Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

The scales of human mobility

Abstract

There is a contradiction at the heart of our current understanding of individual and collective mobility patterns. On the one hand, a highly influential body of literature on human mobility driven by analyses of massive empirical datasets finds that human movements show no evidence of characteristic spatial scales. There, human mobility is described as scale free1,2,3. On the other hand, geographically, the concept of scale—referring to meaningful levels of description from individual buildings to neighbourhoods, cities, regions and countries—is central for the description of various aspects of human behaviour, such as socioeconomic interactions, or political and cultural dynamics4,5. Here we resolve this apparent paradox by showing that day-to-day human mobility does indeed contain meaningful scales, corresponding to spatial ‘containers’ that restrict mobility behaviour. The scale-free results arise from aggregating displacements across containers. We present a simple model—which given a person’s trajectory—infers their neighbourhood, city and so on, as well as the sizes of these geographical containers. We find that the containers—characterizing the trajectories of more than 700,000 individuals—do indeed have typical sizes. We show that our model is also able to generate highly realistic trajectories and provides a way to understand the differences in mobility behaviour across countries, gender groups and urban–rural areas.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: The scales of human mobility.
Fig. 2: The container model generates realistic mobility traces.
Fig. 3: Socio-demographic differences and heterogeneity in scales.

Data availability

Derived data that support the findings of this study are available in DTU Data with the identifier https://doi.org/10.11583/DTU.12941993.v1. Additional data related to this paper may be requested from the authors. Raw data for dataset D1 are not publicly available to preserve individuals’ privacy under the European General Data Protection Regulation. Raw data for dataset D2 are not publicly available due to privacy considerations, but are available to researchers who meet the criteria for access to confidential data, sign a confidentiality agreement and agree to work under supervision in Copenhagen. Please direct your queries to the corresponding author. Source data are provided with this paper.

Code availability

Code is available at https://github.com/lalessan/scales_human_mobility/.

References

  1. 1.

    Brockmann, D., Hufnagel, L. & Geisel, T. The scaling laws of human travel. Nature 439, 462–465 (2006).

    ADS  CAS  PubMed  Google Scholar 

  2. 2.

    González, M. C., Hidalgo, C. A. & Barabási, A.-L. Understanding individual human mobility patterns. Nature 453, 779–782 (2008).

    ADS  PubMed  Google Scholar 

  3. 3.

    Song, C., Koren, T., Wang, P. & Barabási, A.-L. Modelling the scaling properties of human mobility. Nat. Phys. 6, 818–823 (2010).

    CAS  Google Scholar 

  4. 4.

    Paasi, A. Place and region: looking through the prism of scale. Prog. Hum. Geogr. 28, 536–546 (2004).

    Google Scholar 

  5. 5.

    Marston, S. A. The social construction of scale. Prog. Hum. Geogr. 24, 219–242 (2000).

    Google Scholar 

  6. 6.

    Cresswell, T. On the Move: Mobility in the Modern Western World (Taylor & Francis, 2006).

  7. 7.

    Kaluza, P., Kölzsch, A., Gastner, M. T. & Blasius, B. The complex network of global cargo ship movements. J. R. Soc. Interface 7, 1093–1103 (2010).

    PubMed  PubMed Central  Google Scholar 

  8. 8.

    Kraemer, M. U. et al. The effect of human mobility and control measures on the COVID-19 epidemic in China. Science 368, 493–497 (2020).

    ADS  CAS  Google Scholar 

  9. 9.

    Song, X., Zhang, Q., Sekimoto, Y. & Shibasaki, R. Prediction of human emergency behavior and their mobility following large-scale disaster. In Proc. 20th ACM SIGKDD Int. Conf. on Knowledge Discovery and Data Mining 5–14 (ACM, 2014).

  10. 10.

    Becker, F. & Axhausen, K. W. Literature review on surveys investigating the acceptance of automated vehicles. Transportation 44, 1293–1306 (2017).

