摘要
网络的结构复杂性、演化机制及动力学行为是一般输运网络研究的基础性科学问题,对认识大规模交通运输网络具有重要意义。本文从统计物理的观点,运用理论解析及数值模拟等手段,对一般输运网络的演化模型及动力学特征进行了若干探讨和研究。
首先,本文对一般输运网络的加权模型做了一系列的相关研究。根据一般输运网络度、强度、权重分布、强度-度相关性、聚类特性以及相配性等相关统计特征,并结合考虑网络演化的某些特殊规律,提出了若干加权网络模型。这些模型通过引入不同演化机制,不仅可以再现实际输运网络中的多种统计规律,还能够再现小变量饱和、指数甩尾等较为细致的网络统计特征。
在复杂网络科学的研究初期,人们主要关注的是小世界效应、无标度现象、层次性、模块性等网络拓扑结构特征。随着研究的不断深入,人们开始注意到网络的隐含特性和框架结构。本文通过对若干实际网络和模型网络中不同重要程度Hub结点统计规律的研究,从节点重要度的角度研究了输运网络的主要框架。
传播动力学是网络科学中的一类重要的研究分支。本文通过考虑一般输运网络的模块性和小世界特性,分析了模块结构对传播动力学的影响。研究表明网络的模块结构对拥堵传播的传播速度、范围以及传播中出现的同步现象都有较为明显的作用,这在一定程度上揭示了网络模块结构和传播之间的某些内在规律。
级联失效在大规模输运网络上时有发生,且对整个系统的破坏性巨大。如何应对级联失效所引起的大规模故障一直是网络科学的重要课题之一。本文通过对级联失效产生机制较为深入的分析,并结合无标度网络的拓扑特征,提出了利用导航策略来控制级联失效的方法。研究表明,采用合适的导航策略能够有效提高网络稳健性,避免级联失效的发生。
此外,本文通过考虑交通平衡状态下不同拓扑结构输运网络上的流量分配状态,研究了规则网络、随机网络、小世界网络以及无标度网络上流量分布的规律性。认识这些一般输运网络上的流量分布规律,对理解现实世界中的各种交通现象有重要理论价值和实际意义。
To characterize the structure complexity, evolutionary mechanisms and dynamic behaviors of network is the key in researching transportation network. In this paper, by using the theoretical analysis and numerical simulations, we studied the evolution and dynamical behaviors on general transportation networks in the statistical mechanics point of view.
Firstly, we made a series researches on the weighted models of general transportation networks. By considering the statistical properties of transportation networks, such as degree distribution, strength distribution, weight distribution, correlation of strength and degree, clustering coefficient and assortativity, and so on, we evolved some weighted networks models. The different evolution processes can give various statistical features of real transportation systems successfully. Especially, the models can reproduce features such as saturation for small variables, exponential decay, etc.
At the beginning of the research on the complex network science, people focus on the small world effect, scale free phenomena, hierarchy architecture, modularity structure, and so on. With the deeper insight into this field, people begin to realize the hidden features and structures of networks. In this paper, we studied the statistical characteristics of Hub nodes in various important levels, and to reveal the main structures of transportation networks in the nodes' importance point of view.
Epidemic dynamic is one of the most important problems in transpiration network science. By considering the modular structure and small world property of real transportation system, we study the effect of modularity on the epidemic behaviors on networks. It is found that the modularity prevent the rate, extent and the synchronization behavior of the propagation. It reveals some implicit regularities of the relationship between the modularity and the epidemic spreading dynamics.
Cascading failures can take place in large transportation networks, such as power grid, Internet, etc, sometimes. It may bring catastrophe of the whole system. How to defend the cascade break down is one of the most important problems in researching network science. In this paper, we investigate the cascading failure by considering the degree based navigation strategy. It is found that by using the proper navigation strategy, it can reduce the risk of cascading failure considerably.
In addition, we investigate the statistical properties of the traffic dynamics in equilibrium state on transportation networks with different topology, including regular,random, small world and scale free networks. To understand the properties in trafficequilibrium state is important in realizing the traffic phenomena of real transportationsystems.
