The scales of human mobility


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.

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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 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


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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




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.

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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.

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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

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

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Alessandretti, L., Aslak, U. & Lehmann, S. The scales of human mobility. Nature 587, 402–407 (2020).

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