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复杂网络的社团结构分析 Community structure in complex networks

复杂网络的社团结构分析 Community structure in complex networks. 章祥荪. http://zhangroup.aporc.org 中国科学院 数学与系统科学研究院 全国复杂网络会议，苏州大学， 2010 ， 10, 17. Bio-molecular networks （生物分子网）. 许多生物问题 , 特别是人类的疾病 , 在分子层面上都可归于 “ systems problems ” -- Leroy Hood

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复杂网络的社团结构分析 Community structure in complex networks

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  1. 复杂网络的社团结构分析Community structure in complex networks 章祥荪 http://zhangroup.aporc.org 中国科学院 数学与系统科学研究院 全国复杂网络会议，苏州大学， 2010，10, 17

  2. Bio-molecular networks （生物分子网） • 许多生物问题, 特别是人类的疾病, 在分子层面上都可归于 “systems problems” -- Leroy Hood • 许多生物问题可以表达成生物分子网络（bio-molecular networks）的问题。 • 生物分子网络包括：蛋白质相互作用网（ protein interaction networks）, 新陈代谢网（metabolic networks），基因调控网（ gene regulatory networks）, e.t.; 他们都有共同的性质 • 更为有趣的是，许多这样的网是“复杂”网络

  3. 复杂网络的典型代表:生物分子网络之一 ---- 蛋白质相互作用网 （Scale-free) 酵母细胞中的蛋白质相互作用网络 (A.-L. Barabási, NATURE REVIEWS GENETICS, 2004)

  4. Jeong, 2000, Nature 包括太古代（ Archae），细菌（ Becterium), 真核生物（Eukaryote）在内的43个物种的新陈代谢网（ Metabolic network ）都是 Scale-free的。

  5. Protein-protein interaction networks • Rui-Sheng Wang, Yong Wang, Ling-Yun Wu, Xiang-Sun Zhang, Luonan Chen.Analysis on multi-domain cooperation for predicting protein-protein interactions.BMC Bioinformatics, 8:391, 2007 • Shihua Zhang, Xue-Mei Ning and Xiang-Sun Zhang.Identification of functional modules in a PPI network by clique percolation clustering.Computational biology and chemistry, 30(6), 445-451, 2006. • Luonan Chen, Ling-Yun Wu, Yong Wang and Xiang-Sun Zhang.Inferring Protein Interactions from Experimental Data by Association Probabilistic Method.Proteins: Structure, Function, and Bioinformatics, Vol. 62, pp. 833-837, 2006. • Xiang-Sun Zhang, Rui-Sheng Wang, Ling-Yun Wu, Shihua Zhang and Luonan Chen.Inferring Protein-Protein Interactions by Combinatorial Models.IFMBE Proceedings, Vol.14, 2006, 183--186, Springer Berlin Heidelberg. 5

  6. Metabolic and signaling networks • Zhenping Li, Rui-Sheng Wang, Xiang-Sun Zhang and Luonan Chen. Detecting drug targets with minimum side effects in metabolic networks.IET Systems Biology, 3(6), 523-533, 2009 • Zhenping Li, Rui-Sheng Wang, Xiang-Sun Zhang.Mass Flow Model and Essentiality of Enzymes in Metabolic Networks.Lecture Notes in Operations Research, 9, pp. 182-190, World Publishing Corporation, Lijiang, 2008. • Jin G, Zhou X, Wang H, Zhao H, Cui K, Zhang XS, Chen L, Hazen SL, Li K, Wong ST The Knowledge-Integrated Network Biomarkers Discovery for Major Adverse Cardiac Events. J Proteome Res 7(9): 4013-4021,2008 6

  7. Book about Biomolecular networks Luonan Chen, Rui-Sheng Wang, Xiang-Sun Zhang.Biomolecular Networks: Methods and Applications in Systems Biology.John Wiley & Sons, Hoboken, New Jersey. July, 2009. 7

  8. Yeast functional linkage network SCIENCE Vol 306(26) 2004 DNA damage module 可分成564 个模块，由 950 个显著的块间相互作用相连接。

  9. 复杂网络的动态性质研究 • 复杂网络的静态结构研究 • 小世界(Small world)，尺度无关（Scale free)，聚类特性(Clustering)的确切数学模型。 • 社团结构 (Community Structure) • …………

  10. 复杂网络的模块化性质 • 复杂网络中存在模块或者社区结构 (Module or Community structure) • 模块或者社区定义为网络中内部连接稠密，与外部连接稀疏的节点的集合 (Filippo Radicchi et. al. PNAS, Vol.101, No.9, 2658-2663, 2004). • 数学表述: 其中V是子图，K是顶点的度。即子图V是模块的条件是模块内顶点的内部连边的度值之和大于模块内顶点的外部连边的度值之和。 PNAS ---- Proc. Natl. Acad. Sci. USA 美国科学院院刊