引文
[1] Ahmed E, Hegazi A S, Elgazzar A S. 2002. An epidemic model on small-world networks and ring vaccination. International Journal of Modern Physics C, 13: 189-198.
[2] Albert R, Barabasi A L, Jeong H. 2000. Scale-free characteristics of random networks: the topology of the world-wide web. Physica A, 281: 69-77.
[3] Albert R, Barabasi A L. 2000. Topology of evolving networks: local events and universality.Physical Review Letters, 85: 5234-5237.
[4] Albert R, Barabasi A L. 2002. Statistical mechanics of complex networks. Review of Modern Physics, 74: 47-97.
[5] Albert R, Jeong H, Barabasi A. 1999. Diameter ofthe World Wide Web. Nature, 401: 130-133.
[6] Alvarez-Hamelin I, Dall'Asta L, Barrat A, Vespignani A. 2005. K-core decomposition: a tool for the analysis of large scale Internet graphs. arXiv: cs.NI/0511007.
[7] Amaral L A N, Guimera R. 2006. Lies, damned lies and statistics. Nature Physics, 2:75-76.
[8] Amaral L A N, Scala A, Barthelemy M, Stanley H E. 2000. Classes of small-world networks.Proceedings of the National Academy of Sciences USA, 97: 11149-11152.
[9] Arenas A, Danon L, Diaz-Guilera A, Gleiser P M, Guimera R. 2004. Community analysis in social networks. European Physical Journal B, 38: 373-380.
[10] Arenas A, Dias-Guilera A, Guimera R. 2001. Communication in networks with hierarchical branching. Physical Review Letters, 86: 3196.
[11] Axtell R L. 2001. Zipf distribution of U.S. firm sizes. Science, 293: 1818-1820.
[12] Baguna M, Pastor-Satorras R. 2002. Epidemic spreading in correlated complex networks.Physical Review E, 66:047104.
[13] Banavar J R, Maritan A, Rinaldo A. 1999. Size and form in efficient transportation networks.Nature, 399: 130-132.
[14] Barabasi A L, Albert R, Jeong H. 1999. Mean-field theory for scale-free random networks. Physica A,272: 173-187.
[15] Barabasi A L, Albert R. 1999. Emergence of scaling in random networks. Science, 286:509-512.
[16] Barabasi A L, Jeong H, Neda Z, Ravasz E, Schubert A, Vicsek T. 2002. Evolution of the social network of scientific collaborations. Physica A, 311: 590-614.
[17] Barabasi A L, Oltvai Z N. 2004. Network Biology: Understanding the cell's functional organization. Nature reviews genetics, 5: 101-113.
[18] Barrat A, Barthelemy M, Pastor-Satorras R, Vespignani A. 2004a. The architecture of complex weighted networks. Proceedings of the National Academy of Sciences USA, 101: 3747-3752.
[19] Barrat A, Barthelemy M, Vespignani V. 2004b. Modeling the evolution of weighted networks.Physical Review E, 70: 066149.
[20] Barrat A, Barthelemy M, Vespignani V. 2004c. Weighted evolving networks: coupling topology and weight dynamics. Physical Review Letters, 92: 228701.
[21] Barrat A, Weigt M. 2000. On the properties of small world networks. European Physical Journal B, 13:547-560.
[22] Beckmann A B, McGuire C B, Winsten C B. 1956. Studies in the economics of transportation. Yale University Press, New Haven, Connecticut.
[23] Boccalettia S, Latorab V, Morenod Y, Chavezf M, Hwanga D U. 2006. Complex networks:structure and dynamics. Physical Reports, 424: 175-308.
[24] Boguna M, Pastor-Satorras R. 2002. Epidemic spreading in correlated complex networks.Physical Review E, 66:047104.
[25] Bollobas B, Riordan O. 2002. Mathematical results on scale-free random graphs. in: edited by S. Bornholdt and H. G Schuster eds. Handbook of graphs and networks: from the genome to the internet. Berlin: Wiley-VCH, 1-34.
