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Research on cascading failure modes and attack strategies of multimodal transport network
1.  School of Transportation and Logistics, Southwest Jiaotong University, Chengdu 610031, China 
2.  National United Engineering Laboratory of Integrated and Intelligent Transportation, Southwest Jiaotong University, Chengdu 610031, China 
3.  School of Logistics, Chengdu University of Information Technology, Chengdu 610225, China 
Cascading failure overall exists in practical network, which poses a risk of causing significant losses. Studying the effect of different cascading failure modes and attack strategies of the network is conducive to more effectively controlling the network. In the present study, the uniqueness of multimodal transport network is investigated by complying with the percolation theory, and a cascading failure model is built for the multimodal transport network by considering recovery mechanisms and dynamics. Under the three failure modes, i.e., node failure, edge failure and nodeedge failure, nine attack strategies are formulated, consisting of random node attacking strategy (RNAS), highdegree attacking strategy (HDAS), highcloseness attacking strategy (HCAS), random edge attacking strategy (REAS), highimportance attacking strategy (HIAS1), highimportance attacking strategy (HIAS2), random nodeedge attacking strategy (RNEAS), high degreeimportance1 attacking strategy (HDI1AS), as well as high closenessimportance2 attacking strategy (HCI2AS). The effect of network cascading failure is measured at the scale of the affected network that varies with the failure ratio and the network connectivity varying with the step. By conducting a simulation analysis, the results of the two indicators are compared; it is suggested that under the three failure modes, the attack strategies exhibiting high node closeness as the indicator always poses more effective damage to the network. Next, a sensitivity analysis is conducted, and it is concluded that HCAS is the most effective attack strategy. Accordingly, the subsequent study on the cascading failure of multimodal transport network should start with the nodes exhibiting high closeness to optimize the network.
References:
[1] 
Z. J. Bao, Y. J. Cao and L. J. Ding, Comparison of cascading failures in smallworld and scalefree networks subject to vertex and edge attacks, Physica A: Statistical Mechanics and its Applications, 388 (2009), 44914498. 
[2] 
S. R. Broadbent and J. M. Hammersley, Percolation processes, Mathematical Proceedings of the Cambridge Philosophical Society, 53 (1957), 629641. doi: 10.1017/S0305004100032680. 
[3] 
S. V. Buldyrev, R. Parshani and G. Paul, Catastrophic cascade of failures in interdependent networks, Nature, 464 (2009), 10251028. 
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X. Cao, C. Hong and W. Du, Improving the network robustness against cascading failures by adding links, Chaos, Solitions & Fractals, 57 (2013), 3540. 
[5] 
S. Chen, S. Pang and X. Zou, An LCOR model for suppressing cascading failure in weighted complex networks, Chinese Physics B, 22 (2013), 58901. 
[6] 
R. Cohen, K. Erez and D. BenAvraham, Breakdown of the Internet under Intentional Attack, Physical Review Letters, 86 (2001), 36823685. 
[7] 
S. Dong, H. Wang, A. Mostafizi, et al., A networkofnetworks percolation analysis of cascading failures in spatially colocated roadsewer infrastructure networks, Physica A: Statistical Mechanics and its Applications, 538 (2020), 122971. 
[8] 
L. C. Freeman, Centrality in social networks' conceptual clarification, Social Networks, 1 (1979), 215239. 
[9] 
Z. He, J. Guo and J Xu, Cascade failure model in multimodal transport network risk propagation, Mathematical Problems in Engineering, (2019), 1–7. 
[10] 
P. Holme, B. J. Kim and C. N. Yoon, Attack vulnerability of complex networks, Physical Review E Statistical Nonlinear & Soft Matter Physics, 65 (2002), 56109. 
[11] 
Z. Jiang, J. Ma and Y. Shen, Effects of linkorientation methods on robustness against cascading failures in complex networks, Physica A: Statistical Mechanics and its Applications, 457 (2016), 17. 
