# American Institute of Mathematical Sciences

March  2015, 10(1): 127-148. doi: 10.3934/nhm.2015.10.127

## Structural analysis and traffic flow in the transport networks of Madrid

 1 Grupo de Sistemas Complejos. Universidad Politécnica de Madrid, Carretera de Valencia km. 7, 28031 Madrid, Spain 2 Universidad Politécnica de Madrid, Grupo de Sistemas Complejos, E.T.S.I. Agrónomos, Universidad Politécnica de Madrid, 28040 Madrid, Spain

Received  July 2014 Revised  November 2014 Published  February 2015

As the framework to characterize the subway and urban bus networks of Madrid city three topological spaces: geographical stop space, transfer space and route space, are considered. We show that the subway network exhibits better structural parameters than the urban bus network, with higher performance since in average a stop is reachable passing through less number of stops and carrying out less number of transfers between lines. We have found that the cumulative degree distributions of the subway and urban bus networks correspond to an exponential function, while the degree-degree correlations present a power law distributions in both transport systems. The relationship between transport flows and population are also studied at the city level by analyzing the flow between all the district (administrative areas) of Madrid. We prove that these flows can be described by a Gravity Model which takes into account the population from the origin and destination districts as well as the number of sections of a transport line that passes through two different districts.
Citation: Mary Luz Mouronte, Rosa María Benito. Structural analysis and traffic flow in the transport networks of Madrid. Networks and Heterogeneous Media, 2015, 10 (1) : 127-148. doi: 10.3934/nhm.2015.10.127
##### References:
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##### References:
 [1] R. Albert and A. L. Barabási, Statistical mechanics of complex networks, Reviews of Modern Physics, 74 (2002), 47-97. doi: 10.1103/RevModPhys.74.47. [2] R. Albert, H. Jeong and A. Barabasi, Error and attack tolerance of complex networks, Nature, 406 (2000), 378-382. doi: 10.1038/35019019. [3] L. A. N. Amaral, A. Scala, M. Barthélémy and H. E. Stanley, Classes of small-world networks, Proceedings of the National Academy of the United States of America, 97 (2000), 11149-11152. doi: 10.1073/pnas.200327197. [4] K. H. Chang, K. Kim, H. Oshima and S. M. Yoon, Subway networks in cities, Journal of the Korean Physical Society, 48 (2006), S143-S145. [5] Y. Z. Chen, N. Li and D. R. He, A study on some urban bus transport networks, Physica A, 376 (2007), 747-754. doi: 10.1016/j.physa.2006.10.071. [6] J. Hao, J. Yin and B. Zhang, Structural fault tolerance of scale-free networks, Tsinghua Science & Technology, 12 (2007), 246-249. doi: 10.1016/S1007-0214(07)70118-9. [7] B. Jiang, A topological pattern of urban street networks: Universality and peculiarity, Physica A, 384 (2007), 647-655. doi: 10.1016/j.physa.2007.05.064. [8] M. Ke, et al., Power law and small world properties in a comparison of traffic city networks, Chinese Science Vulletin, 56 (2011), 3731-3735. [9] O. Kwon, Intercity express bus flow in Korea and its network analysis, Physica A, 391 (2012), 4261-4265. doi: 10.1016/j.physa.2012.03.031. [10] V. Latora and M. Marchiori, Is the Boston subway a smallworld network?, Physica A, 314 (2002), 109-113. [11] G. Mao and N. Zhang, A Multilevel simplification algorithm for computing the average shortest-path length of scale-free complex network, Journal of Applied Mathematics, (2014), Art. ID 154172, 6 pp. doi: 10.1155/2014/154172. [12] S. Mizokami, R. Kakimoto and J. Hashimoto, A method of line characteristic evaluation and network reorganization planning of bus systems, Journal of Japan Society of Civil Engineers, 793 (1995), 27-39. [13] M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003), 167-256. doi: 10.1137/S003614450342480. [14] M. E. J. Newman, Assortative mixing in networks, Physical Review Letters, 89 (2002), 208701-208705. doi: 10.1103/PhysRevLett.89.208701. [15] S. Ondŏs, I. Paulovičováa, L. Belušák and D. Husendová, Urban heartbeats (daily cycle of public transport intensity), in GIS Ostrava 2014 - Geoinformatics for Intelligent Transportation, 2014, 747-754. [16] J. Sienkiewicz and J. A. Holyst, Statistical analysis of 22 public transport networks in Poland, Physica Review E, 72 (2005), 046127. doi: 10.1103/PhysRevE.72.046127. [17] H. Soh, et al., Weighted complex network analysis of travel routes on the Singapore public transportation system, Physica A, 389 (2010), 5852-5863. [18] Z. Su, et al., Robustness of Interrelated Traffic Networks to Cascading Failures, Scientific Reports, 2014. [19] D. Takeuchi and K. Yamada, Theory of public subsidies for city bus and development of route-potential as a measurement for that decision making, Journal of Infrastructure Planning and Management, 1991 (1995), 183-192. doi: 10.2208/jscej.1991.183. [20] J. Tinbergen, Shaping the World Economy: Suggestions for an International Economic Policy, Twentieth Century Fund, New York, 1962. [21] C. von Ferber, T. Holovatch, Y. Holovatch and V. Palchykov, Public transport networks: empirical analysis and modeling, The European Physical Journal B, 68 (2009), 261-275. [22] D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442. [23] Web site of the Empresa Municipal Transportes (EMT), 2014. Available from: http://www.emtmadrid.es/. [24] Web site of the Metro Madrid (MM), 2014. Available from: http://www.metromadrid.es/. [25] P. Zhang et al., The Robustness of Interdependent Transportation Networks Under Targeted Attack, EPL (Europhysics Letters), 2013. [26] H. Zhang, P. Zhao, J. Gao and X. Yao, The analysis of the properties of bus network topology in Beijing basing on complex networks, Mathematical Problems in Engineering, 2013 (2013), Article ID 694956, 6 pages. doi: 10.1155/2013/694956.
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