The scaling laws of human travel

  • Nature volume 439, pages 462465 (26 January 2006)
  • doi:10.1038/nature04292
  • Download Citation



The dynamic spatial redistribution of individuals is a key driving force of various spatiotemporal phenomena on geographical scales. It can synchronize populations of interacting species, stabilize them, and diversify gene pools1,2,3. Human travel, for example, is responsible for the geographical spread of human infectious disease4,5,6,7,8,9. In the light of increasing international trade, intensified human mobility and the imminent threat of an influenza A epidemic10, the knowledge of dynamical and statistical properties of human travel is of fundamental importance. Despite its crucial role, a quantitative assessment of these properties on geographical scales remains elusive, and the assumption that humans disperse diffusively still prevails in models. Here we report on a solid and quantitative assessment of human travelling statistics by analysing the circulation of bank notes in the United States. Using a comprehensive data set of over a million individual displacements, we find that dispersal is anomalous in two ways. First, the distribution of travelling distances decays as a power law, indicating that trajectories of bank notes are reminiscent of scale-free random walks known as Lévy flights. Second, the probability of remaining in a small, spatially confined region for a time T is dominated by algebraically long tails that attenuate the superdiffusive spread. We show that human travelling behaviour can be described mathematically on many spatiotemporal scales by a two-parameter continuous-time random walk model to a surprising accuracy, and conclude that human travel on geographical scales is an ambivalent and effectively superdiffusive process.

  • Subscribe to Nature for full access:



Additional access options:

Already a subscriber?  Log in  now or  Register  for online access.


  1. 1.

    Bullock, J. M., Kenward, R. E. & Hails, R. S. (eds) Dispersal Ecology (Blackwell, Malden, Massachusetts, 2002)

  2. 2.

    Mathematical Biology (Springer-Verlag, New York, 1993)

  3. 3.

    Clobert, J., Danchin, E., Dhondt, A. A. & Nichols, J. D. (eds) Dispersal (Oxford Univ. Press, Oxford, 2001)

  4. 4.

    & Textbook of Influenza (Blackwell, Malden, Massachusetts, 1998)

  5. 5.

    , & Travelling waves and spatial hierarchies in measles epidemics. Nature 414, 716–723 (2001)

  6. 6.

    et al. Dynamics of the 2001 UK foot and mouth epidemic: stochastic dispersal in a heterogeneous landscape. Science 294, 813–817 (2001)

  7. 7.

    Hudson, P. J., Rizzoli, A., Grenfell, B. T. & Heesterbeek, H. (eds) The Ecology of Wildlife Diseases (Oxford Univ. Press, Oxford, 2002)

  8. 8.

    , & Forecast and control of epidemics in a globalized world. Proc. Natl Acad. Sci. USA 101, 15124–15129 (2004)

  9. 9.

    , & Host immunity and synchronized epidemics of syphilis across the United States. Nature 433, 417–421 (2005)

  10. 10.

    & Are we ready for pandemic influenza? Science 302, 1519–1522 (2003)

  11. 11.

    , & Dispersal data and the spread of invading organisms. Ecology 77, 2027–2042 (1996)

  12. 12.

    Shlesinger, M. F., Zaslavsky, G. M. & Frisch, U. (eds) Lévy Flights and Related Topics in Physics (Springer Verlag, Berlin, 1995)

  13. 13.

    , & Beyond Brownian motion. Phys. Today 49, 33–39 (1996)

  14. 14.

    & Lévy flights in inhomogeneous media. Phys. Rev. Lett. 90, 170601 (2003)

  15. 15.

    & The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

  16. 16.

    , & Random-walks with infinite spatial and temporal moments. J. Stat. Phys. 27, 499–512 (1982)

  17. 17.

    The challenges of studying dispersal. Trends Ecol. Evol. 16, 481–483 (2001)

  18. 18.

    et al. Lévy flight search patterns of wandering albatrosses. Nature 381, 413–415 (1996)

  19. 19.

    , , & Lévy walk patterns in the foraging movements of spider monkeys. Behav. Ecol. Sociobiol. 55, 223–230 (2004)

  20. 20.

    , , & The ecology and evolution of seed dispersal: A theoretical perspective. Annu. Rev. Ecol. Evol. Syst. 34, 575–604 (2003)

  21. 21.

    et al. Mechanisms of long-distance dispersal of seeds by wind. Nature 418, 409–413 (2002)

  22. 22.

    Handbook of Stochastic Methods (Springer Verlag, Berlin, 1985)

  23. 23.

    & Random walks on lattices. J. Math. Phys. 6, 167–181 (1965)

Download references


We would like to thank the initiators of the bill tracking system (www.wheresgeorge.com). We thank cabinetmaker D. Derryberry for discussions and for drawing our attention to the wheresgeorge website, and B. Shraiman, D. Cohen and W. Noyes for critical comments on the manuscript. Author Contributions The project idea was conceived by D.B. and L.H., data pre-processing was done by L.H., data analysis by D.B. and L.H., the theory and model was constructed by D.B., and the manuscript was written by D.B., L.H. and T.G.

Author information


  1. Max-Planck Institute for Dynamics and Self-Organisation, Bunsenstr. 10, 37073 Göttingen, Germany

    • D. Brockmann
    •  & T. Geisel
  2. Department of Physics, University of Göttingen, 37073 Göttingen, Germany

    • D. Brockmann
    •  & T. Geisel
  3. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA

    • L. Hufnagel
  4. Bernstein Center for Computational Neuroscience, 37073 Göttingen, Germany

    • T. Geisel


  1. Search for D. Brockmann in:

  2. Search for L. Hufnagel in:

  3. Search for T. Geisel in:

Competing interests

Reprints and permissions information is available at npg.nature.com/reprintsandpermissions. The authors declare no competing financial interests.

Corresponding author

Correspondence to D. Brockmann.

Supplementary information

PDF files

  1. 1.

    Supplementary Notes 1

    Calibration against independent human travel datasets

  2. 2.

    Supplementary Notes 2

    Theory of Lévy flights and continuous time random walks (CTRW)

  3. 3.

    Supplementary Notes 3

    Relaxation time of two dimensional pure Lévy flights in a confined region

  4. 4.

    Supplementary Notes 4

    Similarities between the dispersal of bank notes and infectious diseases


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.