    Google Scholar 

  11. 11.

    Barbosa, H. et al. Human mobility: models and applications. Phys. Rep. 734, 1–74 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  12. 12.

    Larsen, J. & Urry, J. Mobilities, Networks, Geographies (Routledge, 2016).

  13. 13.

    Hirtle, S. C. & Jonides, J. Evidence of hierarchies in cognitive maps. Mem. Cognit. 13, 208–217 (1985).

    CAS  Google Scholar 

  14. 14.

    Von Thünen, J. H. Der isolierte Staat in Beziehung auf Landwirtschaft und Nationalökonomie Vol. 13 (G Fischer, 1910).

  15. 15.

    Christaller, W. Die zentralen Orte in Süddeutschland: eine ökonomisch-geographische Untersuchung über die Gesetzmässigkeit der Verbreitung und Entwicklung der Siedlungen mit städtischen Funktionen (Wissenschaftliche Buchgesellschaft, 1980).

  16. 16.

    Berry, B. J. L. Geography of Market Centers and Retail Distribution (Prentice Hall, 1967).

  17. 17.

    Alonso, W. et al. Location and Land Use. Toward a General Theory of Land Rent (Harvard Univ. Press, 1964).

  18. 18.

    Cadwallader, M. Migration and Residential Mobility: Macro and Micro Approaches (Univ. Wisconsin Press, 1992).

  19. 19.

    Thiemann, C., Theis, F., Grady, D., Brune, R. & Brockmann, D. The structure of borders in a small world. PLoS ONE 5, e15422 (2010).

    ADS  PubMed  PubMed Central  Google Scholar 

  20. 20.

    Marchetti, C. Anthropological invariants in travel behavior. Technol. Forecast. Soc. Change 47, 75–88 (1994).

    Google Scholar 

  21. 21.

    Noulas, A., Scellato, S., Lambiotte, R., Pontil, M. & Mascolo, C. A tale of many cities: universal patterns in human urban mobility. PLoS ONE 7, e37027 (2012); correction 7, https://doi.org/10.1371/annotation/ca85bf7a-7922-47d5-8bfb-bcdf25af8c72 (2020).

    ADS  CAS  Article  PubMed  PubMed Central  Google Scholar 

  22. 22.

    Alessandretti, L., Sapiezynski, P., Lehmann, S. & Baronchelli, A. Multi-scale spatio-temporal analysis of human mobility. PLoS ONE 12, e0171686 (2017).

    PubMed  PubMed Central  Google Scholar 

  23. 23.

    Newman, M. E. Power laws, Pareto distributions and Zipf’s law. Contemp. Phys. 46, 323–351 (2005).

    ADS  Google Scholar 

  24. 24.

    Liang, X., Zhao, J., Dong, L. & Xu, K. Unraveling the origin of exponential law in intra-urban human mobility. Sci. Rep. 3, 2983 (2013).

    ADS  PubMed  PubMed Central  Google Scholar 

  25. 25.

    Alessandretti, L., Sapiezynski, P., Sekara, V., Lehmann, S. & Baronchelli, A. Evidence for a conserved quantity in human mobility. Nat. Hum. Behav. 2, 485–491 (2018).

    Google Scholar 

  26. 26.

    Gallotti, R., Bazzani, A., Rambaldi, S. & Barthelemy, M. A stochastic model of randomly accelerated walkers for human mobility. Nat. Commun. 7, 12600 (2016).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  27. 27.

    Gheorghiu, S. & Coppens, M.-O. Heterogeneity explains features of “anomalous” thermodynamics and statistics. Proc. Natl Acad. Sci. USA 101, 15852–15856 (2004).

    ADS  CAS  Google Scholar 

  28. 28.

    Amini, A., Kung, K., Kang, C., Sobolevsky, S. & Ratti, C. The impact of social segregation on human mobility in developing and industrialized regions. EPJ Data Sci. 3, 6 (2014).

