, Volume 46, Issue 5, pp 1689–1712 | Cite as

Exploratory analysis of Zipf’s universal power law in activity schedules

  • Wim EctorsEmail author
  • Bruno Kochan
  • Davy Janssens
  • Tom Bellemans
  • Geert Wets


People’s behavior is governed by extremely complex, multidimensional processes. This fact is well-established in the transportation research community, which has been working on travel behavior (travel demand) models for many years. The number of degrees of freedom in a person’s activity schedule is enormous. However, the frequency of occurrence of day-long activity schedules obeys a remarkably simple, scale-free distribution. This particular distribution has been observed in many natural and social processes and is commonly referred to as Zipf’s law, a power law distribution. This research provides evidence that activity schedules from various study areas exhibit a universal power law distribution. To this end, an elaborate analysis using 13 household travel surveys from diverse study areas discusses the effect of proportional outlier removal on the power law’s exponent value. Statistical evidence is provided for the hypothesis that activity schedules in all these datasets exhibit a power law distribution with a common exponent value. The study proposes that a Zipf power law could be used as an additional dimension within a travel demand model’s validation process. Contrary to other validation methods, no new data is required. The observation of a Zipf power law distribution in the generated schedules appears to be a necessary condition. Additionally, the universal activity schedule distribution might enable the full integration of activity schedules in models based on universal mobility patterns.


Zipf’s law Power law Activity schedules Universal distributions 



The authors are deeply grateful to the original data creators, depositors and/or copyright holders for making the microdata available for this research. The copyright and all other intellectual property rights in the data and associated documentation are vested in the original data creators or depositors. The original data creators and analyzers bear no responsibility for the further analysis or interpretation of the data in this research. The authors would like to acknowledge the researches, works, individuals and institutions supporting the following data collections for making this research possible: US NHTS 2009 (US Department of Transportation and Federal Highway Administration 2009), NLD OViN 2013 (Centraal Bureau voor de Statistiek (CBS) and Rijkswaterstaat (RWS) 2014), BEL OVG 3.0--4.5 (Janssens et al. 2014), SVN Ljubljana 2013 (Klemenčič et al. 2014) [Acknowledgements go out to the City Municipality of Ljubljana], GBR NTS 2009--2014 (Department for Transport 2015), KOR Seoul HTS 2010 (Korea Transportation Institute 2011; Metropolitan Transport Authority 2012), DEU Mobidrive 1999 (Chalasani and Axhausen 2004), CHE Thurgau 2003 (Loechl 2005), FRA ENTD 2008 (Armoogum et al. 2011), BEL Beldam 2010 (Cornelis et al. 2012) [financed by BELSPO, FOD Mobiliteit & Vervoer and others. Coordinated by GRT (Université de Namur) in cooperation with IMOB (Universiteit Hasselt) and CES (FUSL)], IRL NTS 2009 (Central Statistics Office 2011) [Accessed via the Irish Social Science Data Archive -], FIN HLT 2010--2011 (Liikennevirasto - Finnish Transport Agency) [Finnish National Travel Survey 2010--2011/Finnish Transport Agency and Wim Ectors], SWE RVU 2011--2014 (Trafik Analys 2015), AUS VISTA 2007 & 2009 (Department of Economic Development; Jobs; Transport and Resources (DEDJTR) 2009; Department of Economic Development; Jobs; Transport and Resources (DEDJTR) 2007). Part of this work was presented at the hEART 2016 conference in Delft, The Netherlands, offering valuable reflections during the preparation of this work. (Ectors et al. 2016a)

Author's contributions

WE: literature Search and Review, manuscript writing and editing; BK: meta-analysis, content planning and detailed manuscript review; DJ: meta-analysis and content planning; TB: meta-analysis and content planning; GW: meta-analysis and content planning.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.