[26] Bollobas B. 2001. Random graphs. 2nd ed. New York: Academic Press.
[27] Braunstein L A, Buldyrev S V, Cohen R, Havlin S, Stanley H E. 2003. Optimal paths in disordered complex networks. Physical Review Letters, 91: 168701.
[28] Broader A Z, Kumar S R, Maghoul F, Raghavan P, Rajagopalan S, Stata R, Tomkins A, Wiener J L. 2000. Graph structure in the web. Computer Networks, 33: 309-320.
[29] Chen Q, Chang H, Govindan R, Jamin S, Shenker S J, Willinger W. 2002. The origin of power laws in Internet topologies revisited. In: Proceding of the IEEE INFOCOM 2002. New York:IEEE Press, 608-617.
[30] Chen Z Y, Wang X F. 2006. Effects of network structure and routing strategy on network capacity. Physical Review E, 73: 036107.
[31] Chowell G, Hyman J M, Eubank S, Castillo-Chavez C. 2003. Scaling laws for the movement of people between locations in a large city. Physical Review E, 68: 066102.
[32] Cliff A, Haggett P. 1984. Island epidemics. Science American, 250:110-117.
[33] Cohen R, Havlin S. 2003. Scale-free networks are ultra small. Physical Review Letters, 90:058701.
[34] Colizza V, Barrat A, Barthelemy M, Vespignani A. 2006a. The role of the airline transportation network in the prediction and predictability of global epidemics. Proceedings of the National Academy of Sciences USA, 102: 2015-2020.
[35] Colizza V, Flammini A, Serrano M A, Vespignani A. 2006b. Detecting rich-club ordering in complex networks. Nature Physics, 2:110-115.
[36] Colizza V, Vespignani A. 2007. Invasion threshold in heterogeneous metapopulation networks.Physical Review Letters, 99: 148701.
[37] Crucitti P, Latora V, Marchiori M. 2004. Model for cascading failures in complex networks.Physical Review E, 69:045104.
[38] Crucitti P, Latora V, Porta S. 2005. Centrality measures in urban networks. arXiv:physics/0504163.
[39] Dafermos S, Nagurney A. 1984. Sensitivity analysis for the asymmetric network equilibrium problem. Mathematical Programming, 28: 174-184.
[40] di Bernardo M, Garofalo F, Manfredi S, Sorrentino F. 2005. Load distribution in small world networks. Physics and Control, 24: 100-105.
[41] Dorogovtsev S N, Mendes J F F, Samukhin A N. 2000. Structure of growing networks with preferential linking. Physical Review Letters, 85: 4633-4636.
[42] Dorogovtsev S N, Mendes J F F. 2001a. Language as an evolving word web. Proceedings of the Royal Society of London B, 268: 2603-2606.
[43] Dorogovtsev S N, Mendes J F F. 2001b. Scaling properties of scale-free evolving networks:continuous approach. Physical Review E, 63: 056125.
[44] Dorogovtsev S N, Goltsev A V, Mendes J F F. 2006a. K-core architecture and k-core percolation on complex networks. Physica D, 224: 7-19.
[45] Dorogovtsev S N, Goltsev A V, Mendes J F F. 2006b. K-Core organization of complex networks. Physical Review Letters, 96: 040601.
[46] Ebel H, Mielsch L I, Borbholdt S. 2002. Scale-free topology of e-mail networks. Physical Review E,66: 035103.
[47] Erdos P, Renyi A. 1959. On random graphs. Publicationes Mathematicae, 6: 290-297.
[48] Erdos P, Renyi A. 1960. On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 5: 17-61.
[49] Erdos P, Renyi A. 1961. On the strength of connectedness of a random graph. Acta Mathematica Scientia Hungary, 12: 261-267.
[50] Eriksen K A, Simonsen I, Maslov S, Sneppen K. 2003. Modularity and extreme edges of the internet. Physical Review Letters, 90: 148701.
