复杂网络建模与典型网络上的动力学过程研究
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摘要
在过去的几年中,有关与网络刻画和理解的研究工作非常活跃。事实上,在许多自然和人造系统中都存在着大量的大规模复杂网络。本论文运用统计物理、运筹学以及计算机模拟等方法,从复杂网络建模和典型复杂网络(即随机网络、小世界网络和无标度网络)上的动力学过程这两个方面进行了相关的分析与研究。重点研究了典型复杂网络上的流量分布与阻塞、级联失效行为以及基于离散时间和离散状态的同步行为。本论文的主要工作和创新点如下：
(1)在复杂网络建模方面,首先介绍了三个典型的复杂网络模型,即Erdos-Renyi随机网络模型、Watts-Strogatz小世界网络模型和Barabasi-Albert无标度网络模型。然后,提出了非对称演化网络模型和基于交通流演化的加权网络模型。在非对称演化网络模型中,引入了节点效用的概念,并且新节点在选择网络中已有节点进行连结时遵从效用偏好的机制,对建立连结的两个节点的效用值以不同概率增长来表征网络的非对称特性。理论分析和数值模拟均表明网络中节点的效用分布服从幂律分布,而度分布则介于指数分布和幂律分布之间。在基于交通流演化的加权网络模型中,交通流的状态被认为是网络中的节点,如果某一个交通流状态能够在一个时间步演化成另一个交通流状态,则在这两个交通流状态(即节点)之间建立连边,而交通流状态在演化过程中传输的交通流量被认为是边上的权重。从理论分析和数值模拟的角度,研究了节点强度和度之间的非线性相关关系。
(2)基于用户均衡模型,研究典型复杂网络(特别是无标度网络)上流量分布的规律,研究发现无标度网络上的流量分布在这种情况下可以呈现出指数分布或者幂律分布的形式。基于元胞传输模型,分析了梯度网络上的阻塞特性,研究发现,随着网络中阻塞程度的增加,阻塞程度在随机网络和无标度网络之间的差值呈现出先增加,后减小,最后又增加的趋势。此外,基于一定的流量演化规则,类似于拥挤条件下的随机游走行为,研究了典型复杂网络上的阻塞消散和流量波动特性,并引入了截流以及截流和诱导两种拓展方式来缓解网络中的局部阻塞,研究发现这两种拓展方式并不会加重网络中的全局阻塞；截流和诱导的方式可以在一定程度上缓解无标度网络(即异质网络)中的全局阻塞,并且可以减少网络中流量的波动特性。
(3)在级联失效方面,本论文将一个基于简单网络的光纤束模型拓展到无标度网络,研究了网络上的边失效行为。理论分析表明,当节点流量和度之间的幂律指数大于度分布的幂律指数时,网络中的平均边失效比例与网络规模之间存在幂律关系,且幂律指数为-1,与度分布的幂律指数无关。基于用户均衡模型,研究了拥挤效应和网络结构对级联失效的影响。研究发现,拥挤效应对级联失效具有一定的正效应,而网络的异质结构对级联失效是负效应。即：适当地增加网络中的拥挤,可以提高网络抗级联失效的能力；度分布指数较小的无标度网络上的级联失效将更加严重。最后,提出了一个较符合城市交通网络中拥堵传播消散特性的级联失效模型,并探讨了反馈效应对级联失效的影响。研究表明,反馈效应可以减少随机网络和无标度网络在抗级联失效方面的差异。
(4)提出了基于离散时间和离散状态的同步模型。为了刻画网络中节点状态的自驱动函数,引入了节点的状态转移矩阵。通过针对典型复杂网络上的数值研究表明,同步指标把耦合强度划分为四个区：递增区、最大区、递减区和振荡区,为复杂网络上同步行为的研究提供了新的视角。
The last few years have witnessed tremendous activities devoted to the characterization and understanding of networked systems. Indeed many large complex networks arise in a vast number of natural and artificial systems. In this thesis, complex network modeling and dynamical processes in typical complex networks (i.e., random networks, small-world networks and scale-free networks) are studied by using statistics physics, operational research and computer simulation. This thesis focuses on investigating load distributions, traffic jamming, cascading failures and synchronization with discrete time and discrete state in typical complex networks. The main contents of this thesis are summarized as follows:
(1) On the issue of complex network modeling, firstly, three typical complex network models, i.e., Erdos-Renyi random network model, Watts-Strogatz small-world network model and Barabasi-Albert scale-free network model are simply introduced. Then, an asymmetrical evolving network model and a weighted model evolution with traffic flow are presented. In the asymmetrical evolving network model, the concept of utility is introduced. The probability for the new node chosing old nodes to be connected is proportional to the utility of the old node. Moreover, the utilities of the connected nodes update asymmetrically. Both theoretical analysis and simulation results show that the distribution of utility follows a power law, and the degree distribution is between the exponential distribution and power law distribution,In the weighted model evolution with traffic flow, the state of traffic flow is considered as a node, an edge between two nodes is created if one state of traffic flow can evolve into another at one time step, and the transferred traffic volume is considered as the weight of the edge. Non-linear relationship between the strength and degree of the node is studied theoretically and by numerical tests.
(2) Based on user equilibrium model, this thesis discusses load distribution in typical complex networks (especially in scale-free networks) under the effect of congestion. It is found that load distribution may follow as exponential distribution or power law distribution in scale-free networks. Based on cell transmission model, traffic jamming in gradient networks is analyzed. The difference of the jamming factor between random networks and scale-free networks is found to increase firstly, and then decrease and finally increase, with the increase of the degree of traffic jamming. In terms of the traffic evolving rule, which is similar to random walks under the condition of congestion, the diffusion of congestion and flow fluctuations in typical complex networks are studied. And two extended cases, i.e., stopping traffic flows and stopping and guiding traffic flows are presented to relieve local congestion. Simulation results show that the two extended cases can not aggravate global congestion, and the second extended case can be used to relieve global congestion and flow flunctions in scale-free networks (i.e., heterogeneous networks).
(3) On the issue of cascading failures, a simple fiber bundle model is extended to scale-free networks to study the behavior of edge failures. Theoretical analysis and simulation results show that, when the exponent of the scaling between the load and degree of the node is larger than the exponent of the degree distribution, there is a scaling relationship between the average rate of failed links and the network size, where the exponent is-1, independent of the exponent of degree distribution. Based on user equilibrium model, the effects of congestion and network structure on cascading failures are investigated. Simulation results show that the effect of congestion has an active effect and the effect of network heterogeneity has a negative effect on influencing the behavior of cascading failures. In other words, the performance of the network against cascading failures can be improved by properly increasing the congestion of the network, and scale-free network with larger value of the exponent of degree distribution is more prone to suffer from cascading failures. Finally, a model for cascading failures fitting urban traffic networks is proposed, and the effect of feedback is also studied. We find that the effect of feedback can reduce the discrepancy between random networks and scale-free networks against cascading failures.
(4) In this thesis, a synchronization model with both discrete time and discrete state is presented. A transfer matrix for the node's state is introduced to determine its self-driven function. Simulation results in typical complex networks show that, according to the synchronization index, the coupling strength is divided into four regions:the increasing region, the maximum region, the decreasing region and the oscillation region. It may shed new insights on investigating synchronization behavior in complex networks.