  1. Adamic, L.A., Huberman, B.A.: Zipf’s law and the internet. Glottometrics 3, 143–150 (2002)Google Scholar
  2. Ahern, A., Weyman, G., Redelbach, M., Schulz, A., Akkermans, L., Vannacci, L., Anoyrkati, E., Von grinsven, A.: Analysis of National Travel Statistics in Europe. European Commission (2013).
  3. Arentze, T., Hofman, F., van Mourik, H., Timmermans, H.J.P.: ALBATROSS: multiagent, rule-based model of activity pattern decisions. Transp. Res. Rec. 1706, 136–144 (2000). CrossRefGoogle Scholar
  4. Armoogum, J., Hubert, J.-P., Francois, D., Roumier, B., Robin, M., Roux, S.: Enquête nationale transports et déplacements 2007–2008 (ENTD 2007–2008) (Rapport technique). (2011)Google Scholar
  5. Auerbach, F.: Das Gesetz der Bevölkerungskonzentration. Petermanns Geogr. Mitt. 59, 74–76 (1913)Google Scholar
  6. Batty, M.: Agents, cells, and cities: new representational models for simulating multiscale urban dynamics. Environ. Plan. A 37, 1373–1394 (2005). CrossRefGoogle Scholar
  7. Batty, M.: The size, scale, and shape of cities. Science 319, 769–771 (2008). CrossRefGoogle Scholar
  8. Boyce, D., Bar-Gera, H.: Validation of multiclass urban travel forecasting models combining origin-destination, mode, and route choices. J. Reg. Sci. 43, 517–540 (2003). CrossRefGoogle Scholar
  9. Brockmann, D., Hufnagel, L., Geisel, T.: The scaling laws of human travel. Nature 439, 462–465 (2006). CrossRefGoogle Scholar
  10. Centraal Bureau voor de Statistiek (CBS), Rijkswaterstaat (RWS): Onderzoek Verplaatsingen in Nederland 2013–OViN 2013,, (2014)
  11. Central Statistics Office: National Travel Survey 2009. Dublin, Ireland (2011)Google Scholar
  12. Chalasani, V.S., Axhausen, K.W.: Mobidrive: a six week travel diary. (2004)
  13. Chen, W.-C.: On the weak form of Zipf’s law. J. Appl. Probab. 17, 611–622 (1980). CrossRefGoogle Scholar
  14. Clauset, A., Shalizi, C.R., Newman, M.E.J.: Power-law distributions in empirical data. SIAM Rev. 51, 661 (2009). CrossRefGoogle Scholar
  15. Cornelis, E., Hubert, M., Hyunen, P., Lebrun, K., Patriarche, G., De Witte, A., Creemers, L., Declercq, K., Janssens, D., Castaigne, M., Hollaert, L., Walle, F.: La mobilité en Belgique en 2010 : résultats de l’enquête BELDAM (2012)Google Scholar
  16. Corominas-Murtra, B., Solé, R.V.: Universality of Zipf’s law. Phys. Rev. E 82, 9 (2010). CrossRefGoogle Scholar
  17. Department for Transport: National Travel Survey, 2002–2014 [computer file]. 9th edn. (2015)Google Scholar
  18. Department of Economic Development; Jobs; Transport and Resources (DEDJTR): Victorian Integrated survey of Travel and Activity 2007, (2007)
  19. Department of Economic Development; Jobs; Transport and Resources (DEDJTR): Victorian Integrated survey of Travel and Activity 2009, (2009)
  20. Doherty, S.T., Axhausen, K.W.: The development of a unified modeling framework for the household activity-travel scheduling process. In: Traffic and Mobility: Simulation-Economics-Environment, pp. 35–36 (1999)Google Scholar
  21. Ectors, W., Kochan, B., Janssens, D., Bellemans, T., Wets, G.: Zipf’s law in activity schedules. In: hEART 2016: The 5th Symposium Arranged by European Association for Research in Transportation, Delft, The Netherlands, 14–16 Sept 2016 (2016a)Google Scholar
  22. Ectors, W., Kochan, B., Knapen, L., Janssens, D., Bellemans, T.: A generic data-driven sequential clustering algorithm determining activity skeletons. In: Procedia Computer Science, pp. 34–41. Elsevier Masson SAS (2016b)Google Scholar
  23. Fujiwara, Y.: Zipf law in firms bankruptcy. Phys. A Stat. Mech. Appl. 337, 219–230 (2004). CrossRefGoogle Scholar
  24. Furusawa, C., Kaneko, K.: Zipf’s law in gene expression. Phys. Rev. Lett. 90, 1–11 (2003). CrossRefGoogle Scholar
  25. Gabaix, X.: Zipf’s law for cities: an explanation. Q. J. Econ. 114, 739–767 (1999)CrossRefGoogle Scholar
  26. Gillespie, C.S.: Fitting heavy tailed distributions: the poweRlaw package. J. Stat. Softw. 64, 1–16 (2015)CrossRefGoogle Scholar
