第6期 方锦清等:网络科学中统一混合理论模型的若干研究进展 从而具有重要的学术价值”.该文对我国网络科学15 Newman M, Barabasia l, Watts d j. The strue 的多个课题研究,诸如:交通流驱动模型、复杂网 Dynamics of Networks. Princeton: Princeton Ur Press. 2006 络的广义同步模型(包括集团同步、部分同步、社16 Newman me j. The structure and function of complex 区网络的同步)及其同步化能力、群集系统中的同 networks. SIAM Review. 2003. 45: 1670256 步和属性连接的网络同步、混沌连接网络以及具 17 Dorogovtsev S, Mendes J. Minimal models of weighted cale-free networks. Arxiv preprint, 2004 d-mat 有小世界和无标度的束流传输网络中的多目标控 18 Look s Barabasi A L, Tu Y. Weighted evolv 制与同步等也都作了好评63.我国整个网络科学 及其应用取得了喜人的进展及其丰硕成果,不少 fanconi G, Barabasi A L. Competition and multiscaling n evolving networks. Eur Phys Lett, 2001, 54: 436--442 成果不仅具有重要的理论价值,而且具有实际意 Zheng D, Trimper S, Zheng B, Hui P M. Stochastic weight 义和应用潜力.但是,仍然需要更高更深更出色的 assignments. Phys ReU E, 2003, 67: 040102 21 Antal P L. Krapivsky, Weight-driven growing networks. 研究. Phys Rev E,2005,71:026103 总之,统一混合网络理论框架提出了3部曲 22 Barrat A, Barthelemy M, Pastor-Satorras R, et al. The ar- lex weighted networks. Proc Natl acad 模型,它符合真实世界具有确定性与随机性和谐 Sci usa,2004,101(11):3747~3752 统一的基本事实,并抓住了实际网络的一些主要 23 Barrat A, Barthelemy M, Vespignani AA. Weighted evolving networks: Coupling topology and weight dynam- 特点,巧妙地引进了4个混合比来统一研究和调 ics. Phys Rev Lett, 2004, 92: 228701 控各类网络的拓扑性质、功能和动力学特性,几乎 plex network model driven by traffic flow. Phys Reve um 4 Wang wx, Wang B HH, Hu B, et al. a weighted co 涵盖了迄今大多数的现有网络模型,因此,统一混 2005.94:188702 合网络理论具有较大的普适性,它既适用于无权 5 Fang J Q, Liang Y. Topological properties and transition 网络,又适用于含权网络,可以应用于设计实际所 e6 Ch in phys Lett, 2005, 22: 2719-272 preferential model 需要的网络,具有应用潜力 Jin Q F, Qiao B, Yong L. Toward a harmonious unifying hybrid model for any evolving complex networks. Ad- 今后,如何精确求解统一混合理论模型和深 vances in Compler Systems, 2007, 10(2): 117 141 入开展复杂网络的各种应用研究,仍然是两大主 27方锦清,毕桥,李永等复杂动态网络的一种和谐统一 的混合择优模型及其普适特性.中国科学G辑,2007 要努力方向和极富挑战性课题,有待国内外学者 进一步开拓创新,更上一层楼 Q, Li Y,et three-power-laws to hybrid ratio in weighted HUHPM. Chi Phys Lett,,2007,24(1):279~282 参考文献 29 Lu X B, Wang X F, Li x, Fang J Q. Topological transi- tion features and synchronizability of a weighted hybrid 1 Wilson E O. Consilience Knopf. New York, 1998. 85 preferential network. Physica A, 2006, 370: 381-389 2欧拉( Leonhard euler,707~1783)简况见:ht 30 Fang JQ, Bi Q, Li Y, et al. A harmonious unifying prefer- 斯:谢克特著我的大开f天才数学家保 tial network model and its universal properties for com- olex dynamical network. Science in China Series G, 2007. 爱多士传奇,王元,李文林译.上海:译文出版社,2002 (3):379~39 3 Erdos P, Renyi A. On the evolution of random graphs. 31 Li Y, Fang Q, Bi Q, Liu Q. Entropy characteristic on har- Publ Math Inst Hung Acad Aci, 1960, 5: 17-6 nonius unifying hybrid preferential networks. Entropy, 4 Watts D J, Strogatz S H. Collective dynamics ofsmall- 2007,9:73~82 world networks. Nature, 1998, 393: 440442 32 Bi Q, Fang J Q. Entropy and HUHPM approach for com- 5 Watts D J. Six Degrees: The Science of a Connected Age plex networks. Physica A, 2007, 383: 753-762 New York: W.W. Norton &z Company, 2003 33 Fang J Q, Bi Q, Li Y. From a harmonious unifying hybrid 6 Watts D.. The"new science of networks. Annual review preferential model toward a large unifying hybrid network of Sociology, 2004, 30: 243-270 nodel. International Journal of Modern Physics B, 2007. 