[51] Faloutsos M, Faloutsos P, Faloutsos C. 1999. On power law relationships of the internet topology. ACM SIGCOMM Computers Communication Review, 29: 251-262.
[52] Frank M, Wolfe P. 1956. An algorithm for quadratic programming. Naval Research Logistics Quarterly, 3: 95-110.
[53] Fronczak A, Fronczak P, Holyst J A. 2003. Mean-field theory for clutsering coefficients in Barabasi-Albert networks. Physical Review E, 68: 046126.
[54] Fu B B, Gao Z Y, Liu F S, Kong X J. 2006. Express passenger transport system as a scale-free network. Modern Physics Letters B, 20: 1755-1761.
[55] Gao Z Y, Li K P. 2005. Evolution of traffic flow with scale free topology. Chinese Physics Letters, 22: 2711-2714.
[56] Gao Z Y, Li K P, Li X G, Huang H J, Mao B H, Zheng J F. 2007. Scaling laws of the network traffic flow. Physica A, 380: 577-584.
[57] Gao Z Y and Song Y F. 2002. A reserve capacity model of optimal signal control with user-equilibrium route choice. Transportation Research part B, 36: 313-323
[58] Gao Z Y, Sun H J and Shan L L. 2004. A continuous equilibrium network design model and its solution algorithm for transit system. Transportation Research part B, 38: 234-250
[59] Girvan M, Newman M E J. 2002. Community structure in social and biological networks.Proceedings of the National Academy of Sciences USA, 99:7821-7826.
[60] Goh K I, Kahng B, Kim D. 2001. Universal behavior of load distribution in scale free networks.Physical Review Letters, 87: 278701.
[61] Goh K I, Noh J D, Kahng B, Kim D. 2005. Load distribution in weighted complex networks.Physical Review E, 72: 017102.
[62] Goh K 1, Salvi G, Kahng B, Kim D. 2006. Skeleton and fractal scaling in complex networks.Physical Review Letters, 96: 018701.
[63] Grassberger P. 1983. On the critical behavior of the general epidemic process and dynamical percolation. Mathematical Biosciences, 63: 157-172.
[64] Guimera R, Amaral L A N. 2004. Modeling the world-wide airport network. European Physical Journal B, 38:381-385.
[65] Guimera R, Dias-Guilera A, Vega-Redondo F, Cabrales A, Arenas A. 2002. Optimal network topologies for local search with congestion. Physical Review Letters, 89: 248701.
[66] Guimera R, Mossa S, Turtschi A, Amaral L A N. 2005. The world-wide air transportation network: anomalous centrality, community structure, and cities' global roles. Proceedings of the National Academy of Sciences USA, 102:7794-7799.
[67] Holme P, Kim B J, Yoon C N, Han S K. 2002. Attack vulnerability of complex networks.Physical Review E, 65: 066109.
[68] Holme P. 2002. Edge overload breakdown in evolving networks. Physical Review E, 66:036119.
[69] Janson S, Luczak T, Rucinski A. 1999. Random Graphs. New York: John Wiley.
[70] Jeong H, Tombor B, Albert R, Oltvai Z N, Barabasi A L. 2000. The large-scale organization of metabolic networks. Nature, 407:651-654.
[71] Jeong H, Mason S P, Barabasi A L, Oltvai Z N. 2001. Lethality and centrality in protein networks. Nature, 411: 41-42.
[72] Jiang B, Claramunt C. 2004. Topological analysis of urban street networks. Environment and Planning B,31: 151-162.
[73] Joo J, Lebowitz J L. 2004. Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation. Physical Review E, 69: 066105.
[74] Karonski M. 1982. A review of random graphs. Journal of Graph Theory, 6: 349-389.
[75] Kernighan B W, Lin S. 1970. An efficient heuristic procedure for partitioning graphs. Bell System Technical Journal, 49: 291-307.
[76] Kim B J, Trusina A, Minnhagen P, Sneppen K. 2005. Self organized scale-free networks from merging and regeneration. European Physical Journal B, 43: 369-372.