  27. González, M.C., Hidalgo, C.A., Barabási, A.-L.: Understanding individual human mobility patterns. Nature 453, 779–782 (2008). CrossRefGoogle Scholar
  28. Guidotti, R., Trasarti, R., Nanni, M.: TOSCA : Two-steps clustering algorithm for personal locations detection. In: Proceedings of the 23nd ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (2015).
  29. Hanel, R., Corominas-Murtra, B., Liu, B., Thurner, S.: Fitting power-laws in empirical data with estimators that work for all exponents. PLoS ONE 12, 1–15 (2017). CrossRefGoogle Scholar
  30. Hupkes, G.: The law of constant travel time and trip-rates. Futures 14, 38–46 (1982). CrossRefGoogle Scholar
  31. Ioannides, Y.M., Overman, H.G.: Zipf’s law for cities: an empirical examintion. Reg. Sci. Urban Econ. 33, 127–137 (2003). CrossRefGoogle Scholar
  32. Janssens, D., Declercq, K., Wets, G.: Onderzoek Verplaatsingsgedrag Vlaanderen 4.5 (2012–2013) (2014)Google Scholar
  33. Jiang, B.: Street hierarchies: a minority of streets account for a majority of traffic flow. Int. J. Geogr. Inf. Sci. 23, 1033–1048 (2009). CrossRefGoogle Scholar
  34. Jiang, B., Jia, T.: Zipf’s law for all the natural cities in the United States: a geospatial perspective. Int. J. Geogr. Inf. Sci. (2010). CrossRefGoogle Scholar
  35. Jiang, S., Yang, Y., Gupta, S., Veneziano, D., Athavale, S., González, M.C.: The TimeGeo modeling framework for urban motility without travel surveys. Proc. Natl. Acad. Sci. 113, E5370–E5378 (2016). CrossRefGoogle Scholar
  36. Ki Baek, S., Bernhardsson, S., Minnhagen, P.: Zipf’s law unzipped. New J. Phys. 13, 043004 (2011). CrossRefGoogle Scholar
  37. Klafter, J., Shlesinger, M.F., Zumofen, G.: Beyond Brownian motion. Phys. Today 49, 33 (1996). CrossRefGoogle Scholar
  38. Klemenčič, M., Lep, M., Mesarec, B., Žnuderl, B.: Potovalne navade prebivalcev v Mestni občini Ljubljana in Ljubljanski urbani regiji (2014)Google Scholar
  39. Korea Transportation Institute: National Transportation Demand Survey and Database Establishment in 2010: Passenger O/D Survey on the National Area. (2011)Google Scholar
  40. Li, S., Lee, D.H.: Learning daily activity patterns with probabilistic grammars. Transportation 44, 49–68 (2017). CrossRefGoogle Scholar
  41. Li, W.: Random texts exhibit Zipf’s-law-like word frequency distribution. IEEE Trans. Inf. Theory 38, 1842–1845 (1992). CrossRefGoogle Scholar
  42. Liikennevirasto - Finnish Transport Agency: National Travel Survey 2010–2011Google Scholar
  43. Loechl, M.: Stability of travel behaviour: Thurgau 2003. (2005)
  44. Lü, L., Zhang, Z.-K., Zhou, T.: Deviation of Zipf’s and Heaps’ laws in human languages with limited dictionary sizes. Sci. Rep. 3, 1082 (2013). CrossRefGoogle Scholar
  45. Ma, D., Sandberg, M., Jiang, B.: A socio-geographic perspective on human activities in social media. Geogr. Anal. 49, 328–342 (2017). CrossRefGoogle Scholar
  46. Maillart, T., Sornette, D., Spaeth, S., von Krogh, G.: Empirical tests of Zipf’s law mechanism in open source Linux distribution. Phys. Rev. Lett. 101, 218701 (2008). CrossRefGoogle Scholar
  47. Makse, H.A., Andrade Jr., J.S., Batty, M., Havlin, S., Eugene Jr., H.E.: Modeling urban growth patterns with correlated percolation. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top 58, 7054–7062 (1998). CrossRefGoogle Scholar
  48. Marsili, M., Zhang, Y.: Interacting individuals leading to Zipf’s law. Phys. Rev. Lett. 80, 2741–2744 (1998). CrossRefGoogle Scholar
  49. Metropolitan Transport Authority: The Report of Household travel survey in Seoul Metropolitan Area [In Korean], Seoul (2012)Google Scholar
  50. Miller, E.J., Farooq, B., Chingcuanco, F., Wang, D.: Historical validation of integrated transport-land use model system. Transp. Res. Rec. J. Transp. Res. Board 2255, 91–99 (2012). CrossRefGoogle Scholar
  51. Newman, M.: Power laws, Pareto distributions and Zipf’s law. Contemp. Phys. 46, 323–351 (2005). CrossRefGoogle Scholar
  52. Nitsch, V.: Zipf zipped. J. Urban Econ. 57, 86–100 (2005). CrossRefGoogle Scholar
