基于复杂网络理论的城市轨道交通网络抗毁性研究
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摘要
目前,城市轨道交通因其具有运量大、安全可靠、速度快、准点舒适等优势成为解决城市交通问题最有效的办法。随着我国城市轨道交通日益网络化及在城市客流运输中所承担客流比例不断加大,近年日常发生的灾害事故和针对城市轨道交通的恐怖袭击都给人们带来巨大的生命和财产损失,同时造成了极坏的社会影响,因此人们对城市轨道交通网络的安全和可靠性提出了更高的要求。
基于复杂网络理论对城市轨道交通网络抗毁性研究,对于定位城市轨道交通网络中关键的枢纽点及轨道区间,对于合理分配有限的防御资源来防备轨道交通网络遭到攻击和破坏,对于制定有效的城市轨道交通应急措施,对于城市轨道交通线网的合理改扩建,提高轨道交通网络可靠性、服务质量和吸引力等都具有极其重要的理论和现实意义。
本文首先分析了城市轨道交通网络结构特点和功能性质,将城市轨道交通网络结构看作三个子网络的集合并认为其主要功能是满足城市客流运输任务,进而提出了城市轨道交通网络抗毁性的定义。然后在复杂网络常用抗毁性测度指标研究的基础上,结合各子网络特点选取三个抗毁性测度指标分别度量三个子网络抗毁性能变化情况,以城市轨道交通网络功能为目标,明确各子网络关系并最终确立了反映整个网络抗毁性测度和性能的指标,接着采用基于不同子网络层面的攻击策略对城市轨道交通网络模拟攻击。在网络结构和功能性质、攻击策略及指标研究基础上,以广州地铁网络为实例建立一个包含物理、逻辑和需求三个子网的有向城市轨道交通分层网络模型,最后利用计算机编程仿真模拟了在不同攻击策略下网络整体和局部各子网络抗毁性能的变化情况并对结果予以分析,同时本文在城市轨道交通网络模型基础上通过计算确定了广州地铁网络站点及区间的重要度,找出了网络中的薄弱环节,最后提出了提高网络抗毁性的几种措施。
The urban transit is the most effective way to solve the problem of urban transportation because of its large capacity,reliablity,comfortablity and so on.With the increasing networking and the proportion of urban transit in urban transportation system, the conventional disaster accident and terrorist attack aimed at urban transit have brought people huge losses of life and property in recent years,so it posed higher demand to the reliability and security of urban transit.
It has theoretical and practical significance to research on invulnerability of urban transit network based on the theory of complex network.It can identify the hub point and key track interval of urban transit network,and can prevent attack and damage to urban transit network through the reasonable distribution of defense resources.The reasearch can also help the decision maker formulate effective emergency measures and has an important impact on expanding,planning and exploring the potential,and on improving the reliability,service,attra ctive of urban transit network.
After analyzing the feature of structrue and function of urban transit network,this paper first defines the task of the network composed of three subnetworks is passenger transportation.Furthermore, the invulnerability of urban transit network is defined.This paper based on common measures of invulnerability of complex network choses respectively three measures of invulnerability for the three subnetworks,then uses attack strategies based on different layered subnetwork to attack the urban transit network with simulation.For the aim of function of the urban transit network, it establishes the measure and performance of invulnerability which reflect the whole invulnerability of the network from the relationship among the three subnetworks.Above all, taking the example of Guangzhou subway,a directed and layered network model of urban transit which contains three subnetworks called respectively physical network,logical network and demand network can be built,and the relationship of three subnetworks are analyzed. The conclusion reflecting the variation of the overall and part of invulnerability can be got and compared by programming.The importance of station and track considered as the weak are calculated and arranged, some improvments are provided for the optimization of invulnerability of network according to the results.