[77] Kim D H, Noh J D, Jeong H. 2004. Scale-free trees: the skeletons of complex networks.Physical Review E, 70: 046126.
[78] Krapivsky P L, Redner S, Leyvraz F. 2000. Connectivity of growing random networks.Physical Review Letters, 85: 4629-4632.
[79] Kuhnert C, Helbing D, West G B. 2006. Scaling laws in urban supply networks. Physica A, 363:96-103.
[80] Kuperman M, Abramson G. 2001. Small world effect in an epidemiological model. Physical Review Letters, 86:2909.
[81] Kuramoto Y. 1984. Chemical oscillations, waves, and turbulence. Springer, Berlin, 1984.
[82] Lai Y C, Motter A E, Nishikawa T. 2004. Attacks and cascades in complex networks. Lecture Notes in Physics, 650: 299-310.
[83] Latora V, Marchiori M. 2001. Efficient behavior of small-world networks. Physical Review Letters, 87: 198701.
[84] Latora V, Marchiori M. 2002. Is the Boston subway a small world network? Physica A, 314:109-113.
[85] LeBlanc L J. 1975. An algorithm for the discrete network design problem. Transportation Science, 9: 183-199.
[86] Levinson D, Yerra B. 2006. Self-organization of surface transportation networks.Transportation Science, 40: 179-188.
[87] Li K P, Gao Z Y. 2006. A topological approach to traffic dynamics. Europhysics Letters, 74:369-375.
[88] Li W, Cai X. 2004. Statistical analysis of airport network of China. Physical Review E, 69:046106.
[89] Marro J, Dickman R. 1999. Nonequalibrium phase transitions in lattice modles. Cambridge: Cambridge University Press.
[90] Mendes J F F, Dorogovtsev S N, Ioffe A F. 2003. Evolution of networks: from biological nets to the Internet and WWW. Oxford: Oxford University Press.
[91] Milo R, Shen-Orr R, Itzkovitz S, Kashtan N, Chklovskii D, Alon U. 2002. Network motifs:simple building blocks of complex networks. Science, 298: 824-827.
[92] Minnhagen P, Rosvall M, Sneppen K, Trusina A. 2004. Self-organization of structures and networks from merging and small-scale fluctuations. Physica A, 340: 725-732.
[93] Monasson R. 1999. Diffusion, localization and dispersion relations on "small-world" lattices.European Physical Journal B, 12: 555-567.
[94] Montis A D, Barthelemy M, Chessa A, Vespignani A. 2005. The structure of inter urban traffic:a weighted network analysis. arXiv: physics/0507106.
[95] Moreno Y, Gomez J B, Pacheco A F. 2002. Instability of scale free networks under node breaking avalanches. Europhysics Letters, 58: 630-636.
[96] Moreno Y, Pastor-Satorras R, Vazquez A, Vespignani A. 2003. Critical load and congetion instabilities in scale free networks. Europhysics Letters, 62: 292-298.
[97] Motter A E, Lai Y C. 2002. Cascade-based attacks on complex networks. Physical Review E,66:065102.
[98] Motter A E. 2004. Cascade control and defense in complex networks. Physical Review Letters,93:098701.
[99] Newman M E J, Forrest S, Balthrop J. 2002. Email networks and the spread of computer viruses. Physical Review E, 66:035101.
[100] Newman M E J, Girvan M. 2004. Finding and evaluating community structure in networks.Physical Review E, 69: 026113.
[101] Newman M E J, Moore C, Watts D J. 2000. Mean field solution of the small-world network model. Physical Review Letters, 84: 3201-3204.
[102] Newman M E J, Watts D J. 1999a. Renormalization group analysis of the small-world network model. Physics Letters A, 263: 341-346.
[103] Newman M E J Watts D J. 1999b. Scaling and percolation in the small-world network model.Physical Review E 60: 7332-7342.
[104] Newman M E J. 2001a. Scientific collaboration networks: I. Network construction and fundamental results. Physical Review E, 64: 016131.