(1)在复杂网络建模方面,首先介绍了三个典型的复杂网络模型,即Erdos-Renyi随机网络模型、Watts-Strogatz小世界网络模型和Barabasi-Albert无标度网络模型。然后,提出了非对称演化网络模型和基于交通流演化的加权网络模型。在非对称演化网络模型中,引入了节点效用的概念,并且新节点在选择网络中已有节点进行连结时遵从效用偏好的机制,对建立连结的两个节点的效用值以不同概率增长来表征网络的非对称特性。理论分析和数值模拟均表明网络中节点的效用分布服从幂律分布,而度分布则介于指数分布和幂律分布之间。在基于交通流演化的加权网络模型中,交通流的状态被认为是网络中的节点,如果某一个交通流状态能够在一个时间步演化成另一个交通流状态,则在这两个交通流状态(即节点)之间建立连边,而交通流状态在演化过程中传输的交通流量被认为是边上的权重。从理论分析和数值模拟的角度,研究了节点强度和度之间的非线性相关关系。
(2)基于用户均衡模型,研究典型复杂网络(特别是无标度网络)上流量分布的规律,研究发现无标度网络上的流量分布在这种情况下可以呈现出指数分布或者幂律分布的形式。基于元胞传输模型,分析了梯度网络上的阻塞特性,研究发现,随着网络中阻塞程度的增加,阻塞程度在随机网络和无标度网络之间的差值呈现出先增加,后减小,最后又增加的趋势。此外,基于一定的流量演化规则,类似于拥挤条件下的随机游走行为,研究了典型复杂网络上的阻塞消散和流量波动特性,并引入了截流以及截流和诱导两种拓展方式来缓解网络中的局部阻塞,研究发现这两种拓展方式并不会加重网络中的全局阻塞；截流和诱导的方式可以在一定程度上缓解无标度网络(即异质网络)中的全局阻塞,并且可以减少网络中流量的波动特性。
(3)在级联失效方面,本论文将一个基于简单网络的光纤束模型拓展到无标度网络,研究了网络上的边失效行为。理论分析表明,当节点流量和度之间的幂律指数大于度分布的幂律指数时,网络中的平均边失效比例与网络规模之间存在幂律关系,且幂律指数为-1,与度分布的幂律指数无关。基于用户均衡模型,研究了拥挤效应和网络结构对级联失效的影响。研究发现,拥挤效应对级联失效具有一定的正效应,而网络的异质结构对级联失效是负效应。即：适当地增加网络中的拥挤,可以提高网络抗级联失效的能力；度分布指数较小的无标度网络上的级联失效将更加严重。最后,提出了一个较符合城市交通网络中拥堵传播消散特性的级联失效模型,并探讨了反馈效应对级联失效的影响。研究表明,反馈效应可以减少随机网络和无标度网络在抗级联失效方面的差异。
(4)提出了基于离散时间和离散状态的同步模型。为了刻画网络中节点状态的自驱动函数,引入了节点的状态转移矩阵。通过针对典型复杂网络上的数值研究表明,同步指标把耦合强度划分为四个区：递增区、最大区、递减区和振荡区,为复杂网络上同步行为的研究提供了新的视角。
The last few years have witnessed tremendous activities devoted to the characterization and understanding of networked systems. Indeed many large complex networks arise in a vast number of natural and artificial systems. In this thesis, complex network modeling and dynamical processes in typical complex networks (i.e., random networks, small-world networks and scale-free networks) are studied by using statistics physics, operational research and computer simulation. This thesis focuses on investigating load distributions, traffic jamming, cascading failures and synchronization with discrete time and discrete state in typical complex networks. The main contents of this thesis are summarized as follows:
(1) On the issue of complex network modeling, firstly, three typical complex network models, i.e., Erdos-Renyi random network model, Watts-Strogatz small-world network model and Barabasi-Albert scale-free network model are simply introduced. Then, an asymmetrical evolving network model and a weighted model evolution with traffic flow are presented. In the asymmetrical evolving network model, the concept of utility is introduced. The probability for the new node chosing old nodes to be connected is proportional to the utility of the old node. Moreover, the utilities of the connected nodes update asymmetrically. Both theoretical analysis and simulation results show that the distribution of utility follows a power law, and the degree distribution is between the exponential distribution and power law distribution,In the weighted model evolution with traffic flow, the state of traffic flow is considered as a node, an edge between two nodes is created if one state of traffic flow can evolve into another at one time step, and the transferred traffic volume is considered as the weight of the edge. Non-linear relationship between the strength and degree of the node is studied theoretically and by numerical tests.
(2) Based on user equilibrium model, this thesis discusses load distribution in typical complex networks (especially in scale-free networks) under the effect of congestion. It is found that load distribution may follow as exponential distribution or power law distribution in scale-free networks. Based on cell transmission model, traffic jamming in gradient networks is analyzed. The difference of the jamming factor between random networks and scale-free networks is found to increase firstly, and then decrease and finally increase, with the increase of the degree of traffic jamming. In terms of the traffic evolving rule, which is similar to random walks under the condition of congestion, the diffusion of congestion and flow fluctuations in typical complex networks are studied. And two extended cases, i.e., stopping traffic flows and stopping and guiding traffic flows are presented to relieve local congestion. Simulation results show that the two extended cases can not aggravate global congestion, and the second extended case can be used to relieve global congestion and flow flunctions in scale-free networks (i.e., heterogeneous networks).
(3) On the issue of cascading failures, a simple fiber bundle model is extended to scale-free networks to study the behavior of edge failures. Theoretical analysis and simulation results show that, when the exponent of the scaling between the load and degree of the node is larger than the exponent of the degree distribution, there is a scaling relationship between the average rate of failed links and the network size, where the exponent is-1, independent of the exponent of degree distribution. Based on user equilibrium model, the effects of congestion and network structure on cascading failures are investigated. Simulation results show that the effect of congestion has an active effect and the effect of network heterogeneity has a negative effect on influencing the behavior of cascading failures. In other words, the performance of the network against cascading failures can be improved by properly increasing the congestion of the network, and scale-free network with larger value of the exponent of degree distribution is more prone to suffer from cascading failures. Finally, a model for cascading failures fitting urban traffic networks is proposed, and the effect of feedback is also studied. We find that the effect of feedback can reduce the discrepancy between random networks and scale-free networks against cascading failures.
(4) In this thesis, a synchronization model with both discrete time and discrete state is presented. A transfer matrix for the node's state is introduced to determine its self-driven function. Simulation results in typical complex networks show that, according to the synchronization index, the coupling strength is divided into four regions:the increasing region, the maximum region, the decreasing region and the oscillation region. It may shed new insights on investigating synchronization behavior in complex networks.
引文
[1]Albert, R., Barabasi, A.L.,2002. Statistical mechanics of complex networks. Reviews of Modern Physics,74(1):47-97
[2]Newman, M.E.J.,2003. The structure and function of complex networks. SIAM Review,45(2): 167-256
[3]Boccalettia, S., Latorab, V., Moreno, Y., Chavezf, M., Hwanga, D.U.,2006. Complex networks: structure and dynamics. Physics Reports,424(4-5):175-308
[4]Strogatz, S.H.,2001. Exploring complex networks. Nature,410(6825):268-276
[5]汪小帆,李翔,陈关荣,2006.复杂网络理论及其应用[M].清华大学出版社
[6]Camacho, J., Guimera, R., Amaral, L.A.N.,2002. Robust patterns in food web structure. Physical Review Letters,88(22):228102
[7]Newman, M.E.J.,2001. The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences, USA,98(2):404-409
[8]Pastor-Satorras, R., Vazquez, A., Vespignani, A.,2001. Dynamical and correlation properties of the Internet. Physical Review Letters,87(25):258701
[9]Albert, R., Jeong, H., Barabasi, A.L.,1999. Diameter of the world wide web. Nature,401(6749): 130-131
[10]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(22):7794-7799
[11]Jeong, H., Mason, S.P., Barabasi, A.L., Oltvai, Z.N.,2001. Lethality and centrality in protein networks. Nature,411(6833):41-42
[12]Watts, D.J., Strogatz, S.H.,1998. Collective dynamics of'small-world'networks. Nature, 393(6684):440-442
[13]Barabasi, A.L., Albert, R.,1999. Emergence of scaling in random networks. Science,286(5439): 509-512
[14]Erdos, P., Renyi, A.,1960. On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Science,5:17-60
[15]Jeong, H., Tombor, B., Albert, R., Oltvai, Z.N., Barabasi, A.L.,2000. The large-scale organization of metabolic networks. Nature,407(6804):651-654
[16]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(21):11149-11152