    Google Scholar 

  29. 29.

    Fotheringham, A. S. A new set of spatial-interaction models: the theory of competing destinations. Environ. Plan. A 15, 15–36 (1983).

    Google Scholar 

  30. 30.

    Saraçli, S., Doğan, N. & Doğan, İ. Comparison of hierarchical cluster analysis methods by cophenetic correlation. J. Inequal. Appl. 2013, 203 (2013).

    MATH  Google Scholar 

  31. 31.

    Barbosa, H., de Lima-Neto, F. B., Evsukoff, A. & Menezes, R. The effect of recency to human mobility. EPJ Data Sci. 4, 21 (2015).

    Google Scholar 

  32. 32.

    Gaddum, J. H. Lognormal distributions. Nature 156, 463–466 (1945).

    ADS  MathSciNet  MATH  Google Scholar 

  33. 33.

    Romeo, M., Da Costa, V. & Bardou, F. Broad distribution effects in sums of lognormal random variables. Eur. Phys. J. B 32, 513–525 (2003).

    ADS  CAS  Google Scholar 

  34. 34.

    Clauset, A., Shalizi, C. R. & Newman, M. E. Power-law distributions in empirical data. SIAM Rev. 51, 661–703 (2009).

    ADS  MathSciNet  MATH  Google Scholar 

  35. 35.

    Song, C., Qu, Z., Blumm, N. & Barabási, A.-L. Limits of predictability in human mobility. Science 327, 1018–1021 (2010).

    ADS  MathSciNet  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  36. 36.

    Gauvin, L. et al. Gender gaps in urban mobility. Humanit. Soc. Sci. Commun. 7, 11 (2020).

    Google Scholar 

  37. 37.

    Breheny, M. The compact city and transport energy consumption. Trans. Inst. Br. Geogr. 20, 81–101 (1995).

    Google Scholar 

  38. 38.

    Carr, L. J., Dunsiger, S. I. & Marcus, B. H. Walk Score™ as a global estimate of neighborhood walkability. Am. J. Prev. Med. 39, 460–463 (2010).

    PubMed  PubMed Central  Google Scholar 

  39. 39.

    Gaye, A. et al. Measuring Key Disparities in Human Development: The Gender Inequality Index Human Development Research Paper 46 (UNDP, 2010).

  40. 40.

    Velaga, N. R., Beecroft, M., Nelson, J. D., Corsar, D. & Edwards, P. Transport poverty meets the digital divide: accessibility and connectivity in rural communities. J. Transp. Geogr. 21, 102–112 (2012).

    Google Scholar 

  41. 41.

    Litman, T. A. Economic value of walkability. Transp. Res. Rec. 1828, 3–11 (2003).

    Google Scholar 

  42. 42.

    Baronchelli, A. & Radicchi, F. Lévy flights in human behavior and cognition. Chaos Solitons Fractals 56, 101–105 (2013).

    ADS  Google Scholar 

  43. 43.

    Han, X.-P., Hao, Q., Wang, B.-H. & Zhou, T. Origin of the scaling law in human mobility: hierarchy of traffic systems. Phys. Rev. E 83, 036117 (2011).

    ADS  Google Scholar 

  44. 44.

    Zhao, K., Musolesi, M., Hui, P., Rao, W. & Tarkoma, S. Explaining the power-law distribution of human mobility through transportation modality decomposition. Sci. Rep. 5, 9136 (2015).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  45. 45.

    Pappalardo, L. et al. Returners and explorers dichotomy in human mobility. Nat. Commun. 6, 8166 (2015).

    ADS  PubMed  PubMed Central  Google Scholar 

  46. 46.

    Szell, M., Sinatra, R., Petri, G., Thurner, S. & Latora, V. Understanding mobility in a social petri dish. Sci. Rep. 2, 457 (2012).

    ADS  PubMed  PubMed Central  Google Scholar 

  47. 47.