基于复杂网络理论对城市轨道交通网络抗毁性研究,对于定位城市轨道交通网络中关键的枢纽点及轨道区间,对于合理分配有限的防御资源来防备轨道交通网络遭到攻击和破坏,对于制定有效的城市轨道交通应急措施,对于城市轨道交通线网的合理改扩建,提高轨道交通网络可靠性、服务质量和吸引力等都具有极其重要的理论和现实意义。
本文首先分析了城市轨道交通网络结构特点和功能性质,将城市轨道交通网络结构看作三个子网络的集合并认为其主要功能是满足城市客流运输任务,进而提出了城市轨道交通网络抗毁性的定义。然后在复杂网络常用抗毁性测度指标研究的基础上,结合各子网络特点选取三个抗毁性测度指标分别度量三个子网络抗毁性能变化情况,以城市轨道交通网络功能为目标,明确各子网络关系并最终确立了反映整个网络抗毁性测度和性能的指标,接着采用基于不同子网络层面的攻击策略对城市轨道交通网络模拟攻击。在网络结构和功能性质、攻击策略及指标研究基础上,以广州地铁网络为实例建立一个包含物理、逻辑和需求三个子网的有向城市轨道交通分层网络模型,最后利用计算机编程仿真模拟了在不同攻击策略下网络整体和局部各子网络抗毁性能的变化情况并对结果予以分析,同时本文在城市轨道交通网络模型基础上通过计算确定了广州地铁网络站点及区间的重要度,找出了网络中的薄弱环节,最后提出了提高网络抗毁性的几种措施。
The urban transit is the most effective way to solve the problem of urban transportation because of its large capacity,reliablity,comfortablity and so on.With the increasing networking and the proportion of urban transit in urban transportation system, the conventional disaster accident and terrorist attack aimed at urban transit have brought people huge losses of life and property in recent years,so it posed higher demand to the reliability and security of urban transit.
It has theoretical and practical significance to research on invulnerability of urban transit network based on the theory of complex network.It can identify the hub point and key track interval of urban transit network,and can prevent attack and damage to urban transit network through the reasonable distribution of defense resources.The reasearch can also help the decision maker formulate effective emergency measures and has an important impact on expanding,planning and exploring the potential,and on improving the reliability,service,attra ctive of urban transit network.
After analyzing the feature of structrue and function of urban transit network,this paper first defines the task of the network composed of three subnetworks is passenger transportation.Furthermore, the invulnerability of urban transit network is defined.This paper based on common measures of invulnerability of complex network choses respectively three measures of invulnerability for the three subnetworks,then uses attack strategies based on different layered subnetwork to attack the urban transit network with simulation.For the aim of function of the urban transit network, it establishes the measure and performance of invulnerability which reflect the whole invulnerability of the network from the relationship among the three subnetworks.Above all, taking the example of Guangzhou subway,a directed and layered network model of urban transit which contains three subnetworks called respectively physical network,logical network and demand network can be built,and the relationship of three subnetworks are analyzed. The conclusion reflecting the variation of the overall and part of invulnerability can be got and compared by programming.The importance of station and track considered as the weak are calculated and arranged, some improvments are provided for the optimization of invulnerability of network according to the results.
引文
[1]沈培钧.地铁的意义[J].综合运输,2007,10：1
[2]应名洪.城市轨道交通网络化建设与运营[M].中国铁道出版社,2007：12-16
[3]耿民,王小平.恐怖袭击下地铁反(防)恐研究概况[J].城市轨道交通研究2008,10：6-11
[4]Albert R, Jeong H, Barabasi A L. Error and attack tolerance of complex network[J]. Nature,2000,406(6794):378-382.
[5]Cohen R, Erez K, Ben- Avrah am D, etal.Resilience of the Internet to random breakdowns [J]. Physical ReviewLetters,2000,85(21):4626-4628.
[6]Magoni D. Tearing down the internet[J]. IEEE J.Sel.Areas Commun,2003,21(6):949-960.
[7]Dunne J A, Williams R J, Martinez N D. Network structure and biodiversity loss in food webs:robustness increases with connectance
[8]Newman M E J,Forrest S,Balthrop J. Email networks and the spread of computer viruses[J].Phys.Rev.E,2002,66(3):035101.
[9]LAI Yingcheng, M otter A, Nishikaw a T, Park Y, ZH AOLiang. Complex networks: Dynamics and security [J].Pramana-Journal of Physics,2005,64(4):483-502.
[10]Motter A E, LaiY C. Cascade based attacks on complex networks[J]. Physical Review E,2002,66(6):065102(1-4).
[11]Moreno Y, Gmez J B, Pacheco A F. Instability of scale free networks under nodebreak ing avalanches[J]. Europhysics Letters, 2002,58 (4):630-636.
[12]Crucitti P, Latora V, Marchiori M. Model for cascading failures in complex networks[J]. Physical Review E,2004,69(4):045104(1-4).
[13]Goh K I, Lee D S, Kahng B, eta.l Cascading toppling dynamics on scale free networks [J]. Physica A, 2005,346 (1/2):93-103.