[17]Latora, V, Marchiori, M.,2002. Is the Boston subway a small-world network?. Physica A, 314(1-4):109-113
[18]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(3):036106
[19]Wu, J.J., Gao, Z.Y., Sun, H.J., Huang, H.J.,2004. Urban transit system as a scale-free network. Modern Physics Letters B,18(19-20):1043-1049
[20]Newman, M.E.J., Watts, D.J.,1999. Renormalization group analysis of the small-world network model. Physics Letters A,263(4-6):341-346
[21]Krapivsky, P.L., Redner, S., Leyvraz, F.,2000. Connectivity of growing random networks. Physical Review Letters,85(21):4629-4632
[22]Onody, R.N., de Castro P.A.,2004. Nonlinear Barabsi-Albert network. Physica A,336(3-4): 491-502
[23]Dorogovtsev, S.N., Mendes, J.F.F., Samukhin, A.N.,2000. Structure of growing networks with preferential linking. Physical Review Letters,85(21):4633-4636
[24]Dorogovstev, S.N., Mendes, J.F.F.,2000. Evolution of networks with aging of sites. Physical Review E,62(2):1842-1845
[25]Albert, R., Barabasi, A.L.,2000. Topology of evolving networks:local events and universality. Physical Review Letters,85(24):5234-5237
[26]Bianconi, G., Barabasi, A.L,2001. Competition and multi-scaling in evolving networks. Europhysics Letters,54(4):436-442
[27]Bianconi, G., Barabasi, A.L,2001. Bose-Einstein condensation in complex networks. Physical Review Letters,86(24):5632-5635
[28]Holme, P., Kim, B.J.,2002. Growing scale-free networks with tunable clustering coefficient. Physical Review E,65(2):026107
[29]Li, X., Chen, G.,2003. A local world evolving network model. Physica A,328(1-2):274-286
[30]Gao, Z.Y., Li, K.P.,2005. Evolution of traffic flow with scale-free topology. Chinese Physics Letters,22(10):2711-2714
[31]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
[32]Li, X.G, Gao, Z.Y., Li, K.P., Zhao, X.M.,2007. Relationship between microscopic dynamics in traffic flow and complexity in networks. Physical Review E,76(1):016110
[33]Wu, J.J., Gao, Z.Y., Sun, H.J., Huang, H.J.,2005. Random and preferential attachment networks with aging. Chinese Physics Letters,22(3):765-768
[34]Wu, J.J., Gao, Z.Y., Sun, H.J.,2006. Unified model for generation complex networks with utility preferential attachment. Communications in Theoretical Physics,46(1):183-186
[35]Zhou, T., Yan, G., Wang, B.H.,2005. Maximal planar networks with large clustering coefficient and power-law degree distribution. Physical Review E,71(4):046141
[36]Guo, Q., Zhou, T., Liu, J.G, Bai, W.J., Wang, B.H., Zhao, M.,2006. Growing scale-free small-world networks with tunable assortative coefficient. Physica A,371(2):814-822
[37]张培培,何阅,周涛,苏蓓蓓,常慧,周月平,汪秉宏,何大韧,2006.一个描述合作网络顶点度分布的模型.物理学报,55(1)：60-67
[38]Bu, S.L., Wang, B.H., Zhou, T.,2007. Gaining scale-free and high clustering complex networks. Physica A,374(2):864-868
[39]Xie, Y.B., Zhou, T., Wang, B.H.,2008. Scale-free networks without growth. Physica A,387(7): 1683-1688
[40]Barrat, A., Barthelemy, M., Pastor-Satorras, R., Vespignani, A.,2004. The architecture of complex weighted networks. Proceedings of the National Academy of Sciences, USA,101(11): 3747-3752
[41]Yook, S.H., Jeong, H., Barabasi, A.L., Tu, Y.,2001. Weighted evolving networks. Physical Review Letters,86(25):5835-5838
[42]Zheng, D., Trimper, S., Zheng, B., Hui, P.M.,2003. Weighted scale-free networks with stochastic weight assignments. Physical Review E,67(4):040102
[43]Barrat, A., Barthelemy, M., Vespignani, A.,2004. Weighted evolving networks:coupling topology and weight dynamics. Physical Review Letters,92(22):228701
[44]Wang, W.X., Wang, B.H., Hu, B., Yan, G., Ou, Q.,2005. General dynamics of topology and traffic on weighted technological networks. Physical Review Letters,94(18):188702
[45]Wang, W.X., Hu, B., Zhou, T., Wang, B.H., Xie, Y.B.,2005. Mutual selection model for weighted networks. Physical Review E,72(4):046140
[46]Wang, W.X., Hu, B., Wang, B.H., Yan, G.,2006. Mutual attraction model for both assortative and disassortative weighted networks. Physical Review E,73(1):016133
[47]Zhao, H., Gao, Z.Y., Wang, W.X., Yan, G.,2006. Self organization of topology and weight dynamics on networks from merging and regeneration. Chinese Physics Letters,23(2):275-278
[48]Wu, J.J., Gao, Z.Y., Sun, H.J.,2007. Strength dynamics of weighted evolving networks. Chinese Physics,16(1):47-50
[49]Li, M.H., Fan, Y, Chen, J.W., Gao, L., Di, Z.R., Wu, J.S.,2005. Weighted networks of scientific communication:the measurement and topological role of weight. Physica A,350(2-4):643-656
[50]Li, M.H., Fan, Y, Wang, D.H., Li, D.Q., Wu, J.S., Di, Z.R.,2007. Small-world effect induced by weight randomization on regular networks. Physics Letters A,364(6):488-493
[51]Li, M.H., Wu, J.S., Wang, D.H., Zhou, T., Di, Z.R., Fan, Y.,2007. Evolving model of weighted networks inspired by scientific collaboration networks. Physica A,375(1):355-364
[52]Fan, Y, Li, M.H., Zhang, P., Wu, J.S., Di, Z.R.,2007. The effect of weight on community structure of networks. Physica A,378(2):583-590
[53]Sole, R.V., Valverde, S.,2001. Information transfer and phase transitions in a model of internet traffic. Physica A,289(3-4):595-605
[54]Arenas, A., Diaz-Guilera, A., Guimera, R.,2001. Communication in networks with hierarchical branching. Physical Review Letters,86(14):3196-3199
[55]Guimera, R., Diaz-Guilera, A., Vega-Redondo, F., Cabrales, A., Arenas, A.,2002. Optimal network topologies for local search with congestion. Physical Review Letters,89(24):248701
[56]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(3): 036102
[57]Tadic, B., Thurner, S.,2004. Information super-diffusion in structured networks. Physica A,332: 566-584
[58]Echenique, P., Gomez-Gardenes, J., Moreno, Y.,2004. Improved routing strategies for Internet traffic delivery. Physical Review E,70(5):056105
[59]Mukherjee, G, Manna, S.S.,2005. Phase transition in a directed traffic flow network. Physical Review E,71(6):066108
[60]Zhao, L., Lai, Y.C., Park, K., Ye, N.,2005. Onset of traffic congestion in complex networks. Physical Review E,71(2):026125
[61]Yan, G., Zhou, T., Hu, B., Fu, Z.Q., Wang, B.H.,2006. Efficient routing on complex networks. Physical Review E,73(4):046108
[62]Wang, W.X., Wang, B.H., Yin, C.Y, Xie, Y.B., Zhou, T.,2006. Traffic dynamics based on local routing protocol on a scale-free network. Physical Review E,73(2):026111
[63]Wang, W.X., Yin, C.Y., Yan, G., Wang, B.H.,2006. Integrating local static and dynamic information for routing traffic. Physical Review E,74(1):016101
[64]Yin, C.Y., Wang, B.H., Wang, W.X., Yan, G., Yang, H.J.,2006. Traffic dynamics based on an efficient routing strategy on scale free networks. European Physics Journal B,49(2):205-211
[65]Yin, C.Y., Wang, B.H., Wang, W.X., Zhou, T., Yang, H.J.,2006. Efficient routing on scale-free networks based on local information. Physics Letters A,351(4-5):220-224
[66]Yang, H.X., Wang, W.X., Wu, Z.X., Wang, B.H.,2008. Traffic dynamics in scale-free networks with limited packet-delivering capacity. Physica A,387(27):6857-6862
[67]Hu, M.B., Wang, W.X., Jiang, R., Wu, Q.S., Wu, Y.H.,2007. Phase transition and hysteresis in scale-free network traffic. Physical Review E,75(3):036102
[68]Liu, Z., Hu, M.B., Jiang, R., Wang, W.X., Wu, Q.S.,2007. Method to enhance traffic capacity for scale-free networks. Physical Review E,76(3):037101
[69]Watts, D.J.,2002. A simple model of global cascades on random networks. Proceedings of the National Academy of Sciences, USA,99(9):5766-5771
[70]Goh, K.I., Lee, D.S., Kahng, B., Kim, D.,2003. Sandpile on scale-free networks. Physical Review Letters,91(14):148701
[71]Motter, A.E., Lai, Y.C.,2002. Cascade-based attacks on complex networks. Physical Review E, 66(6):065102(R)
[72]Motter, A.E.,2004. Cascade control in complex networks. Physical Review Letters,93(9): 098701
[73]Holme, P., Kim, B.J.,2002. Vertex overload breakdown in evolving networks. Physical Review E,65(6):066109
[74]Holme, P.,2002 Edge overload breakdown in evolving networks. Physical Review E,66(3): 036119.
