A universal model for mobility and migration patterns

Journal name:
Date published:
Published online

Introduced in its contemporary form in 1946 (ref. 1), but with roots that go back to the eighteenth century2, the gravity law1, 3, 4 is the prevailing framework with which to predict population movement3, 5, 6, cargo shipping volume7 and inter-city phone calls8, 9, as well as bilateral trade flows between nations10. Despite its widespread use, it relies on adjustable parameters that vary from region to region and suffers from known analytic inconsistencies. Here we introduce a stochastic process capturing local mobility decisions that helps us analytically derive commuting and mobility fluxes that require as input only information on the population distribution. The resulting radiation model predicts mobility patterns in good agreement with mobility and transport patterns observed in a wide range of phenomena, from long-term migration patterns to communication volume between different regions. Given its parameter-free nature, the model can be applied in areas where we lack previous mobility measurements, significantly improving the predictive accuracy of most of the phenomena affected by mobility and transport processes11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23.

At a glance


  1. The radiation model.
    Figure 1: The radiation model.

    a, To demonstrate the limitations of the gravity law we highlight two pairs of counties, one in Utah (UT) and the other in Alabama (AL), with similar origin (m, blue) and destination (n, green) populations and comparable distance r between them (see bottom left table). The gravity law predictions were obtained by fitting equation (1) to the full commuting data set, recovering the parameters [α, β, γ] = [0.30, 0.64, 3.05] for r<119km, and [0.24, 0.14, 0.29] for r>119km of ref. 14. The fluxes predicted by (1) are the same because the two county pairs have similar m, n and r (top right table). Yet the US census 2000 reports a flux that is an order of magnitude greater between the Utah counties, a difference correctly captured by the radiation model (b, c). b, The definition of the radiation model: an individual (for example, living in Saratoga County, New York) applies for jobs in all counties and collects potential employment offers. The number of job opportunities in each county (j) is nj/njobs, chosen to be proportional to the resident population nj. Each offer’s attractiveness (benefit) is represented by a random variable with distribution p(z), the numbers placed in each county representing the best offer among the nj/njobs trials in that area. Each county is marked in green (red) if its best offer is better (lower) than the best offer in the home county (here z = 10). c, An individual accepts the closest job that offers better benefits than his home county. In the shown configuration the individual will commute to Oneida County, New York, the closest county whose benefit z = 13 exceeds the home county benefit z = 10. This process is repeated for each potential commuter, choosing new benefit variables z in each case.

  2. Comparing the predictions of the radiation model and the gravity law.
    Figure 2: Comparing the predictions of the radiation model and the gravity law.

    a, National mobility fluxes with more than ten travellers originating from New York County (left panels) and the high intensity fluxes (over 1,100 travellers) within the state of New York (right panels). Arrows represent commuters fluxes, the colour capturing flux intensity: black, 10 individuals (fluxes below ten travellers are not shown for clarity), white, >10,000 individuals. The top panels display the fluxes reported in US census 2000, the central panels display the fluxes fitted by the gravity law with14 f(r) = rγ, and the bottom panels display the fluxes predicted by the radiation model. bd, Comparing the measured flux, , with the predicted flux, and , for each pair of counties. We compare the census data with two formulations of the gravity law, f(r) = edr (c) and f(r) = rγ (b), and with the radiation model (d). Grey points are scatter plot for each pair of counties. A box is coloured green if the line y = x lies between the 9th and the 91st percentiles in that bin and is red otherwise. The black circles correspond to the mean number of predicted travellers in that bin. e, Probability of a trip between two counties that are at distance r (in km) from each other, Pdist(r). f, Probability of a trip towards a county with population n, Pdest(n). g, The number of commuters in a county, Ti, is proportional to its population, mi. hj, Conditional probability p(T|m,n,r) to observe a flow of T individuals from a location with population m to a location with population n at a distance r for three triplets (m,n,r). The gravity law predicts a highly peaked distribution around the average value , in disagreement with census data and the radiation model, which both show a broad distribution.