7 Watts DJ. Small Worlds: The Dynamics of Networks be- 21(30):5121~5142 tween Order and Randomness. New Jersey: Princeto 34 Fang J Q, Bi Q, Li Y. Advances in theoretical models of University Press, 1999 network science. Front Phys China. 2007. 1: 109 124 8 Newman M E. Moore C. Watts D Mean-field solution 35李永,方锦清,刘强.大统一的混合网络模型中的相称 of the small-world network model. Phys Rev Lett, 2000 性系数转变新特点.科技导报,2007,25(11):23~2 36方锦清.网络科学的理论模型探索及其进展.科技导 9 Newman M E J. Models of the small world. J Stat Phys 报,200,24(12):67~72 2000,101:819~841 37方锦清.非线性网络的动力学复杂性研究的若干进展 10 Buchanan. Nexus M. Small Worlds and the groundbreak- 自然科学进展,2007:17(7:841~857 ing Science of Networks. New York: Ww Norton an 38方锦清,汪小帆,郑志刚等.一门崭新的交叉科学 Company, 2002 物理学进展,2007,27(3):239~343 11 Barabasi A L, Albert R. Emergence of Scaling in Random Networks 络科学(下).物理学进展,2007,27(4) 361~448 12 Strogatz S H. Exploring complex networks. Nature, 200 40 Fang J Q. Theoretical research progress in complexity of 410(8):268~276 complex dynamical networks. Progress in Nature Science, 13 Barabasi A L. The New Science of Networks. Cambridge 2007,17(7):761~74 41 Xul vi-brunet R, Sokolov I M. Reshuffling scale-free net- 14 Albert R, Barabasi A L Statistical mechanics of complex works: From random to assortative. Phys Rev E, 2004 networks. Rev Mod Phys, 2002, 74: 47-98 70(6):066102第 6 期 方锦清等 : 网络科学中统一混合理论模型的若干研究进展 677 从而具有重要的学术价值”. 该文对我国网络科学 的多个课题研究, 诸如: 交通流驱动模型、复杂网 络的广义同步模型 (包括集团同步、部分同步、社 区网络的同步) 及其同步化能力、群集系统中的同 步和属性连接的网络同步、混沌连接网络以及具 有小世界和无标度的束流传输网络中的多目标控 制与同步等也都作了好评 [63] . 我国整个网络科学 及其应用取得了喜人的进展及其丰硕成果, 不少 成果不仅具有重要的理论价值, 而且具有实际意 义和应用潜力. 但是, 仍然需要更高更深更出色的 研究. 总之, 统一混合网络理论框架提出了 3 部曲 模型, 它符合真实世界具有确定性与随机性和谐 统一的基本事实, 并抓住了实际网络的一些主要 特点, 巧妙地引进了 4 个混合比来统一研究和调 控各类网络的拓扑性质、功能和动力学特性, 几乎 涵盖了迄今大多数的现有网络模型, 因此, 统一混 合网络理论具有较大的普适性, 它既适用于无权 网络, 又适用于含权网络, 可以应用于设计实际所 需要的网络, 具有应用潜力. 今后, 如何精确求解统一混合理论模型和深 入开展复杂网络的各种应用研究, 仍然是两大主 要努力方向和极富挑战性课题, 有待国内外学者 进一步开拓创新, 更上一层楼. 参 考 文 献 1 Wilson E O. Consilience Knopf. New York, 1998. 85 2 欧拉 (Leonhard Euler, 1707∼1783) 简况见: http:// www2.zzu.edu.cn/math/classes/2003/y1/oula.htm. 布鲁 斯 · 谢克特著. 我的大脑敞开了 —— 天才数学家保罗 · 爱多士传奇．王元, 李文林译．上海: 译文出版社, 2002 3 Erdos P, Renyi A. On the evolution of random graphs. Publ Math Inst Hung Acad Aci, 1960, 5: 17∼61 4 Watts D J, Strogatz S H. Collective dynamics of ‘small￾world’ networks. Nature, 1998, 393: 440∼442 5 Watts D J. Six Degrees: The Science of a Connected Age. New York: W. W. Norton & Company, 2003 6 Watts D J. The“new” science of networks. Annual Review of Sociology, 2004, 30: 243∼270 7 Watts D J. Small Worlds: The Dynamics of Networks be￾tween Order and Randomness. New Jersey: Princeton University Press, 1999 8 Newman M E J, Moore C, Watts D J. Mean-field solution of the small-world network model. Phys Rev Lett, 2000, 84: 3201∼3204 9 Newman M E J. Models of the small world. J Stat Phys, 2000, 101: 819∼841 10 Buchanan, Nexus M. Small Worlds and the Groundbreak￾ing Science of Networks. New York: W W Norton and Company, 2002 11 Barab´asi A L, Albert R. Emergence of Scaling in Random Networks. Science, 1999, 286: 509∼512 12 Strogatz S H. Exploring complex networks. Nature, 2001, 410(8): 268∼276 13 Barab´asi A L. The New Science of Networks．