[75]Moreno, Y, Gomez, J.B., Pacheco, A.F.,2002. Instability of scale-free networks under node-breaking avalanches Europhysics Letters,58(4):630-636
[76]Moreno, Y, Pastor-Satorras, R., Vazquez, A., Vespignani, A.,2003. Critical load and congestion instabilities in scale-free networks. Europhysics Letters,62(2):292-298
[77]Zhao, L., Park, K., Lai, Y.C.,2004. Attack vulnerability of scale-free networks due to cascading breakdown. Physical Review E,70(3):035101(R)
[78]Zhao, L., Park, K., Lai, Y.C., Ye, N.,2005. Tolerance of scale-free networks against attack-induced cascades. Physical Review E,72(2):025104(R)
[79]Crucitti, P., Latora, V, Marchiori, M.,2004. Model for cascading failures in complex networks. Physical Review E,69(4):045104(R)
[80]Wang, X.F., Xu, J.,2004. Cascading failures in coupled map lattices. Physical Review E,70(5): 056113
[81]Xu, J., Wang, X.F.,2005. Cascading failures in scale-free coupled map lattices. Physica A, 349(3-4):685-692
[82]Lee, E.J., Goh, K.I., Kahng, B., Kim, D.,2005. Robustness of the avalanche dynamics in data-packet transport on scale-free networks. Physical Review E,71(5):056108
[83]Gallos, L.K., Cohen, R., Argyrakis, P., Bunde, A., Havlin, P S.,2005. Stability and topology of scale-free networks under attack and defense strategies. Physical Review Letters,94(18): 188701
[84]Wu, J.J., Gao, Z.Y., Sun, H.J.,2006. Cascading breakdown in scale-free networks with community structure. Physical Review E,74(6):066111
[85]Wu, J.J., Sun, H.J., Gao, Z.Y.,2007. Cascading failures on weighted urban traffic equilibrium networks. Physica A,386(1):407-413
[86]Wu, J.J., Gao, Z.Y., Sun, H.J.,2007. Effects of the cascading failures on scale-free traffic networks. Physica A,387(2):505-511
[87]Zheng, J.F., Gao, Z.Y., Zhao, X.M.,2007. Clustering and congestion effects on cascading failures of scale-free networks. Europhysics Letters,79(5):58002
[88]Zhao, H., Gao, Z.Y.,2007. Cascade defense via navigation in scale-free networks. European Physics Journal B,57(1):95-101
[89]Zhao, X.M., Gao, Z.Y.,2007. How non-uniform tolerance parameter strategy changes the response of scale-free networks to failures. European Physical Journal B,59(1):85-92
[90]Sun, H.J., Zhao, H., Wu, J.J.,2008. A robust matching model of capacity to defense cascading failure on complex networks. Physica A,387(25):6431-6435
[91]Cui, D., Gao, Z.Y., Zheng, J.F.,2009. Tolerance of edge cascades with coupled map lattices methods. Chinese Physics B,18(3):992-996
[92]Wang, B., Kim, B.J.,2007. A high robustness and low cost model for cascading failures. Europhysics Letters,78(4):48001
[93]Wu, J., Deng, H.Z., Tan, Y.J., Zhu, D.Z,2007. Vulnerability of complex networks under intentional attack with incomplete information. Journal of Physics A,40(11):2665-2671
[94]Simonsen, I., Buzna, L., Peters, K., Bornholdt, S., Helbing, D.,2008. Transient dynamics increasing network vulnerability to cascading failures. Physical Review Letters,100(21): 218701
[95]Li, P., Wang, B.H., Sun, H., Gao, P., Zhou, T.,2008. A limited resource model of fault-tolerant capability against cascading failure of complex network. European Physical Journal B,62(1): 101-104
[96]Wang, W.X., Chen, G.R.,2008. Universal robustness characteristic of weighted networks against cascading failure. Physical Review E,77(2):026101
[97]Wang, J.W., Rong, L.L.,2009. A model for cascading failures in scale-free networks with a breakdown probability. Physica A,388(7):1289-1298
[98]Bao, Z.J., Cao, Y.J., Ding, L.J., Wang, G.Z.,2009. Comparison of cascading failures in small-world and scale-free networks subject to vertex and edge attacks. Physica A,388(20): 4491-4498
[99]Wang, J., Liu, Y.H., Jiao, Y, Hu, H.Y.,2009. Cascading dynamics in congested complex networks. European Physical Journal B,67(1):95-100
[100]Pastor-Satorras, R., Vespingnani, A.,2001. Epidemic spreading in scale-free networks. Physical Review Letters,86(4):3200-3203
[101]Pastor-Satorras, R., Vespingnani, A.,2001. Epidemic dynamics and endemic states in complex networks. Physical Review E,63(6):066117
[102]Pastor-Satorras, R., Vespingnani, A.,2002. Epidemic dynamics in finite size scale-free networks. Physical Review E,65(3):035108
[103]Pastor-Satorras, R., Vespingnani, A.,2002. Immunization of complex networks. Physical Review E,65(3):036104
[104]Boguna, M., Pastor-Satorras, R., Vespingnani, A.,2003. Absence of epidemic threshold in scale-free networks with degree correlations. Physical Review Letters,90(2):028701
[105]Cohen, R., Havlin, S., ben-Avraham, D.,2003. Efficient immunization strategies for computer networks and populations. Physical Review Letters,91(24):247901
[106]Barthelemy, M., Barrat, A., Pastor-Satorras, R., Vespingnani, A.,2004. Velocity and hierarchical spread of epidemic outbreaks in scale-free networks. Physical Review Letters, 92(17):178701
[107]Dodds, P.S., Watts, D.J.,2004. Universal behavior in a generalized model of contagion. Physical Review Letters,92(21):218701
[108]Zanette, D.H.,2002. Dynamics of rumor propagation on small world networks. Physical Review E,65(4):041908
[109]Moreno, Y., Nekovee, M., Pacheco, A.F.,2004. Dynamics of rumor spreading in complex networks. Physical Review E,69(6):066130
[110]Balthrop, J., Forrest, S., Newman, M.E.J., Williamson, M.M.,2004. Technological networks and the spread of computer viruses. Science,304(5670):527-529
[111]Zhou, T., Liu, J.G., Bai, W.J., Chen, G.R., Wang, B.H.,2006. Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity. Physical Review E,74(5):056109
[112]Zhou, T., Fu, Z.Q., Wang, B.H.,2006. Epidemic dynamics on complex networks. Progress of Natural Science,16(5):452-457
[113]Yang, R., Wang, B.H., Ren, J., Bai, W.J., Shi, Z.W., Wang, W.X.,2007. Epidemic spreading on heterogeneous networks with identical infectivity. Physics Letters A,364(3-4):189-193
[114]Bai, W.J., Zhou, T., Wang, B.H.,2007. Immunization of susceptible-infected model on scale-free networks. Physica A,384(2):656-662
[115]Kuramoto, Y.,1984. Chemical oscillations, waves and turbulence, Springer-Verlag
[116]Kaneko, K.,1992. Coupled map lattices, Singapore:World Scientific
[117]Neda, Z., Ravasz, E., Brechet, Y., Vicsek, T., Barabasi, A.L.,2000. The sound of many hands clapping. Nature,403(6772):849-850
[118]Wang, X.F., Chen, G.R.,2002. Synchronization in small-world dynamical networks. International Journal of Bifurcation and Chaos,12(1):187-192
[119]Wang, X.F., Chen, G.R.,2002. Synchronization in scale-free dynamical networks:robustness and fragility. IEEE Transactions on Circuits and Systems-I,49(1):54-62
[120]Li, X., Chen, G.R.,2003. Synchronization and desynchronization of complex dynamical networks:an engineering viewpoint. IEEE Transactions on Circuits and Systems-I,50(11): 1381-1390
[121]Lv, J.H., Yu, X., Chen, G.R., Cheng, D.,2004. Characterizing the synchronizability of small-world dynamical networks. IEEE Transactions on Circuits and Systems-I,51(4): 787-796
[122]Lv, J.H., Yu, X., Chen, G.R.,2004. Chaos synchronization of general complex dynamical networks. Physica A,334(1-2):281-302
[123]Li, C.G., Chen, G.R.,2004. Synchronization in general complex dynamical networks with coupling delays. Physica A,343:236-278
[124]Fan, J., Wang, X.F.,2005. On synchronization in scale-free dynamical networks. Physica A, 349(3-4):443-451
[125]Jalan, S., Amritkar, R.E.,2003. Self-organized and driven phase synchronization in coupled maps. Physical Review Letters,90(1):014101
[126]Nishikawa, T., Motter, A.E., Lai, Y.C., Hoppensteadt, F.C.,2003. Heterogeneity in oscillator networks:are smaller worlds easier to synchronize?. Physical Review Letters,91(1):014101
[127]Hong, H., Choi, M.Y., Kim, B.J.,2002. Synchronization on small-world networks. Physical Review E,65(2):026139
[128]Hong, H., Kim, B.J., Choi, M.Y., Park, H.,2004. Factors that predict better synchronizability on complex networks. Physical Review E,69(6):067105