[105] Newman M E J. 2001b. Scientific collaboration networks: II. Shortest paths, weighted networks, and centrality. Physical Review E, 64: 016132.
[106] Newman M E J. 2001c. The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences USA, 98: 404-409.
[107] Newman M E J. 2002a. Assortative mixing in networks. Physical Review Letters, 89: 208701.
[108] Newman M E J. 2002b. Spread of epidemic disease on networks. Physical Review E, 66:016128.
[109] Newman M E J. 2003. The structure and function of networks. SIAM Review, 45: 167-256.
[110] Newman M E J. 2004. Fast algorithm for detecting community structure in networks. Physical Review E,69: 066133.
[111] Olinky R, Stone L. 2004. Unexpected epidemic thresholds in heterogeneous networks: The role of disease transmission. Physical Review E, 70: 030902.
[112] Palla G, Derenyi I, Farkas I, Vicsek T. 2005. Uncovering the overlapping community structure of complex networks in nature and society. Nature, 435: 814-818.
[113] Pastor-Satorras R, Vespignani A. 2001a. Epidemic spreading in scale free networks. Physical Review Letters, 86: 3200-3203.
[114] Pastor-Satorras R, Vespignani A. 2001b. Epidemic dynamics and endemic states in complex networks. Physical Review E, 63: 066117.
[115] Pastor-Satorras R, Vazquez A, Vespignani A. 2001c. Dynamical and correlation properties of the Internet. Physical Review Letters, 87: 258701.
[116] Pastor-Satorras R, Vespignani A. 2004. Evolution and structure of the Internet: a statistical physics approach. Cambridge University Press.
[117] Patriksson M. The traffic assignment problems: models and methods. VSP BV, The Netherlands, 1994.
[118] Peter K H. 2000. Scaling: Rivers, blood and transportation networks. Nature, 408: 159-160.
[119] Pothen A, Simon H, Liou K P. 1990. Partitioning sparse matrices with eigenvectors of graphs.SIAM Journal on Matrix Analysis and Applications, 11:430-452.
[120] Ravasz E, Somera A L, Mongru D A, Oltvai Z N, Barabasi A L. 2002. Hierarchical organization of modularity in metabolic networks. Science, 297: 1551-1556.
[121] Ravasz E, Barabasi A L. 2003. Hierarchical organization in complex networks. Physical Review E,67: 026112.
[122] Rives A W, Galitski T. 2003. Modular organization of cellular networks. Proceedings of the National Academy of Sciences USA, 100:1128-1133.
[123] Rohani P, Earn D J D, Grenfell B T. 1999. Opposite Patterns of Synchrony in Sympatric Disease Metapopulations. Science, 286: 968-971.
[124] Scellato S, Cardillo A, Latora V, Porta S. 2005. The backbone of a city. arXiv:physics/0511063.
[125] Seaton K A, Hackett L M. 2004. Station, trains and small world networks. Physica A, 339:635-644.
[126] Sen P, Dasgupta S, Chatterjee A, Sreeram P A, Mukherjee G, Manna S S. 2003. Small-world properties of the Indian railway network. Physical Review E, 67: 036106.
[127] Sheffi Y. 1985. Urban transportation networks: equilibrium analysis with mathematical programming methods. Prentice-Hall, Englewood Cliffs, New Jersey.
[128] Sienkiewicz J, Holyst J A. 2005a. Public Transport systems in Poland: from Bialystok to Zielona Gora by bus and tram using universal statistics of complex networks. arXiv:physics/0503099.
[129] Sienkiewicz J, Holyst J A. 2005b. Statistical analysis of 22 public transport networks in Poland.arXiv: physics/0506074.
[130] Smith M J. 1979. The existence, uniqueness and stability of traffic equilibrium. Transportation Research part B, 13: 295-304.
[131] Smith M J. 1984. Two alternative definitions of traffic equilibrium. Transportation Research part B, 18:63-65.