  11. 模块划分的重要性 • 许多复杂网络共有的性质。 • 研究模块结构有助于研究整个网络的结构和功能 圣塔菲研究所的科学家合作网：模块代表从事相似领域研究的科学家集合 数学生态学 统计物理

  12. 自然科学论文引用网络：6128期刊, 约600万次引用， Martin Rosvall, Carl T. Bergstrom, PNAS, vol. 105, no.4. 1118-1123, 2007 划分为88个模块和3024条 模块间的连接，刻画了学科之间的联系

  13. W. W. Zachary, An information flow model for conflict and fission in small groups, Journal of Anthropological Research33, 452-473 1977 一个社会网络的例子 • 1970年美国大学里的一个空手道俱乐部关系网络：节点是其34名成员，边是他们两年间的友谊关系，边数为78。俱乐部里的矛盾导致其分裂为两个小的俱乐部。问题是能否用网络的模块结构来重现这个过程？ • 它是模块探测研究中的经典例子。

  14. Importance of the topic • Girvan, M, Newman, M., Proc. Natl. Acad. Sci, 2002 • Ravasz, E, Somera, A, Mongru, D, Oltvai, Z, Barabasi, A., Science, 2002 • Radicchi, F, Castellano, C, Cecconi, F., Proc. Natl. Acad. Sci, 2004 • Guimera, R, Mossa, S, Turtschi, A., Proc. Natl. Acad. Sci, 2005 • Guimera, R, Amaral, L., Nature, 2005 • Newman, M., Proc. Natl. Acad. Sci, 2006 • Rosvall, M, Bergstrom, C.,Proc. Natl. Acad. Sci, 2007 • Fortunato, S, Barthelemy, M., Proc. Natl. Acad. Sci, 2007 • Weinan, E, Li, T, Vanden-Eijnden, E., Proc. Natl. Acad. Sci, 2008 • Rosvall, M, Bergstrom, C., Proc. Natl. Acad. Sci,2008 • Peter J. Mucha, et al., Science2010 • Yong-Yeol Ahn, James P. Bagrow & Sune Lehmann，Nature, 2010 生物信息学与最优化方法

  15. 社团结构探索方法概述 A large number of methods have been developed for detecting communities, which can be generally categorized into local and global methods. • Local methods （局部方法）for community detection identify a subset of nodes as a community according to certain local connection conditions, independently from the structure of the rest of the network. Such methods include clique overlap-based hierarchical clustering, clique percolation method, and sub-graph fitness method. • Global methods （全局方法）for community detection optimize certain global quantitative functions encoding the quality of the overall partition of the network, such as information theoretical method, Potts model, and optimization of modularity measures.

  16. 我们小组在研究这一问题的早期发展了一些基于图论和矩阵谱分解的模块探测算法 (local method) Shihua Zhang, Rui-Sheng Wang, and Xiang-Sun Zhang. Identification of overlapping community structure in complex networks using fuzzy c-means Clustering. Physica A, 2007, 374, 483–490. Shihua Zhang, Rui-Sheng Wang and Xiang-Sun Zhang. Uncovering fuzzy community structure in complex networks. Physical Review E, 76, 046103, 2007 Rui-Sheng Wang, Shihua Zhang, Yong Wang, Xiang-Sun Zhang, Luonan Chen. Clustering complex networks and biological networks by nonnegative matrix factorization with various similarity measures. Neurocomputing, 2007

  17. 衡量网络模块化的指标Q值 • 设网络为 N=（V，E）, Pk = { (V1, E1), …, (Vk, Ek)} 为一个分划。L(Vi, Vj) =|Eij|, i in Vi, j in Vj. • Newman 和 Girvan (Physical Review E, 2004) 提出一种衡量网络社区结构的指标Q 值

  18. 指标Q的问题 (Resolution limit)Fortunato and Barthélemy, PNAS, 2007 • 利用Q划分网络的计算步骤: • 目前很大一部分模块探测的方法集中于利用各种启发式算法来极大化Q值 ，例如模拟退火、遗传算法等(Newman, PNAS, 2006; Guimera, Nature, 2005). • Resolution limit 现象

  19. 极端例子：ring of cliques Fortunato & Barthelemy, Proc. Natl. Acad. Sci. USA 104 (1), 36-41 (2007)