[129]Wei, G.W., Zhan, M., Lai, C.H.,2002. Tailoring wavelets for chaos control. Physical Review Letters,89(28):284103
[130]Dhamala, M., Jirsa, V.K., Ding, M.,2004. Enhancement of neural synchrony by time delay. Physical Review Letters,92(7):074104
[131]Atay, F.M., Jost, J.,2004. Delays, connection topology, and synchronization of coupled chaotic maps. Physical Review Letters,92(14):144101
[132]Chavez, M., Hwang, D.U., Amann, A., Hentschel, H.G.E., Boccaletti, S.,2005. Synchronization is enhanced in weighted complex networks. Physical Review Letters,94(21): 218701
[133]Zhao, M., Zhou, T., Wang, B.H., Wang, W.X.,2005. Enhanced synchronizability by structural perturbations. Physical Review E,72(5):057102
[134]Zhou, T., Zhao, M., Wang, B.H.,2006. Better synchronizability predicted by crossed double cycle. Physical Review E,73(3):037101
[135]Yin, C.Y., Wang, W.X., Chen, G.R., Wang, B.H.,2006. Decoupling process for better synchronizability on scale-free networks. Physical Review E,74(4):047102
[136]Zhao, M., Zhou, T., Wang, B.H., Ou, Q., Ren, J.,2006. Better sychronizability predicted by a new coupling method. European Physical Journal B,53(3):375-379
[137]赵明,汪秉宏,将品群,周涛,2005.复杂网络上动力系统同步研究进展.物理学进展,25(3)：273-295
[138]Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.,2008. Synchronization in complex networks. Physics Reports,469(3):93-153
[139]Kleinberg, J.,2000. Navigation in a small world. Nature,406(6798):845-845
[140]Watts, D.J., Dodds, P.S., Newman, M.E.J.,2002. Identity and search in social networks. Science,296(5571):1302-1305
[141]Adamic, L.A., Lukose, R.M., Puniyani, A.R., Huberman, B.A.,2001. Search in power-law networks. Physical Review E,64(4):046135
[142]Noh, J.D., Rieger, H.,2004. Random walks on complex networks. Physical Review Letters, 92(11):118701
[143]Watts, D.J.,2004. The "new" science of networks. Annual Review of Sociology,30(1): 243-270
[144]Newman, M.E.J.,2002. Assortative mixing in networks. Physical Review Letters,89(20): 208701
[145]Goh K.I., Kahng B., Kim D.,2001. Universal behavior of load distribution in scale-free networks. Physical Review Letters,87(27):278701
[146]Goh K.I., Oh, E., Jeong, H., Kahng B., Kim D.,2002. Classification of scale-free networks. Proceedings of the National Academy of Sciences, USA,99(20):12583-12588
[147]Barthelemy, M.,2004. Betweenness centrality in large complex networks. European Physical Journal B,38(2):163-168
[148]Goh, K.I., Noh, J.D., Kahng, B., Kim, D.,2005. Load distribution in weighted complex networks. Physical Review E,72(1):017102
[149]Holme, P.,2003. Congestion and centrality in traffic flow on complex networks. Advances in Complex System,6(2):163-176
[150]Zheng, J.F., Gao, Z.Y., Zhao, X.M.,2007. Properties of transportation dynamics on scale-free networks. Physica A,373:837-844
[151]Toroczkai, Z., Bassler, K.E.,2004. Network dynamics:jamming is limited in scale-free systems. Nature,428(6984):716-716
[152]Park, K., Lai, Y.C., Zhao, L., Ye, N.,2005. Jamming in complex gradient networks. Physical Review E,71(6):065105(R)
[153]Wu, J.J., Gao, Z.Y., Sun, H.J., Huang, H.J.,2006. Congestion in different topologies of traffic networks. Europhysics Letters,74(3):560-566
[154]Zhao, X.M., Gao, Z.Y.,2007. Topological effects on the performance of transportation networks. Chinese Physics Letters,24(1):283-286
[155]Albert, R., Jeong, H., Barabasi, A.-L.,2000. Attack and error tolerance of complex networks. Nature,406(6794):378-382
[156]陆华普,1997.交通规划理论与方法(第2版)[M].清华大学出版社
[157]Sheffi, Y.,1985. Urban transportation networks:equilibrium analysis with mathematical programming methods. Prentice-Hall, Englewood Cliffs, New Jersey, USA
[158]黄海军,1994.城市交通网络平衡分析——理论与实践[M].人民交通出版社
[159]Wardrop, J.G.,1952. Some theoretical aspects of road traffic research. Proceedings of The Institution of Civil Engineers,1(2):325-378
[160]Beckmann, M.J., McGuire, C.B., Winsten, C.B.,1956. Studies in the Economics of Transportation. Yale University Press, New Haven, Connecticut
[161]Kim, T.J.,1990. Advanced Transport and Spatial Systems Models. Springer-Verlag, New York, USA
[162]Frank, M., Wolfe, P.,1956. An algorithm for quadratic programming. Naval Research Logistics Quarterly,3:95-110
[163]Leblanc, L.J., Morlok, E.K., Pierskalla, W.P.,1975. An efficient approach to solving the road network equilibrium traffic assignment problem. Transportation Research,9(5):309-318
[164]Daganzo, C.F.,1994. The cell transmission model:a simple dynamic representation of highway traffic. Transportation Research Part B,28(4):269-287
[165]Daganzo, C.F.,1995. The cell transmission model, Part Ⅱ:network traffic. Transportation Research Part B,29(2):79-93
[166]Lighthill, M.H., Whitham, G.B.,1955. On kinematics wave:Ⅱ a theory of traffic flow on long crowed roads. Proceedings of the Royal Society, London, Series A,22:317-345
[167]Richards, P.I.,1956. Shock waves on the highway. Operations Research,4:42-51
[168]Barabasi, A.L., Albert, R., Jeong, H.,1999. Mean-field theory for scale-free random networks. Physica A,272(1-2):173-187
[169]贾斌,高自友,李克平,李新刚,2007.基于元胞自动机的交通系统建模与模拟[M].科学 出版社
[170]Nagel, K., Schreckenberg, M.,1992. A cellular automaton model for freeway traffic. Journal de Physique I,2 (12):2221-2228
[171]de Menezes, M.A., Barabasi, A.L.,2004. Fluctuations in network dynamics. Physical Review Letters,92(2):028701
[172]de Menezes, M.A., Barabasi, A.L.,2004. Seperating internal and external dynamics of complex systems. Physical Review Letters,93(6):068701
[173]Meloni, S., Gomez-Gardenes, J., Latora, V., Moreno, Y.,2008. Scaling breakdown in flow fluctuation on complex networks. Physical Review Letters,100(20):028701
[174]Chakrabarti, B.K.,2006. A fiber bundle model of traffic jams. Physica A,372(1):162-166
[175]Cohen, R., Erez, K., ben-Avraham D., Havlin, S.,2000. Resilience of the Internet to random breakdowns. Physical Review Letters,85(21):4626-4628
[176]Newman, M.E.J.,2003. Properties of highly clustered networks. Physical Review E,68(2): 026121
[177]Latora, V., Marchiori, M.,2001. Efficient behavior of small-world networks. Physical Review Letters,87(19):198701
[2]Newman, M.E.J.,2003. The structure and function of complex networks. SIAM Review,45(2): 167-256
[3]Boccalettia, S., Latorab, V., Moreno, Y., Chavezf, M., Hwanga, D.U.,2006. Complex networks: structure and dynamics. Physics Reports,424(4-5):175-308
[4]Strogatz, S.H.,2001. Exploring complex networks. Nature,410(6825):268-276
[5]汪小帆,李翔,陈关荣,2006.复杂网络理论及其应用[M].清华大学出版社
[6]Camacho, J., Guimera, R., Amaral, L.A.N.,2002. Robust patterns in food web structure. Physical Review Letters,88(22):228102
[7]Newman, M.E.J.,2001. The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences, USA,98(2):404-409
[8]Pastor-Satorras, R., Vazquez, A., Vespignani, A.,2001. Dynamical and correlation properties of the Internet. Physical Review Letters,87(25):258701
[9]Albert, R., Jeong, H., Barabasi, A.L.,1999. Diameter of the world wide web. Nature,401(6749): 130-131
[10]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(22):7794-7799
[11]Jeong, H., Mason, S.P., Barabasi, A.L., Oltvai, Z.N.,2001. Lethality and centrality in protein networks. Nature,411(6833):41-42
[12]Watts, D.J., Strogatz, S.H.,1998. Collective dynamics of'small-world'networks. Nature, 393(6684):440-442
[13]Barabasi, A.L., Albert, R.,1999. Emergence of scaling in random networks. Science,286(5439): 509-512
[14]Erdos, P., Renyi, A.,1960. On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Science,5:17-60
[15]Jeong, H., Tombor, B., Albert, R., Oltvai, Z.N., Barabasi, A.L.,2000. The large-scale organization of metabolic networks. Nature,407(6804):651-654
[16]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(21):11149-11152
[17]Latora, V, Marchiori, M.,2002. Is the Boston subway a small-world network?. Physica A, 314(1-4):109-113