[132] Sneppen K, Rosvall M, Trusina A, Minnhagen P. 2004. A simple model for self organization of bipartite networks. Europhysics Letters, 67: 349-354.
[133] Song C, Havlin S, Makse H A. 2005. Self-similarity of complex networks. Nature, 433:392-395.
[134] Sun H J, Gao Z Y. 2007. Dynamical behaviors of epidemics on scale-free networks with community structure. Physica A, 381: 491-496.
[135] Tadic B, Thurner S, Rodgers G J. 2004. Traffic on complex networks: Towards understanding global statistical properties from microscopic density fluctuations. Physical Review E, 69:036102.
[136] Tadic B. 2001. Dynamics of directed graphs: the World Wide Web. PhysicaA, 293: 273-284.
[137] Tyler J R, Wilkinson D M, Huberman B A. 2003. Email as Spectroscopy: Automated Discovery of Community Structure within Organizations. Communities and Technologies,81-96.
[138] Vazquez A, Dobrin R, Sergi D, Eckmann J P, Oltvai Z N, Barabasi A L. 2004. The topological relationship between the large-scale attributes and local interaction patterns of complex networks. Proceedings of the National Academy of Sciences USA, 101:17940-17945.
[139] Verdasca T, Gama M M T da, Nunes A, Bernardino N R, Pacheco J M, and Gomes M C. 2005.Recurrent epidemics in small world networks. Journal of Theoretical Biology, 233: 553-561.
[140] Volchenkov D, Volchenkova L, Blanchard P H. 2002. Epidemic spreading in a variety of scale free networks. Physical Review E, 66: 046137.
[141] Wang W X, Hu B, Zhou T, Wang B H, Xie Y B. 2005a. Mutual selection model for weighted networks. Physical Review E, 72: 046140.
[142] Wang W X, Wang B H, Hu B, Yan G, Ou Q. 2005b. General Dynamics of Topology and Traffic on Weighted Technological Networks. Physical Review Letters, 94: 188702.
[143] Wang W X, Hu B, Wang B H, Yan G 2006a. Mutual attraction model for both assortative and disassortative weighted networks. Physical Review E, 73: 016133.
[144] Wang W X, Wang B H, Yin C Y, Xie Y B, Zhou T. 2006b. Traffic dynamics based on local routing protocol on a scale-free network. Physical Review E, 73: 026111.
[145] Wang W X, Yin C Y, Yan G, Wang B H. 2006c. Integrating local static and dynamic information for routing traffic. Physical Review E, 74: 016101.
[146] Wardrop J G 1952. Some theoretical aspects of road traffic research. Proceedings of Institution of Civil Engineers-Part II, 1: 325-378.
[147] Watts D J, Strogatz S H. 1998. Collective dynamics of 'small-world' networks. Nature, 393:440-442.
[148] Watts D J. 1999. Small Worlds, Princeton: Princeton University Press.
[149] Wilkinson D M, Huberman B A. 2004. A method for finding communities of related genes.Proceedings of the National Academy of Sciences USA, 101: 5241-5248.
[150] Wright C, Roberg P. 1998. The conceptual structure of traffic jams. Transport Policy, 5: 25-35.
[151] Wu J J, Gao Z Y, Sun H J, Huang H J. 2004a. Urban transit system as a scale-free network.Modern Physics Letters B, 18:1043-1049.
[152] Wu J J, Gao Z Y, Sun H J. 2004b. Simulation of traffic congestion with SIR model. Modern Physics Letters B, 18:1537-1542.
[153] Wu J J, Gao Z Y, Sun H J. 2005. Random and preferential attachment networks with aging.Chinese Physics Letters, 22: 765-768.
[154] Wu J J, Gao Z Y, Sun H J. 2006a. Model for dynamic traffic congestion in scale-free networks.Europhysics Letters, 76: 787-793.
[155] Wu J J, Gao Z Y, Sun H J. 2006b. Unified model for generation complex networks with utility preferential attachment. Communications in Theoretical Physics, 46: 183-186.