    Jiang, S. et al. The TimeGeo modeling framework for urban mobility without travel surveys. Proc. Natl Acad. Sci. USA 113, E5370–E5378 (2016); correction 113, E7137 (2016).

  48. 48.

    Pumain, D. in Hierarchy in Natural and Social Sciences (ed. Pumain, D.) 169–222 (Springer, 2006).

  49. 49.

    Batty, M. in Hierarchy in Natural and Social Sciences (ed. Pumain, D.) 143–168 (Springer, 2006).

  50. 50.

    Arcaute, E. et al. Cities and regions in Britain through hierarchical percolation. R. Soc. Open Sci. 3, 150691 (2016).

    ADS  MathSciNet  PubMed  PubMed Central  Google Scholar 

  51. 51.

    Stopczynski, A. et al. Measuring large-scale social networks with high resolution. PLoS ONE 9, e95978 (2014).

    ADS  PubMed  PubMed Central  Google Scholar 

  52. 52.

    Aslak, U. & Alessandretti, L. Infostop: scalable stop-location detection in multi-user mobility data. Preprint at https://arxiv.org/abs/2003.14370 (2020).

  53. 53.

    Pesaresi, M. et al. Operating Procedure for the Production of the Global Human Settlement Layer from Landsat Data of the Epochs 1975, 1990, 2000, and 2014 (Publications Office of the European Union, 2016).

  54. 54.

    Train, K. E. Discrete Choice Methods with Simulation (Cambridge Univ. Press, 2009).

  55. 55.

    Zahavi, Y. & Ryan, J. The stability of travel components over time. Traffic Eng. Control 750, 19–26 (1978).

    Google Scholar 

  56. 56.

    Miller, H. J. Tobler’s first law and spatial analysis. Ann. Assoc. Am. Geogr. 94, 284–289 (2004).

    Google Scholar 

  57. 57.

    Kraemer, M. U. G. et al. Mapping global variation in human mobility. Nat. Hum. Behav. 4, 800–810 (2020).

    Google Scholar 

  58. 58.

    Steele, J. E. et al. Mapping poverty using mobile phone and satellite data. J. R. Soc. Interface 14, 20160690 (2017).

    PubMed  PubMed Central  Google Scholar 

  59. 59.

    Lu, X. et al. Detecting climate adaptation with mobile network data in Bangladesh: anomalies in communication, mobility and consumption patterns during cyclone Mahasen. Climatic Change 138, 505–519 (2016).

    ADS  PubMed  PubMed Central  Google Scholar 

  60. 60.

    Lu, X., Bengtsson, L. & Holme, P. Predictability of population displacement after the 2010 Haiti earthquake. Proc. Natl Acad. Sci. USA 109, 11576–11581 (2012).

    ADS  CAS  Google Scholar 

  61. 61.

    Weiss, D. J. et al. A global map of travel time to cities to assess inequalities in accessibility in 2015. Nature 553, 333–336 (2018).

    ADS  CAS  Google Scholar 

  62. 62.

    Althoff, T. et al. Large-scale physical activity data reveal worldwide activity inequality. Nature 547, 336–339 (2017).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  63. 63.

    Everitt, B. S., Landau, S., Leese, M. & Stahl, D. Hierarchical clustering. Cluster Anal. 5, 71–110 (2011).

    Google Scholar 

  64. 64.

    Sekara, V., Stopczynski, A. & Lehmann, S. Fundamental structures of dynamic social networks. Proc. Natl Acad. Sci. USA 113, 9977–9982 (2016).

    CAS  Google Scholar 

Download references

Acknowledgements

We thank J. Sellergren for being the best; F. Simini and L. K. Hansen for providing insightful comments; and M. C. Gonzalez for help with auxiliary datasets. The work was supported in part by the Villum Foundation and the Danish Council for Independent Research.

Author information

Affiliations

Authors

Contributions

L.A., U.A. and S.L. designed the study and the model. L.A. and U.A. performed the analyses and implemented the model. L.A., U.A. and S.L. analysed the results and wrote the paper.