[18]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(3):036106
[19]Wu, J.J., Gao, Z.Y., Sun, H.J., Huang, H.J.,2004. Urban transit system as a scale-free network. Modern Physics Letters B,18(19-20):1043-1049
[20]Newman, M.E.J., Watts, D.J.,1999. Renormalization group analysis of the small-world network model. Physics Letters A,263(4-6):341-346
[21]Krapivsky, P.L., Redner, S., Leyvraz, F.,2000. Connectivity of growing random networks. Physical Review Letters,85(21):4629-4632
[22]Onody, R.N., de Castro P.A.,2004. Nonlinear Barabsi-Albert network. Physica A,336(3-4): 491-502
[23]Dorogovtsev, S.N., Mendes, J.F.F., Samukhin, A.N.,2000. Structure of growing networks with preferential linking. Physical Review Letters,85(21):4633-4636
[24]Dorogovstev, S.N., Mendes, J.F.F.,2000. Evolution of networks with aging of sites. Physical Review E,62(2):1842-1845
[25]Albert, R., Barabasi, A.L.,2000. Topology of evolving networks:local events and universality. Physical Review Letters,85(24):5234-5237
[26]Bianconi, G., Barabasi, A.L,2001. Competition and multi-scaling in evolving networks. Europhysics Letters,54(4):436-442
[27]Bianconi, G., Barabasi, A.L,2001. Bose-Einstein condensation in complex networks. Physical Review Letters,86(24):5632-5635
[28]Holme, P., Kim, B.J.,2002. Growing scale-free networks with tunable clustering coefficient. Physical Review E,65(2):026107
[29]Li, X., Chen, G.,2003. A local world evolving network model. Physica A,328(1-2):274-286
[30]Gao, Z.Y., Li, K.P.,2005. Evolution of traffic flow with scale-free topology. Chinese Physics Letters,22(10):2711-2714
[31]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
[32]Li, X.G, Gao, Z.Y., Li, K.P., Zhao, X.M.,2007. Relationship between microscopic dynamics in traffic flow and complexity in networks. Physical Review E,76(1):016110
[33]Wu, J.J., Gao, Z.Y., Sun, H.J., Huang, H.J.,2005. Random and preferential attachment networks with aging. Chinese Physics Letters,22(3):765-768
[34]Wu, J.J., Gao, Z.Y., Sun, H.J.,2006. Unified model for generation complex networks with utility preferential attachment. Communications in Theoretical Physics,46(1):183-186
[35]Zhou, T., Yan, G., Wang, B.H.,2005. Maximal planar networks with large clustering coefficient and power-law degree distribution. Physical Review E,71(4):046141
[36]Guo, Q., Zhou, T., Liu, J.G, Bai, W.J., Wang, B.H., Zhao, M.,2006. Growing scale-free small-world networks with tunable assortative coefficient. Physica A,371(2):814-822
[37]张培培,何阅,周涛,苏蓓蓓,常慧,周月平,汪秉宏,何大韧,2006.一个描述合作网络顶点度分布的模型.物理学报,55(1)：60-67
[38]Bu, S.L., Wang, B.H., Zhou, T.,2007. Gaining scale-free and high clustering complex networks. Physica A,374(2):864-868
[39]Xie, Y.B., Zhou, T., Wang, B.H.,2008. Scale-free networks without growth. Physica A,387(7): 1683-1688
[40]Barrat, A., Barthelemy, M., Pastor-Satorras, R., Vespignani, A.,2004. The architecture of complex weighted networks. Proceedings of the National Academy of Sciences, USA,101(11): 3747-3752
[41]Yook, S.H., Jeong, H., Barabasi, A.L., Tu, Y.,2001. Weighted evolving networks. Physical Review Letters,86(25):5835-5838
[42]Zheng, D., Trimper, S., Zheng, B., Hui, P.M.,2003. Weighted scale-free networks with stochastic weight assignments. Physical Review E,67(4):040102
[43]Barrat, A., Barthelemy, M., Vespignani, A.,2004. Weighted evolving networks:coupling topology and weight dynamics. Physical Review Letters,92(22):228701
[44]Wang, W.X., Wang, B.H., Hu, B., Yan, G., Ou, Q.,2005. General dynamics of topology and traffic on weighted technological networks. Physical Review Letters,94(18):188702
[45]Wang, W.X., Hu, B., Zhou, T., Wang, B.H., Xie, Y.B.,2005. Mutual selection model for weighted networks. Physical Review E,72(4):046140
[46]Wang, W.X., Hu, B., Wang, B.H., Yan, G.,2006. Mutual attraction model for both assortative and disassortative weighted networks. Physical Review E,73(1):016133
[47]Zhao, H., Gao, Z.Y., Wang, W.X., Yan, G.,2006. Self organization of topology and weight dynamics on networks from merging and regeneration. Chinese Physics Letters,23(2):275-278
[48]Wu, J.J., Gao, Z.Y., Sun, H.J.,2007. Strength dynamics of weighted evolving networks. Chinese Physics,16(1):47-50
[49]Li, M.H., Fan, Y, Chen, J.W., Gao, L., Di, Z.R., Wu, J.S.,2005. Weighted networks of scientific communication:the measurement and topological role of weight. Physica A,350(2-4):643-656
[50]Li, M.H., Fan, Y, Wang, D.H., Li, D.Q., Wu, J.S., Di, Z.R.,2007. Small-world effect induced by weight randomization on regular networks. Physics Letters A,364(6):488-493
[51]Li, M.H., Wu, J.S., Wang, D.H., Zhou, T., Di, Z.R., Fan, Y.,2007. Evolving model of weighted networks inspired by scientific collaboration networks. Physica A,375(1):355-364
[52]Fan, Y, Li, M.H., Zhang, P., Wu, J.S., Di, Z.R.,2007. The effect of weight on community structure of networks. Physica A,378(2):583-590
[53]Sole, R.V., Valverde, S.,2001. Information transfer and phase transitions in a model of internet traffic. Physica A,289(3-4):595-605
[54]Arenas, A., Diaz-Guilera, A., Guimera, R.,2001. Communication in networks with hierarchical branching. Physical Review Letters,86(14):3196-3199
[55]Guimera, R., Diaz-Guilera, A., Vega-Redondo, F., Cabrales, A., Arenas, A.,2002. Optimal network topologies for local search with congestion. Physical Review Letters,89(24):248701
[56]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(3): 036102
[57]Tadic, B., Thurner, S.,2004. Information super-diffusion in structured networks. Physica A,332: 566-584
[58]Echenique, P., Gomez-Gardenes, J., Moreno, Y.,2004. Improved routing strategies for Internet traffic delivery. Physical Review E,70(5):056105
[59]Mukherjee, G, Manna, S.S.,2005. Phase transition in a directed traffic flow network. Physical Review E,71(6):066108
[60]Zhao, L., Lai, Y.C., Park, K., Ye, N.,2005. Onset of traffic congestion in complex networks. Physical Review E,71(2):026125
[61]Yan, G., Zhou, T., Hu, B., Fu, Z.Q., Wang, B.H.,2006. Efficient routing on complex networks. Physical Review E,73(4):046108
[62]Wang, W.X., Wang, B.H., Yin, C.Y, Xie, Y.B., Zhou, T.,2006. Traffic dynamics based on local routing protocol on a scale-free network. Physical Review E,73(2):026111
[63]Wang, W.X., Yin, C.Y., Yan, G., Wang, B.H.,2006. Integrating local static and dynamic information for routing traffic. Physical Review E,74(1):016101
[64]Yin, C.Y., Wang, B.H., Wang, W.X., Yan, G., Yang, H.J.,2006. Traffic dynamics based on an efficient routing strategy on scale free networks. European Physics Journal B,49(2):205-211
[65]Yin, C.Y., Wang, B.H., Wang, W.X., Zhou, T., Yang, H.J.,2006. Efficient routing on scale-free networks based on local information. Physics Letters A,351(4-5):220-224
[66]Yang, H.X., Wang, W.X., Wu, Z.X., Wang, B.H.,2008. Traffic dynamics in scale-free networks with limited packet-delivering capacity. Physica A,387(27):6857-6862
[67]Hu, M.B., Wang, W.X., Jiang, R., Wu, Q.S., Wu, Y.H.,2007. Phase transition and hysteresis in scale-free network traffic. Physical Review E,75(3):036102
[68]Liu, Z., Hu, M.B., Jiang, R., Wang, W.X., Wu, Q.S.,2007. Method to enhance traffic capacity for scale-free networks. Physical Review E,76(3):037101
[69]Watts, D.J.,2002. A simple model of global cascades on random networks. Proceedings of the National Academy of Sciences, USA,99(9):5766-5771
[70]Goh, K.I., Lee, D.S., Kahng, B., Kim, D.,2003. Sandpile on scale-free networks. Physical Review Letters,91(14):148701
[71]Motter, A.E., Lai, Y.C.,2002. Cascade-based attacks on complex networks. Physical Review E, 66(6):065102(R)
[72]Motter, A.E.,2004. Cascade control in complex networks. Physical Review Letters,93(9): 098701
[73]Holme, P., Kim, B.J.,2002. Vertex overload breakdown in evolving networks. Physical Review E,65(6):066109
[74]Holme, P.,2002 Edge overload breakdown in evolving networks. Physical Review E,66(3): 036119.