  3. Beyond commuting.
    Figure 3: Beyond commuting.

    ac, Testing the radiation model on hourly trips extracted from a mobile phone database of a western European country. The anonymized billing records29, 30 cover the activity of approximately ten million subscribers. We analysed a 6-month period, recording the user locations with tower resolution hourly between 7am and 10pm, identifying all trips between municipalities. a, Probability of a trip between two municipalities at distance r, Pdist(r), shown for 14-hourly time intervals. Radiation model predictions are solid lines; gravity law’s aggregated fit over 24h is a red line with empty squares. b, Probability of a trip towards a municipality with population n, Pdest(n). c, Comparing the measured flux, , with the predicted flux, , for each pair of municipalities with , for commuting trips extracted by identifying each user’s home and workplace from the locations where the user made the most calls. df, Testing (2) on long-term migration patterns, capturing the number of individuals that relocated from one US county to another during tax years 2007–2008 as reported by the US Internal Revenue Service. gi, Phone call volume between municipalities extracted from the anonymized mobile phone database. The number of phone calls between users living in different municipalities during a period of 4weeks resulted in 38,649,153 calls placed by 4,336,217 users. We aggregated the data to obtain the total number of calls between every pair of municipalities. jl, Commodity flows in the US extracted from the Freight Analysis Framework (FAF), which offers a comprehensive picture of freight movement among US states and major metropolitan areas by all modes of transportation. For each data set we measured the quantities discussed in ac.

  4. Unveiling the hidden self-similarity in human mobility.
    Figure 4: Unveiling the hidden self-similarity in human mobility.

    a, The probability ps(>s|m) to observe a trip from a location with population m to a destination in the region beyond a population s from the origin (m varies between 200 and 2,000,000). b, According to the radiation model ps(>s|m) = 1/(1+s/m), a homogeneous function of the ratio s/m. Plotting ps(>s|m) versus s/m, the curves approach the theoretical result y = 1/(1+x). c, The collapse improves if we account for the home field advantage in job search by always adding = 35,000 to the population of the commuter’s home county.


  1. Zipf, G. K. The P1P2/D hypothesis: on the intercity movement of persons. Am. Sociol. Rev. 11, 677686 (1946)
  2. Monge, G. Mémoire sur la Théorie des Déblais et de Remblais. Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année 666704 (De l’Imprimerie Royale, 1781)
  3. Barthélemy, M. Spatial networks. Phys. Rep. 499, 1101 (2010)
  4. Erlander, S. & Stewart, N. F. The Gravity Model in Transportation Analysis: Theory and Extensions (VSP, 1990)
  5. Jung, W. S., Wang, F. & Stanley, H. E. Gravity model in the Korean highway. EPL 81, 48005 (2008)
  6. Thiemann, C., Theis, F., Grady, D., Brune, R. & Brockmann, D. The structure of borders in a small world. PLoS ONE 5, e15422 (2010)
  7. Kaluza, P., Kölzsch, A., Gastner, M. T. & Blasius, B. The complex network of global cargo ship movements. J. R. Soc. Interf. 7 10931103 (2010)
  8. Krings, G., Calabrese, F., Ratti, C. & Blondel, V. D. Urban gravity: a model for inter-city telecommunication flows. J. Stat. Mech. 2009, L07003 (2009)
  9. Expert, P., Evans, T. S., Blondel, V. D. & Lambiotte, R. Uncovering space-independent communities in spatial networks. Proc. Natl Acad. Sci. USA 108, 76637668 (2011)
  10. Pöyhönen, P. A tentative model for the volume of trade between countries. Weltwirtschaftliches Arch. 90, 93100 (1963)
  11. Balcan, D. et al. Multiscale mobility networks and the spatial spreading of infectious diseases. Proc. Natl Acad. Sci. USA 106, 2148421489 (2009)
  12. Helbing, D. Traffic and related self-driven many-particle systems. Rev. Mod. Phys. 73, 10671141 (2001)
  13. Colizza, V., Barrat, A., Barthélemy, M. & Vespignani, A. The role of the airline transportation network in the prediction and predictability of global epidemics. Proc. Natl Acad. Sci. USA 103, 20152020 (2006)
  14. Viboud, C. et al. Synchrony, waves, and spatial hierarchies in the spread of influenza. Science 312, 447451 (2006)
  15. Ferguson, N. M. et al. Strategies for mitigating an influenza pandemic. Nature 442, 448452 (2006)
  16. Xu, X. J., Zhang, X. & Mendes, J. F. F. Impacts of preference and geography on epidemic spreading. Phys. Rev. E 76, 056109 (2007)
  17. Lind, P. G., Da Silva, L. R., Andrade, J. S., Jr & Herrmann, H. J. Spreading gossip in social networks. Phys. Rev. E 76, 036117 (2007)
  18. Roth, C., Kang, S. M., Batty, M. & Barthélemy, M. Structure of urban movements: polycentric activity and entangled hierarchical flows. PLoS ONE 6, e15923 (2011)
  19. Makse, H. A., Havlin, S. & Stanley, H. E. Modelling urban growth patterns. Nature 377, 608612 (1995)
  20. Bettencourt, L. M. A., Lobo, J., Helbing, D., Kühnert, C. & West, G. B. Growth, innovation, scaling, and the pace of life in cities. Proc. Natl Acad. Sci. USA 104, 73017306 (2007)
  21. Batty, M. The size, scale, and shape of cities. Science 319, 769771 (2008)
  22. Garlaschelli, D., Di Matteo, T., Aste, T., Caldarelli, G. & Loffredo, M. I. Interplay between topology and dynamics in the World Trade Web. The Eur. Phys. J. B 57, 159164 (2007)
  23. Eubank, S. et al. Modelling disease outbreaks in realistic urban social networks. Nature 429, 180184 (2004)
  24. Krueckeberg, D. A. & Silvers, A. L. Urban Planning Analysis: Methods and Models (Wiley, 1974)
  25. Wilson, A. G. The use of entropy maximising models in the theory of trip distribution, mode split and route split. J. Transp. Econ. Policy 108126 (1969)
  26. Stouffer, S. A. Intervening opportunities: a theory relating mobility and distance. Am. Sociol. Rev. 5, 845867 (1940)
  27. Block, H. D. & Marschak, J. Random Orderings and Stochastic Theories of Responses (Cowles Foundation, 1960)
  28. Rogerson, P. A. Parameter estimation in the intervening opportunities model. Geogr. Anal. 18, 357360 (1986)
  29. González, M. C., Hidalgo, C. A. & Barabási, A. L. Understanding individual human mobility patterns. Nature 453, 779782 (2008)
  30. Onnela, J. P. et al. Structure and tie strengths in mobile communication networks. Proc. Natl Acad. Sci. USA 104, 73327336 (2007)