[12] 
L. Jin, X. Wang and Y. Zhang, Cascading failure in multilayer networks with dynamic dependency groups, Chinese Physics B), (2018). 
[13] 
Z. Kong and E. M. Yeh, Correlated and cascading node failures in random geometric networks: A percolation view: International Conference on Ubiquitous & Future Networks, IEEE, (2012). 
[14] 
M. Li, R. Liu and C. Jia, Cascading failures on networks with asymmetric dependence, EPL (Europhysics Letters), 108 (2014), 56002. 
[15] 
R. R. Liu, C. X. Jia and Y. C. Lai, Asymmetry in interdependence makes a multilayer system more robust against cascading failure, Phys Rev E, 100 (2019), 52306. 
[16] 
A. E. Motter and Y. C. Lai, Cascadebased attacks on complex networks, Phys Rev E Stat Nonlin Soft Matter Phys, 66 (2002), 65102. 
[17] 
A. E. Motter and Y. Lai, Cascadebased attacks on complex network, Physical Review E Statistical Nonlinear & Soft Matter Physics, 66 (2002), 65102. 
[18] 
R. Parshani, S. V. Buldyrev and S. Havlin, Interdependent networks: Reducing the coupling strength leads to a change from a first to second order percolation transition, Phys Rev Lett, 105 (2010), 48701. 
[19] 
Z. Ruan, C. Song and X. Yang, Empirical analysis of urban road traffic network: A case study in Hangzhou city, China, Physica A: Statistical Mechanics and its Applications, 527 (2019), 121287. 
[20] 
Y. Shen, G. Song and H. Xu, Model of node traffic recovery behavior and cascading congestion analysis in networks, Physica A: Statistical Mechanics and its Applications, (2019), 123422. 
[21] 
J. Song, E. CotillaSanchez and G. Ghanavati, Dynamic Modeling of Cascading Failure in Power Systems, IEEE Transactions on Power Systems), 31 (2014), 20852095. 
[22] 
M. Stippinger and J. Kertész, Universality and scaling laws in the cascading failure model with healing, Physical Review E, 98 (2018). 
[23] 
M. Tian, X. Wang and Z. Dong, Cascading failures of interdependent modular scalefree networks with different coupling preferences, EPL (Europhysics Letters), 111 (2015), 18007. 
[24] 
M. Turalska, K. Burghardt and M. Rohden, Cascading failures in scalefree interdependent networks, Phys Rev E, 99 (2019), 32308. 
[25] 
J. Wang and L. Rong, Cascadebased attack vulnerability on the US power grid, Safety Science, 47 (2009), 13321336. 
[26] 
E. Wang, C. Hong and X. Zhang, Cascading failures with coupled map lattices on WattsStrogatz networks, Physica A: Statistical Mechanics and its Applications, 525 (2019), 10381045. 
[27] 
J. Wang and L. Rong, Effect attack on scalefree networks due to cascading failures, Chinese Physics LETTERS, 25 (2008), 3826. 
[28] 
S. Wang, W. Lv and L. Zhao, Structural and functional robustness of networked critical infrastructure systems under different failure scenarios, Physica A: Statistical Mechanics and its Applications, 523 (2019), 476487. 
[29] 
F. Wang, T. Lixin and R. Du, The robustness of interdependent weighted networks, Physica A Statistical Mechanics & Its Applications, 508 (2018), 675680. 
[30] 
D. Witthaut and M. Timme, Nonlocal effects and countermeasures in cascading failures, Phys Rev E Stat Nonlin Soft Matter Phys, 92 (2015), 32809. 
[31] 
J. J. Wu, H. J. Sun and Z. Y. Gao, Cascading failures on weighted urban traffic equilibrium networks, Physica A: Statistical Mechanics and its Applications, 386 (2007), 407413. 
