# Weighted Complex Network Analysis of the Difference Between Nodal Centralities of the Beijing Subway System

### Abstract

The centrality of stations is one of the most important issues in urban transit systems. The central stations of such networks have often been identified using network to-pological centrality measures. In real networks, passenger flows arise from an interplay between the dynamics of the individual person movements and the underlying physical structure. In this paper, we apply a two-layered model to identify the most central stations in the Beijing Subway System, in which the lower layer is the physical infrastruc-ture and the upper layer represents the passenger flows. We compare various centrality indicators such as degree, strength and betweenness centrality for the two-layered model. To represent the influence of exogenous factors of stations on the subway system, we reference the al-pha centrality. The results show that the central stations in the geographic system in terms of the betweenness are not consistent with the central stations in the network of the flows in terms of the alpha centrality. We clarify this difference by comparing the two centrality measures with the real load, indicating that the alpha centrality approx-imates the real load better than the betweenness, as it can capture the direction and volume of the flows along links and the flows into and out of the systems. The empirical findings can give us some useful insights into the node cen-trality of subway systems.

### References

Newman MEJ. The structure and function of complex networks. SIAM Review. 2003;45(2): 167-256. doi: 10.1137/S003614450342480.

Boccaletti S, et al. Complex networks: Structure and dynamics. Physics Reports. 2006;424(4): 175-308. doi: 10.1016/j.physrep.2005.10.009.

Derrible S, Kennedy C. Applications of graph theory and network science to transit network design. Transport Reviews. 2011;31(4): 495-519. doi: 10.1080/01441647.2010.543709.

Lin J, Ban Y. Complex network topology of transportation systems. Transport Reviews. 2013;33(6): 658-685. doi: 10.1080/01441647.2013.848955.

Xu M, et al. Discovery of critical nodes in road networks through mining from vehicle trajectories. IEEE Transactions on Intelligent Transportation Systems. 2018;20(2): 583-593. doi: 10.1109/TITS.2018.2817282.

Kocur-Bera K. Scale-free network theory in studying the structure of the road network in poland. Promet – Traffic&Transportation. 2014;26(3): 235-242. doi: 10.7307/ptt.v26i3.1316.

Calzada-Infante L, Adenso-Díaz B, Carbajal SG. Analysis of the European international railway network and passenger transfers. Chaos, Solitons & Fractals. 2020;141: 110357. doi: 10.1016/j.chaos.2020.110357.

Wang W, et al. Analysis of the Chinese railway system as a complex network. Chaos, Solitons & Fractals. 2020;130: 109408. doi: 10.1016/j.chaos.2019.109408.

Bombelli A, Santos BF, Tavasszy L. Analysis of the air cargo transport network using a complex network theory perspective. Transportation Research Part E: Logistics and Transportation Review. 2020;138: 101959. doi: 10.1016/j.tre.2020.101959.

Verma T, Araújo NAM, Herrmann HJ. Revealing the structure of the world airline network. Scientific Reports. 2014;4: 5638. doi: 10.1038/srep05638.

Gallotti R, Porter MA, Barthelemy M. Lost in transportation: Information measures and cognitive limits in multilayer navigation. Science Advances. 2016;2(2): e1500445. doi: 10.1126/sciadv.1500445.

Yuan G, Kong DW, Sun LS, Luo W. Connectivity contribution to urban hub network based on super network theory–case study of Beijing. Promet – Traffic&Transportation. 2021;33(1): 35-47. doi: 10.7307/ptt.v33i1.3536.

Wu J, Gao Z, Sun H, Huang H. Urban transit system as a scale-free network. Modern Physics Letters B. 2004;18(19n20): 1043-1049. doi: 10.1142/S021798490400758X.

Louf R, Roth C, Barthelemy M. Scaling in transportation networks. PLoS One. 2014;9(7): e102007. doi: 10.1371/journal.pone.0102007.