Download references

Author information


  1. Center for Complex Network Research and Department of Physics, Biology and Computer Science, Northeastern University, Boston, Massachusetts 02115, USA

    • Filippo Simini &
    • Albert-László Barabási
  2. Dipartimento di Fisica “G. Galilei”, Università di Padova, CNISM and INFN, via Marzolo 8, 35131 Padova, Italy

    • Filippo Simini &
    • Amos Maritan
  3. Institute of Physics, Budapest University of Technology and Economics, Budafoki út 8, Budapest, H-1111, Hungary

    • Filippo Simini
  4. MIT, Department of Civil and Environmental Engineering, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA

    • Marta C. González
  5. Center for Cancer Systems Biology, Dana-Farber Cancer Institute, Boston, Massachusetts 02115, USA

    • Albert-László Barabási
  6. Department of Medicine, Brigham and Women’s Hospital, Harvard Medical School, Boston, Massachusetts 02115, USA

    • Albert-László Barabási


All authors designed and did the research. F.S. analysed the empirical data and performed the numerical calculations. A.M. and F.S. developed the analytical calculations. A.-L.B. was the lead writer of the manuscript.

Competing financial interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to:

Author details

Supplementary information

PDF files

  1. Supplementary Information (4.4M)

    This file contains Supplementary Text and Data 1-9 (see Contents for more details), additional references and Supplementary Figures 1-8 with legends.


  1. Report this comment #42022

    Xinqi Zheng said:

    Special issue: China's mobility and migration.Social mobility is of great significance to achieve social equality. A simple theory can't solve all problems for complex social mobility?Nature, 484, 40-41; 2012?. The radiation model may provide a route to further exploration?Nature?484, 96?100?2012?.
    China's urbanization process is experiencing unprecedented population mobility and migration. Few forces have influenced the modern world economy as much as Chinese migration. The story of migration is the story of modern China, as migrant workers have transformed China?s economy.
    In 2010,China's sixth national population census data show that the number of floating population 221 million, accounts for the population 16.5%(http://www.chinapop.gov.cn/xwzx/rkxw/201104/t20110428_356999.html ). They spend most of the year away from their home town or village. Most migrants across provinces move in search of work. At present, more and more China?s peasant workers are choosing to work near the hometown because of the hometown more attractive to them. The Chinese government?s population-planning commission forecasts another 100 million rural residents could move to cities by 2020(http://www.chinapop.gov.cn/rdzt/zgldrkfzbb/mtjj/201110/t20111012_375789.html ).The Spring Festival every year there are about 2 billion people for the national mobility. China's population mobility and migration is very different than others.
    Facing the situation in China, the radiation model is ragged. China's national condition is different from any other country. A pertinent question for future work is why China and western countries are different. Separating the population mobility and the population migration, and focus on urban and rural and cross provinces should be an entry point for addressing this issue.
    Zheng,Xinqi.China University of Geosciences in Beijing?China.zxqsd@126.com

Subscribe to comments

Additional data