[32] 
J. Wu, J. Zeng and Z. Chen, Effects of traffic generation patterns on the robustness of complex networks, Physica A: Statistical Mechanics and its Applications, 492 (2018), 871877. 
[33] 
X. Wu, R. Gu and Y. Ji, Dynamic behavior analysis of an internet flow interaction model under cascading failures, Phys Rev E, 100 (2019), 22309. 
[34] 
X. Yuan, Y. Hu and H. E. Stanley, Eradicating catastrophic collapse in interdependent networks via reinforced nodes, Proceedings of the National Academy of Sciences, 114 (2017), 33113315. 
[35] 
D. Zhao, L. Wang and S. Li, Immunization of epidemics in multiplex networks, PLoS One, 9 (2014), e112018. 
show all references
References:
[1] 
Z. J. Bao, Y. J. Cao and L. J. Ding, Comparison of cascading failures in smallworld and scalefree networks subject to vertex and edge attacks, Physica A: Statistical Mechanics and its Applications, 388 (2009), 44914498. 
[2] 
S. R. Broadbent and J. M. Hammersley, Percolation processes, Mathematical Proceedings of the Cambridge Philosophical Society, 53 (1957), 629641. doi: 10.1017/S0305004100032680. 
[3] 
S. V. Buldyrev, R. Parshani and G. Paul, Catastrophic cascade of failures in interdependent networks, Nature, 464 (2009), 10251028. 
[4] 
X. Cao, C. Hong and W. Du, Improving the network robustness against cascading failures by adding links, Chaos, Solitions & Fractals, 57 (2013), 3540. 
[5] 
S. Chen, S. Pang and X. Zou, An LCOR model for suppressing cascading failure in weighted complex networks, Chinese Physics B, 22 (2013), 58901. 
[6] 
R. Cohen, K. Erez and D. BenAvraham, Breakdown of the Internet under Intentional Attack, Physical Review Letters, 86 (2001), 36823685. 
[7] 
S. Dong, H. Wang, A. Mostafizi, et al., A networkofnetworks percolation analysis of cascading failures in spatially colocated roadsewer infrastructure networks, Physica A: Statistical Mechanics and its Applications, 538 (2020), 122971. 
[8] 
L. C. Freeman, Centrality in social networks' conceptual clarification, Social Networks, 1 (1979), 215239. 
[9] 
Z. He, J. Guo and J Xu, Cascade failure model in multimodal transport network risk propagation, Mathematical Problems in Engineering, (2019), 1–7. 
[10] 
P. Holme, B. J. Kim and C. N. Yoon, Attack vulnerability of complex networks, Physical Review E Statistical Nonlinear & Soft Matter Physics, 65 (2002), 56109. 
[11] 
Z. Jiang, J. Ma and Y. Shen, Effects of linkorientation methods on robustness against cascading failures in complex networks, Physica A: Statistical Mechanics and its Applications, 457 (2016), 17. 
[12] 
L. Jin, X. Wang and Y. Zhang, Cascading failure in multilayer networks with dynamic dependency groups, Chinese Physics B), (2018). 
[13] 
Z. Kong and E. M. Yeh, Correlated and cascading node failures in random geometric networks: A percolation view: International Conference on Ubiquitous & Future Networks, IEEE, (2012). 
[14] 
M. Li, R. Liu and C. Jia, Cascading failures on networks with asymmetric dependence, EPL (Europhysics Letters), 108 (2014), 56002. 
[15] 
R. R. Liu, C. X. Jia and Y. C. Lai, Asymmetry in interdependence makes a multilayer system more robust against cascading failure, Phys Rev E, 100 (2019), 52306. 
[16] 
A. E. Motter and Y. C. Lai, Cascadebased attacks on complex networks, Phys Rev E Stat Nonlin Soft Matter Phys, 66 (2002), 65102. 
[17] 
A. E. Motter and Y. Lai, Cascadebased attacks on complex network, Physical Review E Statistical Nonlinear & Soft Matter Physics, 66 (2002), 65102. 