Latora V, Marchiori M. Is the Boston subway a small-world network?. Physica A: Statistical Mechanics and its Applications. 2002;314(1-4): 109-113. doi: 10.1016/S0378-4371(02)01089-0.

Derrible S, Kennedy C. The complexity and robustness of metro networks. Physica A: Statistical Mechanics and its Applications. 2010;389(17): 3678-3691. doi: 10.1016/j.physa.2010.04.008.

Latora V, Marchiori M. Efficient behavior of small-world networks. Physical Review Letters. 2001;87(19): 198701. doi: 10.1103/PhysRevLett.87.198701.

Ek B, VerSchneider C, Narayan DA. Efficiency of star-like graphs and the Atlanta subway network. Physica A: Statistical Mechanics and its Applications. 2013;392(21): 5481-5489. doi: 10.1016/j.physa.2013.06.055.

Latora V, Marchiori M. Vulnerability and protection of infrastructure networks. Physical Review E. 2005;71(1): 015103. doi: 10.1103/PhysRevE.71.015103.

Yang Y, et al. Robustness assessment of urban rail transit based on complex network theory: A case study of the Beijing Subway. Safety Science. 2015;79: 149-162. doi: 10.1016/j.ssci.2015.06.006.

Roth C, Kang SM, Batty M, Barthelemy M. A long-time limit for world subway networks. Journal of The Royal Society Interface. 2012;9(75): 2540-2550. doi: 10.1098/rsif.2012.0259.

Leng B, Zhao X, Xiong Z. Evaluating the evolution of subway networks: Evidence from Beijing subway network. Europhysics Letters. 2014;105(5): 58004. doi: 10.1209/0295-5075/105/58004.

Freeman LC. Centrality in social networks conceptual clarification. Social Networks. 1978;1(3): 215-239. doi: 10.1016/0378-8733(78)90021-7.

Wang J, Li C, Xia C. Improved centrality indicators to characterize the nodal spreading capability in complex networks. Applied Mathematics and Computation. 2018;334: 388-400. doi: 10.1016/j.amc.2018.04.028.

Newman MEJ. A measure of betweenness centrality based on random walks. Social Networks. 2005;27(1): 39-54. doi: 10.1016/j.socnet.2004.11.009.

Derrible S. Network centrality of metro systems. PLoS ONE. 2012;7(7): e40575. doi: 10.1371/journal.pone.0040575.

Tang J, Li Z, Gao F, Zong F. Identifying critical metro stations in multiplex network based on DS D–S evidence theory. Physica A: Statistical Mechanics and its Applications. 2021;574: 126018. doi: 10.1016/j.physa.2021.126018.

Kurant M, Thiran P. Layered complex networks. Physical Review Letters. 2006;96(13): 138701. doi: 10.1103/PhysRevLett.96.138701.

Ramli MA, Monterola CP, Khoon GLK, Guang THG. A method to ascertain rapid transit systems’ throughput distribution using network analysis. Procedia Computer Science. 2014;29: 1621-1630. doi: 10.1016/j.procs.2014.05.147.

Crucitti P, Latora V, Porta S. Centrality measures in spatial networks of urban streets. Physical Review E. 2006;73(3): 036125. doi: 10.1103/PhysRevE.73.036125.

Liao C, Dai T, Zhao P, Ding T. Weighted centrality and retail store locations in Beijing, China: A temporal perspective from dynamic public transport flow networks. Applied Sciences. 2021;11(19): 9069. doi: 10.3390/app11199069.

Wang Z, Li J, Huang L, Yang Z. Discovering the evolution of Beijing Rail Network in fifty years. Modern Physics Letters B. 2020;34(21): 2050212. doi: 10.1142/S0217984920502127.

Xiao X, Jia L, Wang Y, Zhang C. Topological characteristics of metro networks based on transfer constraint. Physica A: Statistical Mechanics and its Applications. 2019;532: 121811. doi: 10.1016/j.physa.2019.121811.