[14]Panagiotis Angeloudis, David Fisk.Large Subway Systems As Complex network[J]. Physica A, 2006,367:553-558
[15]Maciej Kurant,PatrickThiran.Layered Complex Networks[J].arXiv:physics/0510194 v1,2005
[16]Maciej Kurant, Patrick Thiran.Error and Attack Tolerance of Layered Complex Networks[J]. arXiv: physics/ 0610018v1,2006
[17]李英,周伟,郭世进.上海公共交通网络复杂性分析.系统工程,2007,25(1)：38-43
[18]叶婷婷.基于复杂网络理论的全国铁路网络连通可靠性研究[D].北京交通大学硕士学位论文,2009.6
[19]刘宏鳃,周涛.中国城市航空网络的实证研究与分析[J].物理报,2007,56(1)：107-114
[20]丁琳,谭敏生,肖炜.复杂网络抗毁性研究综述[J].电脑知识与技术,2009,1：51-53
[21]谭跃进,吕欣,吴俊,邓宏钟.复杂网络抗毁性研究的主要科学问题[C].和谐发展与系统工程——中国系统工程学会第十五届年会论文集,2008：107-114
[22]Xiao Zhong-zhe,Dong Zai-Wang.Improved GIB synchronization method for OFDM system [J].IEEE Telecommunications,2003,2 (8):1417-1421.
[23]Lu Hui, Dong Zai-wang.Carrier frequency offset estimation for DAB receiver based onphase reference symbol [J].IEEE Transactions on Consumer Electronics,2000,46 (1):127-130.
[24]王云琴.基于复杂网络理论的城市轨道交通网络连通可靠性研究[D].北京交通大学硕士学位论文,2008.6
[25]李进.交通网络复杂性及其优化研究[D].天津大学博士学位论文,2009,2：46-53
[26]Wang B,Tang H W,Guo C H, Entropy Optimization of Scale-free Network Robustness to Random Failures[J].Physica A,2005,363:591-596
[27]Duncan J.Watts,Steve Strogatz. Collective dynamics of small-world' network[J]. Nature,1998,393:440-442
[28]Albert-Laszlo Barabasi Reka Albert.Emergence of Scaling in Random Networks[J]. Science,1999,286:509-512
[29]汪小帆,李翔,陈关荣,复杂网络理论及其应用[M].清华大学出版社,2006：2-3
[30]周涛,柏文洁,汪秉宏,刘之景,严钢,复杂网络研究概述[J].物理,2005,1：31-36
[31]曾宪钊,网络科学[M].军事科学出版社,2006：24
[32]Linton C.Freeman, A Set of measures of Centrality Based on Betweenness[J]. Sociometry,1977,30:35-41
[33]Marc Barthelemy, betweenness Centrality in Large Complex Networks[J]. The European Physical Jounral B,2004,38:163-168
[34]M. E. J. Newman.The structure and function of complex networks[M]. SIAM Review.2003,45(2):167-256
[35]Rodrigue J Petal. The Geography of Transport Systems[M].Hofstra University, Department of Economics & Geogra-phy, 2006
[36]汪涛,方志耕,吴卉,吴琳丽.城市地铁网络的复杂性分析[J].军事交通学院学报,20083：24-28
[37]刘志谦,宋瑞.基于复杂网络理论的广州轨道交通网络可靠性研究[J].交通运输系统工程与信息,2010,10：194-200
[38]李进,马军海.城市地铁网络复杂性研究[J].西安电子科技大学学报(社会科学版),2009,3：51-55
[39]Vito Latoraa, Massimo Marchiori.Is the Boston Subway a Small-Worl network[J]. Physica A, 314(2002):109-113
[40]周妍.网络化城市轨道交通行车组织相关问题研究[D].北京交通大学硕士学位论文,2008.5
[41]方蕾,庞志显.城市轨道交通客流与行车组织分析[J].城市轨道交通研究2004,10：42-44
[42]徐新玉.城市轨道交通行车组织交路形式分析[J].铁道运输与经济,2010,9：55-58
[43]严波.城市轨道交通行车间隔时间优化模型研究[J].城市轨道交通研究,2008,6：53--57
[44]郭钰.城市轨道交通列车停站方案优化研究[D].北京交通大学硕士学位论文,2009.