[18] 
R. Parshani, S. V. Buldyrev and S. Havlin, Interdependent networks: Reducing the coupling strength leads to a change from a first to second order percolation transition, Phys Rev Lett, 105 (2010), 48701. 
[19] 
Z. Ruan, C. Song and X. Yang, Empirical analysis of urban road traffic network: A case study in Hangzhou city, China, Physica A: Statistical Mechanics and its Applications, 527 (2019), 121287. 
[20] 
Y. Shen, G. Song and H. Xu, Model of node traffic recovery behavior and cascading congestion analysis in networks, Physica A: Statistical Mechanics and its Applications, (2019), 123422. 
[21] 
J. Song, E. CotillaSanchez and G. Ghanavati, Dynamic Modeling of Cascading Failure in Power Systems, IEEE Transactions on Power Systems), 31 (2014), 20852095. 
[22] 
M. Stippinger and J. Kertész, Universality and scaling laws in the cascading failure model with healing, Physical Review E, 98 (2018). 
[23] 
M. Tian, X. Wang and Z. Dong, Cascading failures of interdependent modular scalefree networks with different coupling preferences, EPL (Europhysics Letters), 111 (2015), 18007. 
[24] 
M. Turalska, K. Burghardt and M. Rohden, Cascading failures in scalefree interdependent networks, Phys Rev E, 99 (2019), 32308. 
[25] 
J. Wang and L. Rong, Cascadebased attack vulnerability on the US power grid, Safety Science, 47 (2009), 13321336. 
[26] 
E. Wang, C. Hong and X. Zhang, Cascading failures with coupled map lattices on WattsStrogatz networks, Physica A: Statistical Mechanics and its Applications, 525 (2019), 10381045. 
[27] 
J. Wang and L. Rong, Effect attack on scalefree networks due to cascading failures, Chinese Physics LETTERS, 25 (2008), 3826. 
[28] 
S. Wang, W. Lv and L. Zhao, Structural and functional robustness of networked critical infrastructure systems under different failure scenarios, Physica A: Statistical Mechanics and its Applications, 523 (2019), 476487. 
[29] 
F. Wang, T. Lixin and R. Du, The robustness of interdependent weighted networks, Physica A Statistical Mechanics & Its Applications, 508 (2018), 675680. 
[30] 
D. Witthaut and M. Timme, Nonlocal effects and countermeasures in cascading failures, Phys Rev E Stat Nonlin Soft Matter Phys, 92 (2015), 32809. 
[31] 
J. J. Wu, H. J. Sun and Z. Y. Gao, Cascading failures on weighted urban traffic equilibrium networks, Physica A: Statistical Mechanics and its Applications, 386 (2007), 407413. 
[32] 
J. Wu, J. Zeng and Z. Chen, Effects of traffic generation patterns on the robustness of complex networks, Physica A: Statistical Mechanics and its Applications, 492 (2018), 871877. 
[33] 
X. Wu, R. Gu and Y. Ji, Dynamic behavior analysis of an internet flow interaction model under cascading failures, Phys Rev E, 100 (2019), 22309. 
[34] 
X. Yuan, Y. Hu and H. E. Stanley, Eradicating catastrophic collapse in interdependent networks via reinforced nodes, Proceedings of the National Academy of Sciences, 114 (2017), 33113315. 
[35] 
D. Zhao, L. Wang and S. Li, Immunization of epidemics in multiplex networks, PLoS One, 9 (2014), e112018. 
Failure mode  Attack strategy  
Node failure  RNAS  HDAS  HCAS 
Edge failure  REAS  HIAS1  HIAS2 
Nodeedge failure  RNEAS  HDI1AS  HCI2AS 
Failure mode  Attack strategy  
Node failure  RNAS  HDAS  HCAS 
Edge failure  REAS  HIAS1  HIAS2 
Nodeedge failure  RNEAS  HDI1AS  HCI2AS 
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