[75]Moreno, Y, Gomez, J.B., Pacheco, A.F.,2002. Instability of scale-free networks under node-breaking avalanches Europhysics Letters,58(4):630-636
[76]Moreno, Y, Pastor-Satorras, R., Vazquez, A., Vespignani, A.,2003. Critical load and congestion instabilities in scale-free networks. Europhysics Letters,62(2):292-298
[77]Zhao, L., Park, K., Lai, Y.C.,2004. Attack vulnerability of scale-free networks due to cascading breakdown. Physical Review E,70(3):035101(R)
[78]Zhao, L., Park, K., Lai, Y.C., Ye, N.,2005. Tolerance of scale-free networks against attack-induced cascades. Physical Review E,72(2):025104(R)
[79]Crucitti, P., Latora, V, Marchiori, M.,2004. Model for cascading failures in complex networks. Physical Review E,69(4):045104(R)
[80]Wang, X.F., Xu, J.,2004. Cascading failures in coupled map lattices. Physical Review E,70(5): 056113
[81]Xu, J., Wang, X.F.,2005. Cascading failures in scale-free coupled map lattices. Physica A, 349(3-4):685-692
[82]Lee, E.J., Goh, K.I., Kahng, B., Kim, D.,2005. Robustness of the avalanche dynamics in data-packet transport on scale-free networks. Physical Review E,71(5):056108
[83]Gallos, L.K., Cohen, R., Argyrakis, P., Bunde, A., Havlin, P S.,2005. Stability and topology of scale-free networks under attack and defense strategies. Physical Review Letters,94(18): 188701
[84]Wu, J.J., Gao, Z.Y., Sun, H.J.,2006. Cascading breakdown in scale-free networks with community structure. Physical Review E,74(6):066111
[85]Wu, J.J., Sun, H.J., Gao, Z.Y.,2007. Cascading failures on weighted urban traffic equilibrium networks. Physica A,386(1):407-413
[86]Wu, J.J., Gao, Z.Y., Sun, H.J.,2007. Effects of the cascading failures on scale-free traffic networks. Physica A,387(2):505-511
[87]Zheng, J.F., Gao, Z.Y., Zhao, X.M.,2007. Clustering and congestion effects on cascading failures of scale-free networks. Europhysics Letters,79(5):58002
[88]Zhao, H., Gao, Z.Y.,2007. Cascade defense via navigation in scale-free networks. European Physics Journal B,57(1):95-101
[89]Zhao, X.M., Gao, Z.Y.,2007. How non-uniform tolerance parameter strategy changes the response of scale-free networks to failures. European Physical Journal B,59(1):85-92
[90]Sun, H.J., Zhao, H., Wu, J.J.,2008. A robust matching model of capacity to defense cascading failure on complex networks. Physica A,387(25):6431-6435
[91]Cui, D., Gao, Z.Y., Zheng, J.F.,2009. Tolerance of edge cascades with coupled map lattices methods. Chinese Physics B,18(3):992-996
[92]Wang, B., Kim, B.J.,2007. A high robustness and low cost model for cascading failures. Europhysics Letters,78(4):48001
[93]Wu, J., Deng, H.Z., Tan, Y.J., Zhu, D.Z,2007. Vulnerability of complex networks under intentional attack with incomplete information. Journal of Physics A,40(11):2665-2671
[94]Simonsen, I., Buzna, L., Peters, K., Bornholdt, S., Helbing, D.,2008. Transient dynamics increasing network vulnerability to cascading failures. Physical Review Letters,100(21): 218701
[95]Li, P., Wang, B.H., Sun, H., Gao, P., Zhou, T.,2008. A limited resource model of fault-tolerant capability against cascading failure of complex network. European Physical Journal B,62(1): 101-104
[96]Wang, W.X., Chen, G.R.,2008. Universal robustness characteristic of weighted networks against cascading failure. Physical Review E,77(2):026101
[97]Wang, J.W., Rong, L.L.,2009. A model for cascading failures in scale-free networks with a breakdown probability. Physica A,388(7):1289-1298
[98]Bao, Z.J., Cao, Y.J., Ding, L.J., Wang, G.Z.,2009. Comparison of cascading failures in small-world and scale-free networks subject to vertex and edge attacks. Physica A,388(20): 4491-4498
[99]Wang, J., Liu, Y.H., Jiao, Y, Hu, H.Y.,2009. Cascading dynamics in congested complex networks. European Physical Journal B,67(1):95-100
[100]Pastor-Satorras, R., Vespingnani, A.,2001. Epidemic spreading in scale-free networks. Physical Review Letters,86(4):3200-3203
[101]Pastor-Satorras, R., Vespingnani, A.,2001. Epidemic dynamics and endemic states in complex networks. Physical Review E,63(6):066117
[102]Pastor-Satorras, R., Vespingnani, A.,2002. Epidemic dynamics in finite size scale-free networks. Physical Review E,65(3):035108
[103]Pastor-Satorras, R., Vespingnani, A.,2002. Immunization of complex networks. Physical Review E,65(3):036104
[104]Boguna, M., Pastor-Satorras, R., Vespingnani, A.,2003. Absence of epidemic threshold in scale-free networks with degree correlations. Physical Review Letters,90(2):028701
[105]Cohen, R., Havlin, S., ben-Avraham, D.,2003. Efficient immunization strategies for computer networks and populations. Physical Review Letters,91(24):247901
[106]Barthelemy, M., Barrat, A., Pastor-Satorras, R., Vespingnani, A.,2004. Velocity and hierarchical spread of epidemic outbreaks in scale-free networks. Physical Review Letters, 92(17):178701
[107]Dodds, P.S., Watts, D.J.,2004. Universal behavior in a generalized model of contagion. Physical Review Letters,92(21):218701
[108]Zanette, D.H.,2002. Dynamics of rumor propagation on small world networks. Physical Review E,65(4):041908
[109]Moreno, Y., Nekovee, M., Pacheco, A.F.,2004. Dynamics of rumor spreading in complex networks. Physical Review E,69(6):066130
[110]Balthrop, J., Forrest, S., Newman, M.E.J., Williamson, M.M.,2004. Technological networks and the spread of computer viruses. Science,304(5670):527-529
[111]Zhou, T., Liu, J.G., Bai, W.J., Chen, G.R., Wang, B.H.,2006. Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity. Physical Review E,74(5):056109
[112]Zhou, T., Fu, Z.Q., Wang, B.H.,2006. Epidemic dynamics on complex networks. Progress of Natural Science,16(5):452-457
[113]Yang, R., Wang, B.H., Ren, J., Bai, W.J., Shi, Z.W., Wang, W.X.,2007. Epidemic spreading on heterogeneous networks with identical infectivity. Physics Letters A,364(3-4):189-193
[114]Bai, W.J., Zhou, T., Wang, B.H.,2007. Immunization of susceptible-infected model on scale-free networks. Physica A,384(2):656-662
[115]Kuramoto, Y.,1984. Chemical oscillations, waves and turbulence, Springer-Verlag
[116]Kaneko, K.,1992. Coupled map lattices, Singapore:World Scientific
[117]Neda, Z., Ravasz, E., Brechet, Y., Vicsek, T., Barabasi, A.L.,2000. The sound of many hands clapping. Nature,403(6772):849-850
[118]Wang, X.F., Chen, G.R.,2002. Synchronization in small-world dynamical networks. International Journal of Bifurcation and Chaos,12(1):187-192
[119]Wang, X.F., Chen, G.R.,2002. Synchronization in scale-free dynamical networks:robustness and fragility. IEEE Transactions on Circuits and Systems-I,49(1):54-62
[120]Li, X., Chen, G.R.,2003. Synchronization and desynchronization of complex dynamical networks:an engineering viewpoint. IEEE Transactions on Circuits and Systems-I,50(11): 1381-1390
[121]Lv, J.H., Yu, X., Chen, G.R., Cheng, D.,2004. Characterizing the synchronizability of small-world dynamical networks. IEEE Transactions on Circuits and Systems-I,51(4): 787-796