  20. 提出新的模块化指标D值 • 模块化密度函数 D: Zhenping Li, Shihua Zhang, Rui-Sheng Wang, Xiang-Sun Zhang, Luonan Chen, Quantitative function for community detection. Physical Review E, 77, 036109, 2008

  21. D值克服了Q值存在的 resolution limit 问题

  22. 结果 D值 划分正确的顶点的比例 Q值

  23. 错分现象---Misidentification • 用Q或D作优化可能得到不满足定义的模块 Q partitions the network into three communities (two Kn and one K5) when n>=16 (respectively, n>=21), in which K5 is a sub-graph violating all reasonable community definition. Xiang-Sun Zhang, Rui-Sheng Wang, Yong Wang, Ji-Guang Wang, Yu-Qing Qiu, Lin Wang, and Luonan Chen. Modularity optimization in community detection of complex networks. Europhysics Letters (EPL), 87, 2009. 被评为 EPL 2009 best paper 23

  24. 该文的主要贡献是用离散凸规划的概念对两个重要问题进行解析分析该文的主要贡献是用离散凸规划的概念对两个重要问题进行解析分析 • Q 值和D 值的最优化模型都是非线性整数规划 • 目标函数的凸性和凹性无法解析得到 • 对两个具有特殊结构的网络进行分析 • 引入离散凸规划（变量是离散的，可以嵌入一个连续的凸规划）的概念进行分析, 得到解析解

  25. 所有对modularity进行研究的论文(指上面所列的的PNAS,Nature,Sience文章)都是试题论证的,即没有解析的证明.所有对modularity进行研究的论文(指上面所列的的PNAS,Nature,Sience文章)都是试题论证的,即没有解析的证明. • 为了彻底分析resolution limit和Misidentification现象，我们对两类典型网络建立了优化模型,引入了离散凸分析技术,得到了两类问题的解析解. 生物信息学与最优化方法

  26. 基于特殊结构的凸分析 • 这两个例子出现在PNAS中几乎所有讨论网络模块探测的论文里 ring of dense lumps ad hoc network

  27. Finding 1 对

  28. Finding 2 生物信息学与最优化方法

  29. Finding 3 • 解析解表明，对这两个经典的算例，Q和D都有Resolution limit和Misidentification的现象产生，所以Q 和D均只是近似的定量评估函数。 • 网络社团划分的问题可以用一个优化问题来精确 描述，我们证明了这一模型是NP-hard的。 • 我们相信用优化理论可以彻底解决网络社团划分 的问题。网络科学是运筹学的下一个热点。

  30. 直接得到团环网络的社团结构数值解

  31. 直接得到ad hoc网络的社团结构数值解

  32. 为了彻底解决这些问题 • 提出一个新的 OR 模型和相应的算法，这一算法不会产生resolution limit 和 mis-identification 现象 • 关键思路：模块分划质量函数的定义要包含社团定义。 Xiang-Sun Zhang, Zhenping Li, Rui-Sheng Wang, Yong Wang. A combinatorial model and algorithm for globally searching community structure in complex networks Journal of Combinatorial Optimization (JCO), 2010. DOI: 10.1007/s10878-010-9356-0

  33. A new OR model • Problem definition: Given a network, the community identification problem is to partition the network into as many non-overlapping sub-networks as possible such that each sub-network satisfies a given community definition. 给定一个网络和一个社团的定义，社团结构识别的问题就是将整个网络分成尽可能多的满足社团定义的子网络。 34

  34. 以上文字定义可以用一个整数线性规划来描述 • 我们证明了这个模型是 NP-hard . 35

  35. A qualified min-cut (QMC) algorithm • A heuristic principle is given to find a feasible partition with the largest number of communities. • It is realized by a min-cut operation: A min-cut operation is calledqualified if the two resulting sub-networks satisfy the module definition. • The community identification problem can be solved based on a series of qualified min-cut operations. 36

  36. Experiment results (artificial networks) Rings of cliques Uneven ad-hoc network 37

  37. Experiment results (real networks) Football team network Jazz musician network 38

  38. 学术性的，实用性的问题远远没有解决 • Yong-Yeol Ahn, James P. Bagrow & Sune Lehmann，Nature, 2010 Link communities reveal multiscale complexity in networks

  39. 致谢 This work is cooperated with Dr. 李珍萍，Dr. 王瑞省，Dr. 王勇，Dr. 张世华， Dr. 王吉光，Dr. 张俊华 This work is supported by 国家自然科学重点基金10631070 973项目2066CB503905 国家自然科学基金项目60873205

  40. Thanks Welcome to visit us at http://zhangroup.aporc.org

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