustering coefficient, average path length, which characterize the network topologic structure, are defined. Topology models, such as ER model, WS model and BA model are introduced.
(2) The degree-rank function is proposed as a statistic characteristic of complex network and the mathematical relationship between degree-rank function and degree distribution is derived. Then the degree-rank function of scale-free network and exponential network is given. Make use of the degree-rank function, the maximum degree and average degree of scale-free network is studied.
(3) A method of constructing complex networks with arbitrary degree distributions is proposed. Take scale-free network and exponential network as examples, the efficiency of the method is verified.
(4) Topology builder of complex network is designed and the function of topology builder is validated by example.
主要研究内容如下:
(1)本文首先对复杂网络及其相关领域作了概述,给出了描述复杂网络结构特性的几个重要统计特征:度分布、最短路径和集团系数。同时介绍了ER模型、WS模型和BA模型等著名的网络拓扑模型。
(2)给出了度秩函数的概念,推导出了网络节点度分布和度秩函数的数学关系并给出了详细的理论证明,给出了具有代表性的无标度网络和指数网络的度秩函数的数学表达式并进行了仿真分析。利用度秩函数,分析研究了无标度网络在各种标度指数下网络节点的最大度和平均度并进行了仿真。
(3)建立了具有任意度分布的复杂网络拓扑结构模型并用无标度网络及指数网络进行了验证。
(4)构造了复杂网络拓扑结构生成器,用实例展现了拓扑生成器的功能及效果。
Complex networks describe a wide range of systems in nature and society. Frequently cited examples include Internet, WWW, a network of chemicals linked by chemical reactions, social relationship networks, citation networks, etc. In recent years, the research on complex network has been developing rapidly and is extended to many science fields, such as biology, physics and even social science. It boils down to the first reason is with the improvement of computing capability, people can do research in various realistic networks including multimillion nodes which could not be realized in the past, the second reason is they also need to recognize various networks urgently to find out instructional rules. Make use of graph theory, statistical theory and computer simulation etc, this thesis could get the network topology with arbitrary degree distribution. The main contents are outlined as follows:
(1) We first introduced the recent progress in the study of complex networks. The basic concepts, such as degree distribution, clustering coefficient, average path length, which characterize the network topologic structure, are defined. Topology models, such as ER model, WS model and BA model are introduced.
(2) The degree-rank function is proposed as a statistic characteristic of complex network and the mathematical relationship between degree-rank function and degree distribution is derived. Then the degree-rank function of scale-free network and exponential network is given. Make use of the degree-rank function, the maximum degree and average degree of scale-free network is studied.
(3) A method of constructing complex networks with arbitrary degree distributions is proposed. Take scale-free network and exponential network as examples, the efficiency of the method is verified.
(4) Topology builder of complex network is designed and the function of topology builder is validated by example.
引文
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[3] Scott J. Social Network Analysis: A Handbook [M]. London: Sage Publications, 2000.
[4] Wasserman S, Faust K. Social Network Analysis: Methods and Applications[M]. Cambridge: Cambridge University Press, 1994.
[5] Sporns O, Tononi G, Edelman G M. Theoretical neuroanatomy: Relating anatomical and functional connectivity in graphs and cortical connection matrices[J]. Cerebral Cortex, 2000, 10 127-141.
[6] Vázquez A, Pastor-Satorras R, Vespignani A. Large-scale topological and dynamical properties of the Internet[J]. Phys. Rev. E, 2002, 65 (6): 066130.
[7] Faloutsos M, Faloutsos P, Faloutsos C. On power-law relations of the Internet topology[J]. Comput. Commun. Rev., 1999, 29 251-262.
[8] Barabási A-L, Albert R, Jeong H. Scale-free characteristics of random networks: the topology of the World Wide Web[J]. Physica A, 2000, 281 69-77.
[9] Adamic L A, Huberman B A. Power-law distribution of the world wide web[J]. Science, 2000, 287 2115.
[10] Huberman B A. The Laws of the Web[M]. Cambridge,: MIT Press, 2001.
[11] Serrano M A, Bogu?á M. Topology of the world trade web[J]. Phys. Rev. E, 2003, 68 (1): 015101.
[12] Andersson C, Hellervik A, Lindgren K. Urban economy as a scale-free network[J]. Phys. Rev. E, 2003, 68 (3): 036124.
[13] Barabási A-L, Albert R. Emergence of scaling in random networks[J]. Science, 1999, 286 509-512.
[14] 车宏安, 顾基发. 无标度网络及其系统科学意义[J]. 系统工程理论与实践, 2004, 24 (4): 11-16.
[15] 姜璐, 刘琼慧. 系统科学与复杂网络研究[J]. 系统辩证学学报, 2005, 13 (4): 14-17.
[16] 吴金闪, 狄增如. 从统计物理学看复杂网络研究[J]. 物理学进展, 2004, 24 (1): 18-46.
[17] 汪秉宏, 周涛, 何大韧. 统计物理学与复杂系统研究最新发展趋势分析[J]. 中国基础科学, 2005, 7 (3): 37-43.
[18] Watts D J, Strogatz S H. Collective dynamics of 'small-world' networks[J]. Nature, 1998,393 (6684): 440-442.
[19] 张凤林. 物流保障网络可用性技术研究[D]. 长沙: 国防科学技术大学, 2003.