Cats O, Jenelius E. Dynamic vulnerability analysis of public transport networks: Mitigation effects of real-time information. Networks and Spatial Economics. 2014;14(3): 435-463. doi: 10.1007/s11067-014-9237-7.

Luo D, Cats O, Lint HV. Can passenger flow distribution be estimated solely based on network properties in public transport systems?. Transportation. 2020;47(6): 2757-2776. doi: 10.1007/s11116-019-09990-w.

Gao S, Wang Y, Gao Y, Liu Y. Understanding urban traffic-flow characteristics: A rethinking of betweenness centrality. Environment and Planning B: Planning and Design. 2013;40(1): 135-153. doi: 10.1068/b38141.

Feng J, et al. Weighted complex network analysis of the Beijing subway system: Train and passenger flows. Physica A: Statistical Mechanics and its Applications. 2017;474: 213-223. doi: 10.1016/j.physa.2017.01.085.

Soh H, et al. Weighted complex network analysis of travel routes on the Singapore public transportation system. Physica A: Statistical Mechanics and its Applications. 2010;389(24): 5852-5863. doi: 10.1016/j.physa.2010.08.015.

Gonzalez MC, Hidalgo CA, Barabasi AL. Understanding individual human mobility patterns. Nature. 2008;453(7196): 779-782. doi: 10.1038/nature06958.

Sun SW, Li HY, Xu XY. A key station identification method for urban rail transit: A case study of Beijing subway. Promet – Traffic&Transportation. 2017;29(3): 267-273. doi: 10.7307/ptt.v29i3.2133.

Lee K, Jung WS, Park JS, Choi MY. Statistical analysis of the Metropolitan Seoul Subway system: Network structure and passenger flows. Physica A: Statistical Mechanics and its Applications. 2008;387(24): 6231-6234. doi: 10.1016/j.physa.2008.06.035.

Sun L, Lee DH, Erath A, Huang X. Using smart card data to extract passenger's spatio-temporal density and train's trajectory of MRT system. Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, 12 August 2012. 2012. p. 142-148.

Hasan S, Schneider CM, Ukkusuri SV, Gonzalez MC. Spatiotemporal patterns of urban human mobility. Journal of Statistical Physics. 2013;151(1-2): 304-318. doi: 10.1007/s10955-012-0645-0.

Roth C, Kang SM, Batty M, Barthelemy M. Structure of urban movements: Polycentric activity and entangled hierarchical flows. PLoS ONE. 2011;6(1): e15923. doi: 10.1371/journal.pone.0015923.

Bonacich P, Lloyd P. Eigenvector-like measures of centrality for asymmetric relations. Social Networks. 2001;23(3): 191-201. doi: 10.1016/S0378-8733(01)00038-7.

Opsahl T, Agneessens F, Skvoretz J. Node centrality in weighted networks: Generalizing degree and shortest paths. Social Networks. 2010;32(3): 245-251. doi: 10.1016/j.socnet.2010.03.006.

Barrat A, Barthelemy M, Pastor-Satorras R, Vespignani A. The architecture of complex weighted networks. Proceedings of the National Academy of Sciences. 2004;101(11): 3747-3752. doi: 10.1073/pnas.0400087101.

Li C, Wang L, Sun S, Xia CY. Identification of influential spreaders based on classified neighbors in real-world complex networks. Applied Mathematics and Computation. 2018;320: 512-523. doi: 10.1016/j.amc.2017.10.001.

Albert R, Albert I, Nakarado GL. Structural vulnerability of the North American power grid. Physical Review E. 2004; 69(2): 025103. doi: 10.1103/PhysRevE.69.025103.

Huang A, et al. Cascading failures in weighted complex networks of transit systems based on coupled map lattices. Mathematical Problems in Engineering. 2015;940795. doi: 10.1155/2015/940795.

Copyright (c) 2022 Ruiyong TONG, Qi XU, Runbin WEI, Junsheng HUANG, Zhongsheng XIAO

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors who publish with this journal agree to the following terms:

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).