[156] Wu J J, Gao Z Y, Sun H J. 2006c. Cascade and breakdown in scale-free networks with community Structure.Physical Review E,74:066111-066115.
[157]Wu J J,Gao Z Y,Sun H J.2006d.Complexity and efficiency of Beijing transit network.International Journal of Modern Physics B,20:2129-2136.
[158]Wu J J,Gao Z Y,Sun H J,Huang H J.2006e.Congestion in different topologies of traffic networks.Europhysics Letters,74:560-566.
[159]Wu J J,Gao Z Y,Sun H J.2007a.Effects of the cascading failures on scale-free traffic networks.PhysicaA,387:505-511.
[160]Wu J J,Gao Z Y,Sun H J.2007b,Strength dynamics of weighted evolving networks.Chinese Physics,16:47-50.
[161]Xiong S J.2004.Dynamics and asymptotical behavior of spreading processes in a closed system.Physical Review E,69:066102.
[162]Yan G,Zhou T,Hu B,Fu Z Q,Wang B H.2006.Efficient routing on complex networks.Physical Review E,73:046108.
[163]Yan G,Fu Z Q,Ren J,Wang W X.2007.Collective synchronization induced by epidemic dynamics on complex networks with communities.Physical Review E,75:016108.
[164]Yang H,Yagar S.1994.Traffic assignment and traffic control in general freeway-arterial corridor system.Transportation Research part B,28:463-486.
[165]Yang H J,Zhou T,Wang W X,Wang B H,Zhao F C.2005.Load distribution on small-world networks,arXiv:cond-mat/0509354.
[166]Yin C Y,Wang B H,Wang W X,Yan G,Yang H J.2006a.Traffic dynamics based on an efficient routing strategy on scale free networks.European Physical Journal B,49:205-211.
[167]Yin C Y,Wang B H,Wang W X,Zhou T,Yang H J.2006b.Efficient routing on scale-free networks based on local information.Physics Letters A,351:220-224.
[168]Yook S H,Radicchi F,Meyer-Ortmanns H.2005.Self-similar scale-free networks and disassortativity.Physical Review E,72:045105.
[169]Zhao L,Park K,Lai Y C.2004.Attack vulnerability of scale-free networks due to cascading breakdown.Physical Review E,70:035101.
[170]Zhao L,Park K,Lai Y C,Ye N.2005a.Tolerance of scale-free networks against attack-induced cascades.Physical Review E,72:025104.
[171]Zhao L,Lai Y C,Park K,Ye N.2005b.Onset of traffic congestion in complex networks.Physical Review E,71:026125.
[172]Zheng J F,Gao Z Y,Zhao H.2007a.Properties of asymmetrical evolving networks.Physica A,376:719-724
[173]Zheng J F,Gao Z Y,Zhao X M.2007b,Clustering and congestion effects on cascading failures of scale-free networks.Europhysics letters,79:58002-58007.
[174]Zheng J F,Gao Z Y,Zhao X M.2007c,Modeling cascading failures in congested complex networks.Physica A,385:700-706.
[175]Zheng J F,Gao Z Y,Zhao X M.2007d,Properties of transportation dynamics on scale free networks.PhysicaA,373:837-844.
[176]Zhou S,Mondragon R J.2004.The rich-club phenomenon in the lnternet topology.IEEE Communication Letters,8:180-182.
[177]汪小帆,陈关荣,李翔.2006.复杂网络理论及其应用.北京:清华大学出版社.
[178]祁国宁,徐福缘,王恒山,车宏安.2004.复杂网络-系统结构研究文集.浙江大学现代制造工程研究所,上海大学理工学院系统工程研究所.
[179]高白友,任华玲.2005.城市动态交通流分配模型与算法.北京:人民交通出版社.
[180]高自友,宋一凡,四兵锋.2000.城市交通连续平衡网络设计-理论与方法.北京:中国铁道出版社.
[181]高自友,赵小梅,黄海军.2006.城市交通网络的复杂性.复杂网络.上海科技教育出版社