Corresponding author

Correspondence to Sune Lehmann.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Elsa Arcaute, Denise Pumain and José Ramasco for their contribution to the peer review of this work. Peer review reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 The D1 dataset.

a, Number of individuals for each gender. b, Number of individuals per age group. c, Number of individuals per country (see colour scale). We considered the 600,817 individuals in our dataset with at least one year of data, and whose time coverage (the fraction of time an individual position is known) was higher than 50% at any given day. For these individuals, we considered one year of data with highest median time coverage. Map data from the GADM Database of Global Administrative Areas, version 3.6, available at http://www.gadm.org.

Extended Data Fig. 2 Distribution of container sizes at different levels.

ah, Distribution of individual container sizes at hierarchical levels 2 (a, e), 3 (b, f), 4 (c, g) and 5 (d, h) (black line) and the corresponding lognormal (blue line) and truncated power-law (orange line) fits. Results are shown for the D1 (ad) and D2 (eh) datasets.

Extended Data Fig. 3 Schematic description and validation of the likelihood optimization algorithm.

We find the hierarchical partitioning corresponding to a sequence of locations as follows. a, Individual locations are iteratively merged to form clusters via the complete linkage algorithm. Here the output of the algorithm is visualized as a dendogram. b, We add levels to the hierarchical partition by maximizing the likelihood of the container model one level at a time: at the first iteration (top), we find the container size (x axis) corresponding to the dendogram cut (dashed line) that minimizes the negative likelihood (y axis), if any. We proceed by adding more dendogram cuts (middle and bottom), and thus hierarchical levels, until the likelihood can not be further improved. c, The dendogram cuts correspond to a hierarchical partitioning of individual locations. We evaluate the ability of the algorithm to recover the original parameters using 5,000 synthetic traces of 3,000 locations. d, Distribution of the difference between the number of recovered and original levels. The difference is 0 in 70% of the cases. e, Probability density associated with the cophenetic similarity between the original and recovered hierarchical structure. The dashed line corresponds to the median value. f, Probability density associated to the relative difference |popr|/pr between original (po) and recovered (pr) entries of the matrix p.

Extended Data Fig. 4 The container model generates realistic synthetic traces.

a, e, The distribution of displacements for the entire population. b, f, The median individual radius of gyration versus the number of displacements. c, g, The average visitation frequency versus the rank of individuals’ locations. d, h, The distribution of the difference between the real entropy Stemp and the uncorrelated entropy Sunc across individuals. Results are shown for real traces (black line, dots), and traces generated by various models (see legend), for dataset D1 (ad) and D2 (eh). In a, c, d, e, g and h, the filled areas for the synthetic traces include two standard deviations around the mean computed across 1,000 simulations for each user. In b and f, the filled areas include the interquartile range. For each individual, we fitted the models considering a training period of one year. The data used here for validation corresponds to the 50 individual displacements following the training period.

Extended Data Fig. 5 Number of hierarchical levels recovered from traces.

Distribution of the number of hierarchical levels found by the container model for trajectories in the D1 dataset (plain black line), the D2 dataset (dashed black line), and 1,000 synthetic traces generated by the EPR model9 (blue line) and the memory EPR (m-EPR) model (green line).

Extended Data Table 1 The distribution of container sizes is not scale free
Extended Data Table 2 The distribution of time spent within container is not scale free
Extended Data Table 3 Characteristics of the lognormal distributions of container sizes
Extended Data Table 4 The container model describes unseen data better than other individual mobility models

Supplementary information

Supplementary Information

This file contains Supplementary Notes 1–4 and Supplementary Tables S1–S3.

Peer Review File

Source data

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alessandretti, L., Aslak, U. & Lehmann, S. The scales of human mobility. Nature 587, 402–407 (2020). https://doi.org/10.1038/s41586-020-2909-1

Download citation

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing