678 力 展 2008年第38卷 42 Newmanm E J. Assortative mixing in networks. Phys Rev 2005,72:016123 Let,2002,89(20):208701 54 Sen P. Accelerated growth in outgoing links in evolving 43N M EJ, Park J. Why social networks are dif- networks: Deterministic versus stochastic picture. Phys ferent from other types of networks. Phys ReU E, 2003 RevE,2004,69:046107 68(3):036122 44 Soon-hyung Y, Filippo R, Hildegard M O. Self-similar Fang J Q. Evolution features of large unifying hybrid net- scale-free networks and disassortativity. Phys Rev E, work model with a variable growing speeds. In: The 4th nternational 2005,72(4) 105 Physics, 2007-11-09 12. Hangzhou, Zhejiang, 2007 45 Sumiyoshi A, Norikazu S Complex earthquake networks Hierarchical organization and assortative mixing. Phys 56 Fang J Q. Some progresses in theoretical model of nol RetE,2006,74(2):026113 linear dynamical complex networks. In: 2008 National Physics Conference, 2008-09-18N10-02, Nanji 46 Wang w x, Wang B H, Hu B, et al. General dynamics of 57 Fang J Q, Li Y Large unified hybrid network model with ology and traffic on weighted technological networks. Phys Rev Lett2005,94(18):188702 ble growth. Chi Phys Lett, 2008 47 Grabowski A. Kosinski R A. Evolution of a social network 58 Laherere J, Sornette D. Stretched exponential distribu- The role of cultural diversity. Phys Rev E, 2006, 73(1) tions in nature and economy:"fat tail"with characteristic 16135 scales. Eur Phys J B, 1998, 2: 525/539 48 Wang Wx, Hu B, Wang B H, et al. Mutual attraction 59方锦清,李永,毕桥统一混合变速增长网络模型及其 model for both assortative and disassortative weighted 特性转变复杂系统与复杂性科学,2008,5(4) networks. Phys Rev E, 2006, 73(1):016133 60方锦清.略论复杂网络金字塔及其普适性与复杂性 49 Julian S, Janusz A H. Statistical analysis of 22 public 见:第十五届全国凝聚态理论与统计物理学术会议 transport networks in Poland. Phys Rev E, 2005, 72(4) 50 Li CG, Philip k M Complex networks generated by the 61 61方锦清.试论四大层次的高科技网络的若干特点和思 Penna bit-string model: Emergence of small-world and 考.见:陈关荣,许晓,百家鸣编.复杂网络理论与应用 assortative mixing. Phys Rev E, 2005, 72(4):04 三届全国复杂网络学术会议文集.上海:上海 51 Michele C, Guido C, Luciano P. Assortative model for 系统科学出版社.2008.1~4 cial networks. Phys Rev E, 2004, 70(3):037101 62李永等.从网络科学视角探索企业合作网络.见:第四 52 Mattick J S, Gagen M J. Accelerating networks. Science, 届全国网络科学论坛文集.中国高等科学技术 2005,307:856~858 青岛,2008 53 Gagen G M, Mattick J S. Accelerating, hyperaccelerat- 63刘作仪.复杂网络理论及相关管理复杂性研究的资助 ing and decelerating probabilistic networks. Phys Reu E, 进展中国科学基金,2008,1:13~17 ADVANCES IN UNIFIED HYBRID THEORETICAL MODEL OF NETWORK SCIENCE FANG Jingingt LI Yon China Institute of Atomic Energy, Beijing 102413, China Abstract Theoretical model has been one of the most significant issues in the network sciences. First of all three milestones in this area are reviewed as breakthroughs in theoretical models, and progress in weighted evolution network model is presented. In order to describe the determinist and randomness of the harmony and unity of the real world and to mirror overall property of the real-world networks, the unified hybrid network theoretical frame, which formed the so-called hy brid network theoretical model trilogy, are reviewed and summarized in the main part of this article: the first of the trilogy is a harmonious unification hybrid preferential model(HUHPM), the second the large unification hybrid network model (LUHNM) and the third the large unified hybrid variable growth model (LUHVGm). Main features and topological properties of the hybrid network trilogy are then summarized, and diversity and complexity of the complex networks are revealed as well as their complicated transition relationship. Possible applications of these theories to actual network re finally mentioned Keywords network science, unified hybrid network theoretical model trilogy, complexity, universality, small- world. scale-free The project supported by the National Nature Science Foundation of China(70431002, 60874087) and Grand Theory Physics Special Project of NSFC (10647001) Er-mail:fjq96@126.com678 力 学 进 展 2008 年 第 38 卷 42 Newmanm E J. Assortative mixing in networks. Phys Rev Lett, 2002, 