Skip to main content
SpringerLink
Log in

A fractional perspective to the modelling of Lisbon’s public transportation network

  • Published:
Transportation Aims and scope Submit manuscript

Abstract

Urban growth originates multiscale spatial patterns, such as those of transportation networks. Here, the public transportation network (PTN) of the city of Lisbon is analysed from 1901 to 2015, employing several mathematical tools. In a first stage, the fractal dimension and the fractional entropy are used to quantify the evolution of the structure of the PTN in space and time. In a second stage, the PTN is analysed adopting additional information, namely considering different levels of the network based on transportation schedule and passenger capacity, and studying the significance of the distance between stops. Both the fractal dimension and the fractional entropy reveal time patterns compatible with known historical events, showing them to be appropriate for quantifying the growth of the PTN. When the routes’ schedules are used to stratify the PTN, not only the fractal behaviour is observed at different levels, but also the evolution of the network in respect to the homogenization of the capacity of different routes. Finally, when the distance between consecutive stops is analysed, a power law behaviour is revealed, as expected from the fractal geometry of the network. This result is then confirmed using the ht-index.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Notes

  1. Another way of describing these maps is to consider them as level curves of 3-D representations of the PTN. The z-axis is the passenger capacity of each route. The bottom level curve of this 3-D representation is the entire PTN for the year. Other level curves will include only some of the routes, leaving out those with a low passenger capacity. A level curve at height \((100\%-p)M_i\) will include all routes that have a passenger capacity of \((100\%-p)M\) or more. The 3-D representation of a PTN is in fact a layered network (Boccaletti et al. 2014), in which routes with less (more) capacity exist in few (many) layers, and the routes with capacity \(M_i\) exists in all layers.

References

  • Batty, M.: Building a science of cities. Cities 29, S9–S16 (2012)

    Article  Google Scholar 

  • Batty, M., Kim, K.: Form follows function: reformulating urban population density functions. Urb. Stud. 29(7), 1043–1070 (1992)

    Article  Google Scholar 

  • Batty, M., Longley, P.A.: Fractal Cities: A Geometry of Form and Function. Academic Press, London (1994)

    Google Scholar 

  • Benguigui, L.: The fractal dimension of some railway networks. J. Phys. Fr. 2(4), 385–388 (1992)

    Google Scholar 

  • Benguigui, L.: A fractal analysis of the public transportation system of Paris. Environ. Plan. A 27(7), 1147–1161 (1995)

    Article  Google Scholar 

  • Benguigui, L., Daoud, M.: Is the suburban railway system a fractal? Geogr. Anal. 23(4), 362–368 (1991)

    Article  Google Scholar 

  • Berry, M.V.: Diffractals. J. Phys. A Math. Gen. 12(6), 781–798 (1979)

    Article  Google Scholar 

  • Boccaletti, S., Bianconi, G., Criado, R., del Genio, C., Nes, J.G.G., Romance, M., Nadal, I.S., Wang, Z., Zanin, M.: The structure and dynamics of multilayer networks. Phys. Rep. 544, 1–122 (2014)

    Article  Google Scholar 

  • Borak, S., Härdle, W., Weron, R.: Stable Distributions. Springer, Berlin (2005)

    Google Scholar 

  • Carr, J.C., Fright, W.R., Beatson, R.K.: Surface interpolation with radial basis functions for medical imaging. IEEE Trans. Med. Imaging 16(1), 96–107 (1997)

    Article  Google Scholar 

  • Carris: História da Carris. http://carris.transporteslisboa.pt/pt/historia/. Accessed in February (2016)

  • Chechkin, A.V., Metzler, R., Klafter, J., Gonchar, V.Y.: Introduction to the theory of Lévy flights. In: Klages, R., Radons, G., Sokolov, I.M. (eds.) Anomalous Transport: Foundations and Applications. Wiley, Hoboken (2008)

    Google Scholar 

  • Chen, Y.: Multi-scaling allometric analysis for urban and regional development. Phys. A Stat. Mech. Appl. 465, 673–689 (2017)