  53. Noulas, A., Scellato, S., Lambiotte, R., Pontil, M., Mascolo, C.: A tale of many cities: universal patterns in human urban mobility. PLoS ONE (2012). CrossRefGoogle Scholar
  54. Okuyama, K., Takayasu, M., Takayasu, H.: Zipf’s law in income distribution of companies. Phys. A Stat. Mech. Appl. 269, 125–131 (1999). CrossRefGoogle Scholar
  55. Paleari, S., Redondi, R., Malighetti, P.: A comparative study of airport connectivity in China, Europe and US: Which network provides the best service to passengers? Transp. Res. Part E Logist. Transp. Rev. 46, 198–210 (2010). CrossRefGoogle Scholar
  56. Pappalardo, L., Simini, F.: Data-driven generation of spatio-temporal routines in human mobility. Springer, Berlin (2017)Google Scholar
  57. Park, B., Schneeberger, J.D.: Microscopic simulation model calibration and validation: case study of VISSIM simulation model for a coordinated actuated signal system. Transp. Res. Rec. 1856, 185–192 (2003). CrossRefGoogle Scholar
  58. Reed, W.J.: The Pareto, Zipf and other power laws. Econ. Lett. 74, 15–19 (2001). CrossRefGoogle Scholar
  59. Riccardo, G., Armando, B., Sandro, R.: Towards a statistical physics of human mobility. Int. J. Mod. Phys. C 23, 1250061 (2012). CrossRefGoogle Scholar
  60. Saberi, M., Mahmassani, H.S., Brockmann, D., Hosseini, A.: A complex network perspective for characterizing urban travel demand patterns: graph theoretical analysis of large-scale origin–destination demand networks. Transportation 44, 1383–1402 (2017). CrossRefGoogle Scholar
  61. Schneider, C.M., Belik, V., Couronné, T., Smoreda, Z., González, M.C.: Unravelling daily human mobility motifs. J. R. Soc. Interface R. Soc. 10, 20130246 (2013). CrossRefGoogle Scholar
  62. Shen, Y., Karimi, K.: Urban function connectivity: characterisation of functional urban streets with social media check-in data. Cities 55, 9–21 (2016). CrossRefGoogle Scholar
  63. Simini, F., González, M.C., Maritan, A., Barabási, A.-L.: A universal model for mobility and migration patterns. Nature 484, 96–100 (2012). CrossRefGoogle Scholar
  64. Song, C., Koren, T., Wang, P., Barabasi, A.-L.: Modelling the scaling properties of human mobility. Nat. Phys. 6, 1–6 (2010). CrossRefGoogle Scholar
  65. Soo, K.T.: Zipf’s law for cities: a cross-country investigation. Reg. Sci. Urban Econ. 35, 239–263 (2005). CrossRefGoogle Scholar
  66. Toledo, T., Koutsopoulos, H.: Statistical validation of traffic simulation models. Transp. Res. Rec. 1876, 142–150 (2004). CrossRefGoogle Scholar
  67. Trafik Analys: RVU Sverige 2011–2014—Den nationella resvaneundersökningen (RVU Sweden 2011–2014—national travel survey). Stockholm (2015)Google Scholar
  68. U.S. Department of Transportation, Federal Highway Administration: 2009 National Household Travel Survey. (2009)
  69. Urzúa, C.M.: Testing for Zipf’s law: a common pitfall. Econ. Lett. 112, 254–255 (2011). CrossRefGoogle Scholar
  70. Volchenkov, D., Blanchard, P.: Scaling and universality in city space syntax: between Zipf and Matthew. Phys. A Stat. Mech. Appl. 387, 2353–2364 (2008). CrossRefGoogle Scholar
  71. Wang, G., Zhong, Y., Teo, C.-P., Liu, Q.: Flow-based accessibility measurement: The Place Rank approach. Transp. Res. Part C Emerg. Technol. 56, 335–345 (2015). CrossRefGoogle Scholar
  72. Willis, J.C.: Age and Area. Cambridge University Press, Cambridge (1922)Google Scholar
  73. Yang, X.H., Chen, G., Chen, S.Y., Wang, W.L., Wang, L.: Study on some bus transport networks in China with considering spatial characteristics. Transp. Res. Part A Policy Pract. 69, 1–10 (2014). CrossRefGoogle Scholar
  74. Zheng, Z., Rasouli, S., Timmermans, H.: Two-regime pattern in human mobility: evidence from GPS taxi trajectory data. Geogr. Anal. 48, 157–175 (2016). CrossRefGoogle Scholar
  75. Zipf, G.K.: Human Behaviour and the Principle of Least Effort. Addison-Wesley, Reading (1949)Google Scholar

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Authors and Affiliations

  1. 1.Transportation Research Institute (IMOB)UHasselt – Universiteit HasseltDiepenbeekBelgium

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