[45]徐瑞华,,江志彬,朱效洁,吴强.城市轨道交通列车运行图计算机编制的关键问题研究[J].城市轨道交通研究,2005,10：44-48
[46]葛喜俊,刘凯,贾传峻,王峰.城市群物流需求网络结构特征分析方法研究[J]物流技术,2007,23：1-4：
[47]陆锡明.地铁客运量差异新析[J].交通与运输,1998,12：7
[48]Irina Petreska, Igor Tomovskia, Eugenio Gutierrezc, Ljupco Kocareva, Flavio Bonoc, Karmen Poljansek.Application of modal analysis in assessing attack vulnerability of complex networks[J]. Commun Nonlinear Sci Numer Simulat.2010
[49]张琨,谈革新,庄克琛,赵荣生.复杂网络抗毁性测度研究综述[J].计算机时代,2010,5：4-7
[50]刘啸林,王能.通信网络抗毁性量度研究[J].上海师范大学学报(自然科学版),2006,10：38-41
[51]Petter Holme, Beom Jun Kim.Attack vulnerability of complex networks[J]. Phys.Rev. E, 2002, 65(5):056109.
[52]George Leu,Akira Namatame. Analysis of attacking strategies in Weighted Complex Networks[J].Advances in Studies on Risk Analysis and Crisis Response, 2004:363-368
[53]刘天顺,朱效洁,徐瑞华.城市轨道交通系统运营安全和可靠性分析[J].城市轨道交通,2006,1：15-17
[54]刘晓燕.上海轨道交通应急机制研究[D].上海交通大学硕士学位论文,2008.6
[55]吴俊,谭跃进,邓宏钟,李勇,刘斌.基于不等概率抽样的不完全信息条件下复杂网络抗毁性模型[J].系统工程理论与实践,2010,7：1207-1227
[56]吴俊,谭跃进.复杂网络抗毁性测度研究[J].系统工程学报,2005,4：128-131
[57]金雷.基于复杂网络的地域公路交通网抗毁性分析[D].国防科学技术大学研究生院工程硕士学位论文,2008.11
[58]Anna Nagurney,Qiang Qiang.A Transportation Network Efficiency Measure that Captures Flows, Behavior, and Costs with Applications to Network Component Importance Identification and Vulnerability[J]. SOCIAL SCIENCE RESEARCH NETWORK,2007: 22-27
[59]王志强,徐瑞华.基于复杂网络的轨道交通路网可靠性仿真分析[J].系统仿真学报,2009,10：6670-6674
[60]陈晔,赵金超,齐欢.分层复杂网络的鲁棒性研究[J].兵工学报,2009.6：839-843
[61]高洁,施其洲.城市轨道网络抗毁可靠性定义及评价指标模型研究[J].铁道学报,2007,6：29-33
[62]Maciej Kurant,Patrick Thiran.Trainspotting Extraction and Analysis of Traffic and Topologies of Transportation[J]. Physica,2005,74-83
[63]黄一华.城市轨道交通客流分配模型与算法的研究[D].北京交通大学硕士学位论文,2010.6
[64]DeCea J,Fernandez E.Transit assignment for congestedpublic transportation system :An equilibrium model[J].Transportation Science,1993,27 (2):1332147.
[65]四兵锋,毛保华,刘智丽.无缝换乘条件下城市轨道交通网络客流分配模型及算法[J].铁道学报,2007,12：12-18
[66]苏娟.城市轨道交通客流分配研究[D].北京交通大学硕士学位论文,2009.6
[67]李金海.城市轨道交通网络客流分配模型与算法研究[D].北京交通大学硕士学位论文,2010.6
[68]刘会灯,朱飞MATLAB编程基础与典型应用[M].人民邮电出版社,2008.7
[69]Vito Latora Massimo Marchiori. Vulnerability and protection of infrastructure networks[J].10.1103/PhysRevE.2005,71:015103
[2]应名洪.城市轨道交通网络化建设与运营[M].中国铁道出版社,2007：12-16
[3]耿民,王小平.恐怖袭击下地铁反(防)恐研究概况[J].城市轨道交通研究2008,10：6-11
[4]Albert R, Jeong H, Barabasi A L. Error and attack tolerance of complex network[J]. Nature,2000,406(6794):378-382.
[5]Cohen R, Erez K, Ben- Avrah am D, etal.Resilience of the Internet to random breakdowns [J]. Physical ReviewLetters,2000,85(21):4626-4628.
[6]Magoni D. Tearing down the internet[J]. IEEE J.Sel.Areas Commun,2003,21(6):949-960.
[7]Dunne J A, Williams R J, Martinez N D. Network structure and biodiversity loss in food webs:robustness increases with connectance
[8]Newman M E J,Forrest S,Balthrop J. Email networks and the spread of computer viruses[J].Phys.Rev.E,2002,66(3):035101.