    Article  Google Scholar 

  • Chen, Y., Jiang, S.: Modeling fractal structure of systems of cities using spatial correlation function. Int. J. Artif. Life Res. 1(1), 12–34 (2010)

    Article  Google Scholar 

  • Chen, Y., Wang, J., Feng, J.: Understanding the fractal dimensions of urban forms through spatial entropy. Entropy 19(600), 1–18 (2017)

    Google Scholar 

  • Comboios de Portugal: Comboios de Portugal. https://pt.wikipedia.org/wiki/Comboios_de_Portugal. Accessed in February (2016)

  • Cruz-Filipe, L.: História das carreiras da Carris. http://historiaccfl.webatu.com/. Accessed in February (2013)

  • Direção-Geral do Território.: Carta administrativa oficial de portugal (CAOP). http://www.dgterritorio.pt/cartografia_e_geodesia/cartografia/carta_administrativa_oficial_de_portugal__caop_. Accessed in September (2017)

  • Feng, J., Chen, Y.: Spatiotemporal evolution of urban form and land-use structure in Hangzhou, China: evidence from fractals. Environ. Plan. B Plan. Des. 37(5), 838–856 (2010)

    Article  Google Scholar 

  • Fleckinger-Pelle, J., Lapidus, M.: Tambour fractal: Vers une résolution de la conjecture de Weyl–Berry pour les valeurs propres du Laplacien. C. R. Acad. Sci. Paris Ser. I Math. 306(4), 171–175 (1988)

    Google Scholar 

  • Frankhauser, P.: Aspects fractals des structures urbaines. Espace Geogr. 19(1), 45–69 (1990)

    Article  Google Scholar 

  • Gao, P., Liu, Z., Xie, M., Tian, K., Liu, G.: CRG index: a more sensitive ht-index for enabling dynamic views of geographic features. Prof. Geogr. 68(4), 533–545 (2016)

    Article  Google Scholar 

  • Jiang, B.: Head/tail breaks for visualization of city structure and dynamics. Cities 43, 69–77 (2015)

    Article  Google Scholar 

  • Jiang, B., Ma, D.: How complex is a fractal? Head/tail breaks and fractional hierarchy. J. Geovis. Spatial Anal. 2(1), 6 (2018)

    Article  Google Scholar 

  • Jiang, B., Yin, J.: Ht-index for quantifying the fractal or scaling structure of geographic features. Ann. Assoc. Am. Geogr. 104(3), 530–540 (2014)

    Article  Google Scholar 

  • Kim, K.S., Benguigui, L., Marinov, M.: The fractal structure of Seoul’s public transportation system. Cities 20(1), 31–39 (2003)

    Article  Google Scholar 

  • Koutrouvelis, I.A.: Regression-type estimation of the parameters of stable laws. J. Am. Stat. Assoc. 75(372), 918–928 (1980)

    Article  Google Scholar 

  • Koutrouvelis, I.A.: An iterative procedure for the estimation of the parameters of stable laws. Commun. Stat. Simul. Comput. 10(1), 17–28 (1981)

    Article  Google Scholar 

  • Lu, Y., Tang, J.: Fractal dimension of a transportation network and its relationship with urban growth: a study of the Dallas–Fort Worth area. Environ. Plan. B Plan. Des. 31(6), 895–911 (2004)

    Article  Google Scholar 

  • Machado, J.A.T.: Fractional order generalized information. Entropy 16(4), 2350–2361 (2014)

    Article  Google Scholar 

  • Maps, G.: Google Maps. https://www.google.pt/maps. Accessed in February (2016)

  • Metropolitano, de Lisboa: Cronologia do Metro. http://metro.transporteslisboa.pt/empresa/um-pouco-de-historia/cronologia/. Accessed in February (2016)

  • Rinaldo, A., Rodriguez-Iturbe, I., Rigon, R., Bras, R.L., Ijjasz-Vasquez, E., Marani, A.: Minimum energy and fractal structures of drainage networks. Water Resour. Res. 28(9), 2183–2195 (1992)