[122]Lv, J.H., Yu, X., Chen, G.R.,2004. Chaos synchronization of general complex dynamical networks. Physica A,334(1-2):281-302
[123]Li, C.G., Chen, G.R.,2004. Synchronization in general complex dynamical networks with coupling delays. Physica A,343:236-278
[124]Fan, J., Wang, X.F.,2005. On synchronization in scale-free dynamical networks. Physica A, 349(3-4):443-451
[125]Jalan, S., Amritkar, R.E.,2003. Self-organized and driven phase synchronization in coupled maps. Physical Review Letters,90(1):014101
[126]Nishikawa, T., Motter, A.E., Lai, Y.C., Hoppensteadt, F.C.,2003. Heterogeneity in oscillator networks:are smaller worlds easier to synchronize?. Physical Review Letters,91(1):014101
[127]Hong, H., Choi, M.Y., Kim, B.J.,2002. Synchronization on small-world networks. Physical Review E,65(2):026139
[128]Hong, H., Kim, B.J., Choi, M.Y., Park, H.,2004. Factors that predict better synchronizability on complex networks. Physical Review E,69(6):067105
[129]Wei, G.W., Zhan, M., Lai, C.H.,2002. Tailoring wavelets for chaos control. Physical Review Letters,89(28):284103
[130]Dhamala, M., Jirsa, V.K., Ding, M.,2004. Enhancement of neural synchrony by time delay. Physical Review Letters,92(7):074104
[131]Atay, F.M., Jost, J.,2004. Delays, connection topology, and synchronization of coupled chaotic maps. Physical Review Letters,92(14):144101
[132]Chavez, M., Hwang, D.U., Amann, A., Hentschel, H.G.E., Boccaletti, S.,2005. Synchronization is enhanced in weighted complex networks. Physical Review Letters,94(21): 218701
[133]Zhao, M., Zhou, T., Wang, B.H., Wang, W.X.,2005. Enhanced synchronizability by structural perturbations. Physical Review E,72(5):057102
[134]Zhou, T., Zhao, M., Wang, B.H.,2006. Better synchronizability predicted by crossed double cycle. Physical Review E,73(3):037101
[135]Yin, C.Y., Wang, W.X., Chen, G.R., Wang, B.H.,2006. Decoupling process for better synchronizability on scale-free networks. Physical Review E,74(4):047102
[136]Zhao, M., Zhou, T., Wang, B.H., Ou, Q., Ren, J.,2006. Better sychronizability predicted by a new coupling method. European Physical Journal B,53(3):375-379
[137]赵明,汪秉宏,将品群,周涛,2005.复杂网络上动力系统同步研究进展.物理学进展,25(3)：273-295
[138]Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.,2008. Synchronization in complex networks. Physics Reports,469(3):93-153
[139]Kleinberg, J.,2000. Navigation in a small world. Nature,406(6798):845-845
[140]Watts, D.J., Dodds, P.S., Newman, M.E.J.,2002. Identity and search in social networks. Science,296(5571):1302-1305
[141]Adamic, L.A., Lukose, R.M., Puniyani, A.R., Huberman, B.A.,2001. Search in power-law networks. Physical Review E,64(4):046135
[142]Noh, J.D., Rieger, H.,2004. Random walks on complex networks. Physical Review Letters, 92(11):118701
[143]Watts, D.J.,2004. The "new" science of networks. Annual Review of Sociology,30(1): 243-270
[144]Newman, M.E.J.,2002. Assortative mixing in networks. Physical Review Letters,89(20): 208701
[145]Goh K.I., Kahng B., Kim D.,2001. Universal behavior of load distribution in scale-free networks. Physical Review Letters,87(27):278701
[146]Goh K.I., Oh, E., Jeong, H., Kahng B., Kim D.,2002. Classification of scale-free networks. Proceedings of the National Academy of Sciences, USA,99(20):12583-12588
[147]Barthelemy, M.,2004. Betweenness centrality in large complex networks. European Physical Journal B,38(2):163-168
[148]Goh, K.I., Noh, J.D., Kahng, B., Kim, D.,2005. Load distribution in weighted complex networks. Physical Review E,72(1):017102
[149]Holme, P.,2003. Congestion and centrality in traffic flow on complex networks. Advances in Complex System,6(2):163-176
[150]Zheng, J.F., Gao, Z.Y., Zhao, X.M.,2007. Properties of transportation dynamics on scale-free networks. Physica A,373:837-844
[151]Toroczkai, Z., Bassler, K.E.,2004. Network dynamics:jamming is limited in scale-free systems. Nature,428(6984):716-716
[152]Park, K., Lai, Y.C., Zhao, L., Ye, N.,2005. Jamming in complex gradient networks. Physical Review E,71(6):065105(R)
[153]Wu, J.J., Gao, Z.Y., Sun, H.J., Huang, H.J.,2006. Congestion in different topologies of traffic networks. Europhysics Letters,74(3):560-566
[154]Zhao, X.M., Gao, Z.Y.,2007. Topological effects on the performance of transportation networks. Chinese Physics Letters,24(1):283-286
[155]Albert, R., Jeong, H., Barabasi, A.-L.,2000. Attack and error tolerance of complex networks. Nature,406(6794):378-382
[156]陆华普,1997.交通规划理论与方法(第2版)[M].清华大学出版社
[157]Sheffi, Y.,1985. Urban transportation networks:equilibrium analysis with mathematical programming methods. Prentice-Hall, Englewood Cliffs, New Jersey, USA
[158]黄海军,1994.城市交通网络平衡分析——理论与实践[M].人民交通出版社
[159]Wardrop, J.G.,1952. Some theoretical aspects of road traffic research. Proceedings of The Institution of Civil Engineers,1(2):325-378
[160]Beckmann, M.J., McGuire, C.B., Winsten, C.B.,1956. Studies in the Economics of Transportation. Yale University Press, New Haven, Connecticut
[161]Kim, T.J.,1990. Advanced Transport and Spatial Systems Models. Springer-Verlag, New York, USA
[162]Frank, M., Wolfe, P.,1956. An algorithm for quadratic programming. Naval Research Logistics Quarterly,3:95-110
[163]Leblanc, L.J., Morlok, E.K., Pierskalla, W.P.,1975. An efficient approach to solving the road network equilibrium traffic assignment problem. Transportation Research,9(5):309-318
[164]Daganzo, C.F.,1994. The cell transmission model:a simple dynamic representation of highway traffic. Transportation Research Part B,28(4):269-287
[165]Daganzo, C.F.,1995. The cell transmission model, Part Ⅱ:network traffic. Transportation Research Part B,29(2):79-93
[166]Lighthill, M.H., Whitham, G.B.,1955. On kinematics wave:Ⅱ a theory of traffic flow on long crowed roads. Proceedings of the Royal Society, London, Series A,22:317-345
[167]Richards, P.I.,1956. Shock waves on the highway. Operations Research,4:42-51
[168]Barabasi, A.L., Albert, R., Jeong, H.,1999. Mean-field theory for scale-free random networks. Physica A,272(1-2):173-187
[169]贾斌,高自友,李克平,李新刚,2007.基于元胞自动机的交通系统建模与模拟[M].科学 出版社
[170]Nagel, K., Schreckenberg, M.,1992. A cellular automaton model for freeway traffic. Journal de Physique I,2 (12):2221-2228
[171]de Menezes, M.A., Barabasi, A.L.,2004. Fluctuations in network dynamics. Physical Review Letters,92(2):028701
[172]de Menezes, M.A., Barabasi, A.L.,2004. Seperating internal and external dynamics of complex systems. Physical Review Letters,93(6):068701
[173]Meloni, S., Gomez-Gardenes, J., Latora, V., Moreno, Y.,2008. Scaling breakdown in flow fluctuation on complex networks. Physical Review Letters,100(20):028701
[174]Chakrabarti, B.K.,2006. A fiber bundle model of traffic jams. Physica A,372(1):162-166
[175]Cohen, R., Erez, K., ben-Avraham D., Havlin, S.,2000. Resilience of the Internet to random breakdowns. Physical Review Letters,85(21):4626-4628
[176]Newman, M.E.J.,2003. Properties of highly clustered networks. Physical Review E,68(2): 026121
[177]Latora, V., Marchiori, M.,2001. Efficient behavior of small-world networks. Physical Review Letters,87(19):198701