[9]LAI Yingcheng, M otter A, Nishikaw a T, Park Y, ZH AOLiang. Complex networks: Dynamics and security [J].Pramana-Journal of Physics,2005,64(4):483-502.
[10]Motter A E, LaiY C. Cascade based attacks on complex networks[J]. Physical Review E,2002,66(6):065102(1-4).
[11]Moreno Y, Gmez J B, Pacheco A F. Instability of scale free networks under nodebreak ing avalanches[J]. Europhysics Letters, 2002,58 (4):630-636.
[12]Crucitti P, Latora V, Marchiori M. Model for cascading failures in complex networks[J]. Physical Review E,2004,69(4):045104(1-4).
[13]Goh K I, Lee D S, Kahng B, eta.l Cascading toppling dynamics on scale free networks [J]. Physica A, 2005,346 (1/2):93-103.
[14]Panagiotis Angeloudis, David Fisk.Large Subway Systems As Complex network[J]. Physica A, 2006,367:553-558
[15]Maciej Kurant,PatrickThiran.Layered Complex Networks[J].arXiv:physics/0510194 v1,2005
[16]Maciej Kurant, Patrick Thiran.Error and Attack Tolerance of Layered Complex Networks[J]. arXiv: physics/ 0610018v1,2006
[17]李英,周伟,郭世进.上海公共交通网络复杂性分析.系统工程,2007,25(1)：38-43
[18]叶婷婷.基于复杂网络理论的全国铁路网络连通可靠性研究[D].北京交通大学硕士学位论文,2009.6
[19]刘宏鳃,周涛.中国城市航空网络的实证研究与分析[J].物理报,2007,56(1)：107-114
[20]丁琳,谭敏生,肖炜.复杂网络抗毁性研究综述[J].电脑知识与技术,2009,1：51-53
[21]谭跃进,吕欣,吴俊,邓宏钟.复杂网络抗毁性研究的主要科学问题[C].和谐发展与系统工程——中国系统工程学会第十五届年会论文集,2008：107-114
[22]Xiao Zhong-zhe,Dong Zai-Wang.Improved GIB synchronization method for OFDM system [J].IEEE Telecommunications,2003,2 (8):1417-1421.
[23]Lu Hui, Dong Zai-wang.Carrier frequency offset estimation for DAB receiver based onphase reference symbol [J].IEEE Transactions on Consumer Electronics,2000,46 (1):127-130.
[24]王云琴.基于复杂网络理论的城市轨道交通网络连通可靠性研究[D].北京交通大学硕士学位论文,2008.6
[25]李进.交通网络复杂性及其优化研究[D].天津大学博士学位论文,2009,2：46-53
[26]Wang B,Tang H W,Guo C H, Entropy Optimization of Scale-free Network Robustness to Random Failures[J].Physica A,2005,363:591-596
[27]Duncan J.Watts,Steve Strogatz. Collective dynamics of small-world' network[J]. Nature,1998,393:440-442
[28]Albert-Laszlo Barabasi Reka Albert.Emergence of Scaling in Random Networks[J]. Science,1999,286:509-512
[29]汪小帆,李翔,陈关荣,复杂网络理论及其应用[M].清华大学出版社,2006：2-3
[30]周涛,柏文洁,汪秉宏,刘之景,严钢,复杂网络研究概述[J].物理,2005,1：31-36
[31]曾宪钊,网络科学[M].军事科学出版社,2006：24
[32]Linton C.Freeman, A Set of measures of Centrality Based on Betweenness[J]. Sociometry,1977,30:35-41
[33]Marc Barthelemy, betweenness Centrality in Large Complex Networks[J]. The European Physical Jounral B,2004,38:163-168
[34]M. E. J. Newman.The structure and function of complex networks[M]. SIAM Review.2003,45(2):167-256
[35]Rodrigue J Petal. The Geography of Transport Systems[M].Hofstra University, Department of Economics & Geogra-phy, 2006
[36]汪涛,方志耕,吴卉,吴琳丽.城市地铁网络的复杂性分析[J].军事交通学院学报,20083：24-28
[37]刘志谦,宋瑞.基于复杂网络理论的广州轨道交通网络可靠性研究[J].交通运输系统工程与信息,2010,10：194-200
[38]李进,马军海.城市地铁网络复杂性研究[J].西安电子科技大学学报(社会科学版),2009,3：51-55
[39]Vito Latoraa, Massimo Marchiori.Is the Boston Subway a Small-Worl network[J]. Physica A, 314(2002):109-113
[40]周妍.网络化城市轨道交通行车组织相关问题研究[D].北京交通大学硕士学位论文,2008.5
[41]方蕾,庞志显.城市轨道交通客流与行车组织分析[J].城市轨道交通研究2004,10：42-44
[42]徐新玉.城市轨道交通行车组织交路形式分析[J].铁道运输与经济,2010,9：55-58
[43]严波.城市轨道交通行车间隔时间优化模型研究[J].城市轨道交通研究,2008,6：53--57
[44]郭钰.城市轨道交通列车停站方案优化研究[D].北京交通大学硕士学位论文,2009.