    Article  Google Scholar 

  • Santos, A.D.F., Valério, D., Machado, J.A.T., Lopes, A.M.: Data for Santos, Valério, Tenreiro Machado, Mendes Lopes. https://docs.google.com/spreadsheets/d/e/2PACX-1vRSH4S-d1mJfgXei414Konib0PKirtYrfOSO586Q3bTmZwxZP4MIgBQmeJvjbiXg9Mb-mcmtp8LqHU6/pubhtml. Accessed in August (2018)

  • Schroeder, M.: Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. Dover, New York (1991)

    Google Scholar 

  • Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(379–423), 623–656 (1948)

    Article  Google Scholar 

  • Shen, G.: Fractal dimension and fractal growth of urbanized areas. Int. J. Geogr. Inf. Sci. 16(5), 419–437 (2002)

    Article  Google Scholar 

  • Tannier, C., Thomas, I.: Defining and characterizing urban boundaries: a fractal analysis of theoretical cities and Belgian cities. Comput. Environ. Urban Syst. 41, 234–248 (2013)

    Article  Google Scholar 

  • Tannier, C., Thomas, I., Vuidel, G., Frankhauser, P.: A fractal approach to identifying urban boundaries. Geogr. Anal. 43(2), 211–227 (2011)

    Article  Google Scholar 

  • Thomas, I., Frankhauser, P.: Fractal dimensions of the built-up footprint: buildings versus roads. Fractal evidence from Antwerp (Belgium). Environ. Plan. B Plan. Des. 40(2), 310–329 (2013)

    Article  Google Scholar 

  • Thomas, I., Frankhauser, P., Biernacki, C.: The morphology of built-up landscapes in Wallonia (Belgium): a classification using fractal indices. Landsc. Urban Plan. 84(2), 99–115 (2008)

    Article  Google Scholar 

  • Ubriaco, M.R.: Entropies based on fractional calculus. Phys. Lett. A 373(30), 2516–2519 (2009)

    Article  Google Scholar 

  • Valério, D., Lopes, A.M., Machado, J.A.T.: Entropy analysis of a railway network complexity. Entropy 18(11), 388 (2016)

    Article  Google Scholar 

  • Veillette, M.: Alpha-stable distributions in MATLAB. http://math.bu.edu/people/mveillet/html/alphastablepub.html. Accessed in August (2016)

  • von Ferber, C., Holovatch, Y.: Fractal transit networks: self-avoiding walks and Lévy flights. Eur. Phys. J. Spec. Top. 216(1), 49–55 (2013)

    Article  Google Scholar 

  • Wang, H., Luo, S., Luo, T.: Fractal characteristics of urban surface transit and road networks: case study of Strasbourg, France. Adv. Mech. Eng. 9(2), 1–12 (2017)

    Google Scholar 

  • Zhuangzhi, S.: The study of fractal approach to measure urban rail transit network morphology. J. Transp. Syst. Eng. Inf. Technol. 7(1), 29 (2007)

    Google Scholar 

Download references

Acknowledgements

This work was supported by FCT, through IDMEC, under LAETA, Project UID/EMS/50022/2013.

Author information

Authors and Affiliations

  1. IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001, Lisbon, Portugal

    António Dinis F. Santos & Duarte Valério

  2. Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Rua Dr. António Bernardino de Almeida, 431, 4249-015, Porto, Portugal

    J. A. Tenreiro Machado

  3. Faculty of Engineering, UISPA - LAETA/INEGI, University of Porto, Rua Dr. Roberto Frias, 4200-465, Porto, Portugal

    António M. Lopes

Authors
  1. António Dinis F. Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Duarte Valério
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. A. Tenreiro Machado
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. António M. Lopes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Duarte Valério.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Santos, A.D.F., Valério, D., Tenreiro Machado, J.A. et al. A fractional perspective to the modelling of Lisbon’s public transportation network. Transportation 46, 1893–1913 (2019). https://doi.org/10.1007/s11116-018-9906-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11116-018-9906-3

Keywords

Search

Navigation