, Volume 46, Issue 5, pp 1893–1913 | Cite as

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

  • António Dinis F. Santos
  • Duarte ValérioEmail author
  • J. A. Tenreiro Machado
  • António M. Lopes


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.


Public transportation network Fractal dimension Entropy Lisbon 



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

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.IDMEC, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  2. 2.Department of Electrical EngineeringInstitute of Engineering, Polytechnic of PortoPortoPortugal
  3. 3.Faculty of Engineering, UISPA - LAETA/INEGIUniversity of PortoPortoPortugal

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