[45]徐瑞华,,江志彬,朱效洁,吴强.城市轨道交通列车运行图计算机编制的关键问题研究[J].城市轨道交通研究,2005,10：44-48
[46]葛喜俊,刘凯,贾传峻,王峰.城市群物流需求网络结构特征分析方法研究[J]物流技术,2007,23：1-4：
[47]陆锡明.地铁客运量差异新析[J].交通与运输,1998,12：7
[48]Irina Petreska, Igor Tomovskia, Eugenio Gutierrezc, Ljupco Kocareva, Flavio Bonoc, Karmen Poljansek.Application of modal analysis in assessing attack vulnerability of complex networks[J]. Commun Nonlinear Sci Numer Simulat.2010
[49]张琨,谈革新,庄克琛,赵荣生.复杂网络抗毁性测度研究综述[J].计算机时代,2010,5：4-7
[50]刘啸林,王能.通信网络抗毁性量度研究[J].上海师范大学学报(自然科学版),2006,10：38-41
[51]Petter Holme, Beom Jun Kim.Attack vulnerability of complex networks[J]. Phys.Rev. E, 2002, 65(5):056109.
[52]George Leu,Akira Namatame. Analysis of attacking strategies in Weighted Complex Networks[J].Advances in Studies on Risk Analysis and Crisis Response, 2004:363-368
[53]刘天顺,朱效洁,徐瑞华.城市轨道交通系统运营安全和可靠性分析[J].城市轨道交通,2006,1：15-17
[54]刘晓燕.上海轨道交通应急机制研究[D].上海交通大学硕士学位论文,2008.6
[55]吴俊,谭跃进,邓宏钟,李勇,刘斌.基于不等概率抽样的不完全信息条件下复杂网络抗毁性模型[J].系统工程理论与实践,2010,7：1207-1227
[56]吴俊,谭跃进.复杂网络抗毁性测度研究[J].系统工程学报,2005,4：128-131
[57]金雷.基于复杂网络的地域公路交通网抗毁性分析[D].国防科学技术大学研究生院工程硕士学位论文,2008.11
[58]Anna Nagurney,Qiang Qiang.A Transportation Network Efficiency Measure that Captures Flows, Behavior, and Costs with Applications to Network Component Importance Identification and Vulnerability[J]. SOCIAL SCIENCE RESEARCH NETWORK,2007: 22-27
[59]王志强,徐瑞华.基于复杂网络的轨道交通路网可靠性仿真分析[J].系统仿真学报,2009,10：6670-6674
[60]陈晔,赵金超,齐欢.分层复杂网络的鲁棒性研究[J].兵工学报,2009.6：839-843
[61]高洁,施其洲.城市轨道网络抗毁可靠性定义及评价指标模型研究[J].铁道学报,2007,6：29-33
[62]Maciej Kurant,Patrick Thiran.Trainspotting Extraction and Analysis of Traffic and Topologies of Transportation[J]. Physica,2005,74-83
[63]黄一华.城市轨道交通客流分配模型与算法的研究[D].北京交通大学硕士学位论文,2010.6
[64]DeCea J,Fernandez E.Transit assignment for congestedpublic transportation system :An equilibrium model[J].Transportation Science,1993,27 (2):1332147.
[65]四兵锋,毛保华,刘智丽.无缝换乘条件下城市轨道交通网络客流分配模型及算法[J].铁道学报,2007,12：12-18
[66]苏娟.城市轨道交通客流分配研究[D].北京交通大学硕士学位论文,2009.6
[67]李金海.城市轨道交通网络客流分配模型与算法研究[D].北京交通大学硕士学位论文,2010.6
[68]刘会灯,朱飞MATLAB编程基础与典型应用[M].人民邮电出版社,2008.7
[69]Vito Latora Massimo Marchiori. Vulnerability and protection of infrastructure networks[J].10.1103/PhysRevE.2005,71:015103