[20] 王盛明主编. 战役后勤战例研究[M]. 北京: 解放军出版社, 2002.
[21] Tangmunarunkit H, Govindan R, Shenker S, et al. Proceedings of IEEE Infocom [C]. Alaska: 2001.
[22] Erd?s P, Rényi A. On random graphs[J]. Publ. Math., 1959, 6 290-297.
[23] Erd?s P, Rényi A. On the evolution of random graphs[J]. Publ. Math. Inst. Hung. Acad. Sci., 1960, 5 17-60.
[24] Albert R, Jeong H, Barabási A-L. Diameter of the world-wide web[J]. Nature, 1999, 401 130-131.
[25] Jeong H, Tombor B, Albert R, et al. The large-scale organization of metabolic networks[J]. Nature, 2000, 407 651-654.
[26] Amaral L A N, Scala A, Barthélémy M, et al. Classes of behavior of small-world networks[J]. Proc. Natl. Acad. Sci. U.S.A, 2000, 97 11149-11152.
[27] Ito T, Chiba T, Ozawa R, et al. A comprehensive two-hybrid analysis to explore the yeast protein interactome[J]. Proc. Natl. Acad. Sci. USA, 2001, 98 (8): 4569-4574.
[28] Dodds P S, Rothman D H. Geometry of river networks[J]. Phys. Rev. E, 2001, 63 016115
[29] Barabási A-L, Jeong H, Ravasz E, et al. Evolution of the social network of scientific collaborations[J]. Physica A, 2002, 311 590-614.
[30] Aiello W, Chung F, Lu L. Random evolution of massive graphs[M]. Dordrecht: Kluwer, 2002.
[31] Newman M E J, Forrest S, Balthrop J. Email networks and the spread of computer viruses[J]. Phys. Rev. E, 2002, 66 (3): 035101.
[32] Latora V, Marchiori M. Is the Boston subway a small-world network?[J]. Physica A, 2002, 314 109-113.
[33] Montoya J M, Solé R V. Small world patterns in food webs[J]. J. Theor. Bio, 2002, 214 405-412.
[34] Sporns O. Network analysis, complexity, and brain function[J]. Complexity, 2002, 8 (1): 56-60.
[35] Farkas I J, Jeong H, Vicsek T, et al. The topology of the transcription regulatory network in the yeast, Saccharomyces cerevisiae[J]. Physica A, 2003, 381 601-612.
[36] Price D J d S. Networks of scientific papers[J]. Science, 1965, 149 510-515.
[37] Milgram S. The small world problem[J]. Psychology Today, 1967, 2 60-67.
[38] Barrat A, Weigt M. On the properties of small world networks[J]. Eur. Phys. J. B 2000, 13 547-560.
[39] Bianconi G, Capocci A. Number of loops of size h in growing scale-free networks[J]. Phys. Rev. Lett., 2003, 90 078701.
[40] Albert R, Jeong H, Barabási A-L. Error and attack tolerance of complex networks [J]. Nature, 2000, 406 378-382.
[41] Pastor-Satorras R, Vázquez A, Vespignani A. Dynamical and Correlation Properties of the Internet[J]. Phys. Rev. Lett., 2001, 87 (25): 258701.
[42] Maslov S, Sneppen K, Zaliznyak A. Pattern detection in complex networks: Correlation profile of the Internet[J]. cond-mat/0205379, 2002.
[43] Adamic L A, Lukose R M, Huberman B A. Handbook of graphs and networks[M]. Berlin: Wiley-VCH, 2003.
[44] Adamic L A, Lukose R M, Puniyani A R, et al. Search in power-law networks[J]. Phys. Rev. E, 2001, 64 046135.
[45] Kleinberg J M. Navigation in a small world[J]. Nature, 2000, 406 845.
[46] Goh K-I, Oh E, Jeong H, et al. Classification of scale-free networks[J]. Proc. Natl. Acad.Sci. USA, 2002, 99 12583-12588.
[47] Flake G W, Lawrence S R, Giles C L, et al. Self-organization and identification of Web communities[J]. IEEE Comput., 2002, 35 66-71.
[48] Girvan M, Newman M E J. Community structure in social and biological networks[J]. Proc. Natl. Acad.Sci. USA, 2002, 99 8271-8276.
[49] Moody J. Race school integration, and friendship segregation in America[J]. Am. J. Sociol., 2001, 107 679-716.
[50] Newman M E J. Mixing patterns in networks[J]. Phys. Rev. E, 2003, 67 (2): 026126
[51] Price D J d S, Amer. J. A general theory of bibliometric and other cumulative advantage processes[J]. Soc. Inform. Sci., 1976, 27 292-306.
[52] Kleinberg J M, Kumar S R, Raghavan P, et al. Proceedings of the International Conference on Combinatorics and Computing [C]. Berlin: Springer, 1999.
[53] Cohen R, Erez K, ben-Avraham D. Breakdown of the Internet under Intentional Attack[J]. Phys. Rev. Lett., 2001, 86 (16): 3682-3685.
[54] Callaway D S, Newman M E J, Strogatz S H, et al. Network robustness and fragility: percolation on random graphs[J]. Phys. Rev. Lett., 2000, 85 (25): 5468-5471.
[55] Moore C, Newman M E J. Epidemics and percolation in small-world networks[J]. Phys. Rev. E, 2000, 61 5678-5682.
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