89(20): 208701 43 Newmanm M E J, Park J. Why social networks are dif￾ferent from other types of networks. Phys Rev E, 2003, 68(3): 036122 44 Soon-hyung Y, Filippo R, Hildegard M O. Self-similar scale-free networks and disassortativity. Phys Rev E, 2005, 72(4): 045105 45 Sumiyoshi A, Norikazu S. Complex earthquake networks: Hierarchical organization and assortative mixing. Phys Rev E, 2006, 74(2): 026113 46 Wang W X, Wang B H, Hu B, et al. General dynamics of topology and traffic on weighted technological networks. Phys Rev Lett, 2005, 94(18): 188702 47 Grabowski A, Kosinski R A. Evolution of a social network: The role of cultural diversity. Phys Rev E, 2006, 73(1): 016135 48 Wang W X, Hu B, Wang B H, et al. Mutual attraction model for both assortative and disassortative weighted networks. Phys Rev E, 2006, 73(1): 016133 49 Julian S, Janusz A H. Statistical analysis of 22 public transport networks in Poland. Phys Rev E, 2005, 72(4): 046127 50 Li C G, Philip K M. Complex networks generated by the Penna bit-string model: Emergence of small-world and assortative mixing. Phys Rev E, 2005, 72(4): 045102 51 Michele C, Guido C, Luciano P. Assortative model for so￾cial networks. Phys Rev E, 2004, 70(3): 037101 52 Mattick J S, Gagen M J. Accelerating networks. Science, 2005, 307: 856∼858 53 Gagen G M, Mattick J S. Accelerating, hyperaccelerat￾ing and decelerating probabilistic networks. Phys Rev E, 2005, 72: 016123 54 Sen P. Accelerated growth in outgoing links in evolving networks: Deterministic versus stochastic picture. Phys Rev E, 2004, 69: 046107 55 Fang J Q. Evolution features of large unifying hybrid net￾work model with a variable growing speeds. In: The 4th International Workshop Hangzhou 2007 on Simulational Physics, 2007-11-09∼12. Hangzhou, Zhejiang, 2007 56 Fang J Q. Some progresses in theoretical model of non￾linear dynamical complex networks. In: 2008 National Physics Conference, 2008-09-18∼10-02,Nanjing, 2008 57 Fang J Q, Li Y. Large unified hybrid network model with variable growth. Chi Phys Lett, 2008 58 Laher`ere J, Sornette D. Stretched exponential distribu￾tions in nature and economy: “fat tail” with characteristic scales. Eur Phys J B, 1998, 2: 525∼539 59 方锦清, 李永, 毕桥. 统一混合变速增长网络模型及其 特性转变. 复杂系统与复杂性科学, 2008, 5(4) 60 方锦清. 略论复杂网络金字塔及其普适性与复杂性. 见：第十五届全国凝聚态理论与统计物理学术会议. 合肥, 2008 61 方锦清. 试论四大层次的高科技网络的若干特点和思 考. 见: 陈关荣, 许晓, 百家鸣编. 复杂网络理论与应用 —— 第三届全国复杂网络学术会议文集. 上海: 上海 系统科学出版社. 2008. 1∼43 62 李永等. 从网络科学视角探索企业合作网络. 见: 第四 届全国网络科学论坛文集. 中国高等科学技术中心, 青岛, 2008 63 刘作仪. 复杂网络理论及相关管理复杂性研究的资助 进展. 中国科学基金, 2008, 1: 13∼17 ADVANCES IN UNIFIED HYBRID THEORETICAL MODEL OF NETWORK SCIENCE* FANG Jinqing† LI Yong China Institute of Atomic Energy, Beijing 102413, China Abstract Theoretical model has been one of the most significant issues in the network sciences. First of all, three milestones in this area are reviewed as breakthroughs in theoretical models, and progress in weighted evolution network model is presented. In order to describe the determinist and randomness of the harmony and unity of the real world and to mirror overall property of the real-world networks, the unified hybrid network theoretical frame, which formed the so-called hybrid network theoretical model trilogy, are reviewed and summarized in the main part of this article: the first of the trilogy is a harmonious unification hybrid preferential model (HUHPM), the second the large unification hybrid network model (LUHNM) and the third the large unified hybrid variable growth model (LUHVGM). Main features and topological properties of the hybrid network trilogy are then summarized, and diversity and complexity of the complex networks are revealed as well as their complicated transition relationship. Possible applications of these theories to actual networks are finally mentioned. Keywords network science, unified hybrid network theoretical model trilogy, complexity, universality, small￾world, scale-free ∗ The project supported by the National Nature Science Foundation of China (70431002, 60874087) and Grand Theory Physics Special Project of NSFC (10647001) † E-mail: fjq96@126.com