Cambridge: Prerseus, 2002 14 Albert R, Barab´asi A L. Statistical mechanics of complex networks. Rev Mod Phys, 2002, 74: 47∼98 15 Newman M, Barab´asi A L, Watts D J. The Structure and Dynamics of Networks. Princeton: Princeton University Press, 2006 16 Newman M E J. The structure and function of complex networks. SIAM Review, 2003, 45: 167∼256 17 Dorogovtsev S, Mendes J. Minimal models of weighted scale-free networks. Arxiv preprint, 2004, cond-mat/ 0408343 18 Yook S H, Jeong H, Barab´asi A L, Tu Y. Weighted evolv￾ing networks. Phys Rev Lett, 2001, 86(25): 5835∼5838 19 Bianconi G, Barab´asi A L. Competition and multiscaling in evolving networks. Eur Phys Lett, 2001, 54: 436∼442 20 Zheng D, Trimper S, Zheng B, Hui P M. Stochastic weight assignments. Phys Rev E, 2003, 67: 040102 21 Antal P L. Krapivsky, Weight-driven growing networks. Phys Rev E, 2005, 71: 026103 22 Barrat A, Barth´elemy M, Pastor-Satorras R, et al. The ar￾chitecture of complex weighted networks. Proc Natl Acad Sci USA, 2004, 101(11): 3747∼3752 23 Barrat A, Barth´elemy M, Vespignani A A. Weighted evolving networks: Coupling topology and weight dynam￾ics. Phys Rev Lett, 2004, 92: 228701 24 Wang W X, Wang B H H, Hu B, et al. A weighted com￾plex network model driven by traffic flow. Phys Rev Lett, 2005, 94: 188702 25 Fang J Q, Liang Y. Topological properties and transition features generated by a new hybrid preferential model. Chin Phys Lett, 2005, 22: 2719∼2722 26 Jin Q F, Qiao B, Yong L. Toward a harmonious unifying hybrid model for any evolving complex networks. Ad￾vances in Complex Systems, 2007, 10(2): 117∼141 27 方锦清, 毕桥, 李永等. 复杂动态网络的一种和谐统一 的混合择优模型及其普适特性. 中国科学 G 辑, 2007, 3(2): 230∼249 28 Fang J Q, Bi Q, Li Y, et al. Sensitivity of exponents of three-power-laws to hybrid ratio in weighted HUHPM. Chi Phys Lett, 2007, 24(1): 279∼282 29 Lu X B, Wang X F, Li X, Fang J Q. Topological transi￾tion features and synchronizability of a weighted hybrid preferential network. Physica A, 2006, 370: 381∼389 30 Fang J Q, Bi Q, Li Y, et al. A harmonious unifying prefer￾ential network model and its universal properties for com￾plex dynamical network. Science in China Series G, 2007, 50(3): 379∼396 31 Li Y, Fang J Q, Bi Q, Liu Q. Entropy characteristic on har￾monious unifying hybrid preferential networks. Entropy, 2007, 9: 73∼82 32 Bi Q, Fang J Q. Entropy and HUHPM approach for com￾plex networks. Physica A, 2007, 383: 753∼762 33 Fang J Q, Bi Q, Li Y. From a harmonious unifying hybrid preferential model toward a large unifying hybrid network model. International Journal of Modern Physics B, 2007, 21(30): 5121∼5142 34 Fang J Q, Bi Q, Li Y. Advances in theoretical models of network science. Front Phys China, 2007, 1: 109∼124 35 李永, 方锦清, 刘强. 大统一的混合网络模型中的相称 性系数转变新特点. 科技导报, 2007, 25(11): 23∼29 36 方锦清. 网络科学的理论模型探索及其进展. 科技导 报, 2006, 24(12): 67∼72 37 方锦清. 非线性网络的动力学复杂性研究的若干进展. 自然科学进展, 2007, 17(7): 841∼857 38 方锦清, 汪小帆, 郑志刚等. 一门崭新的交叉科学 —— 网络科学 (上). 物理学进展, 2007, 27(3): 239∼343 39 方锦清, 汪小帆, 郑志刚等. 一门崭新的交叉科学 —— 网络科学 (下). 物理学进展, 2007, 27(4): 361∼448 40 Fang J Q. Theoretical research progress in complexity of complex dynamical networks. Progress in Nature Science, 2007, 17(7): 761∼774 41 Xul vi-brunet R, Sokolov I M. Reshuffling scale-free net￾works: From random to assortative. Phys Rev E, 2004, 70(6): 066102