Abstract
Human mobility impacts many aspects of a city, from its spatial structure1,2,3 to its response to an epidemic4,5,6,7. It is also ultimately key to social interactions8, innovation9,10 and productivity11. However, our quantitative understanding of the aggregate movements of individuals remains incomplete. Existing models—such as the gravity law12,13 or the radiation model14—concentrate on the purely spatial dependence of mobility flows and do not capture the varying frequencies of recurrent visits to the same locations. Here we reveal a simple and robust scaling law that captures the temporal and spatial spectrum of population movement on the basis of large-scale mobility data from diverse cities around the globe. According to this law, the number of visitors to any location decreases as the inverse square of the product of their visiting frequency and travel distance. We further show that the spatio-temporal flows to different locations give rise to prominent spatial clusters with an area distribution that follows Zipf’s law15. Finally, we build an individual mobility model based on exploration and preferential return to provide a mechanistic explanation for the discovered scaling law and the emerging spatial structure. Our findings corroborate long-standing conjectures in human geography (such as central place theory16 and Weber’s theory of emergent optimality10) and allow for predictions of recurrent flows, providing a basis for applications in urban planning, traffic engineering and the mitigation of epidemic diseases.
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Data availability
Raw mobility data are not publicly available to preserve privacy. Grid-cell-level data to reproduce the findings of this study can be requested from the corresponding author.
Code availability
The code to replicate this research can be requested from the corresponding author.
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Acknowledgements
We thank L. M. A. Bettencourt, R. Sinatra, P. Herthogs, G. Du, Y. Qin and Y. Liu for helpful discussions; W. Cao for assistance with performing the CCA analysis; and S. Grauwin for providing the MATLAB code for the radiation model. We acknowledge Airsage, ORANGE/SONATEL and Singtel for providing the data. This work was supported by the National Science Foundation (grant number PHY1838420), the AT&T Foundation, the MIT SMART programme, the MIT CCES programme, Audi Volkswagen, BBVA, Ericsson, Ferrovial, GE, the MIT Senseable City Lab Consortium, the John Templeton Foundation (grant number 15705), the Eugene and Clare Thaw Charitable Trust, Toby Shannan, the Charities Aid Foundation of Canada, the US Army Research Office Minerva Programme (grant number W911NF-12-1-0097), the Singapore National Research Foundation (FI 370074016) and the National Natural Science Foundation of China (grant number 41801299).
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G.B.W., C.R., M. Schläpfer, L.D., K.O. and P.S. designed the research. L.D., M. Schläpfer, K.O. and M.V. processed and analysed the data. G.B.W. and M. Schläpfer identified the data collapse to a single, universal curve. K.O., L.D., S.A. and P.S. developed the PEPR model. H.S., M. Schläpfer and G.B.W. developed the theoretical argument. M.V. and P.S. tested the Fermat–Toricelli–Weber metric. L.D., M. Schläpfer, K.O., M. Szell, P.S., G.B.W. and C.R. wrote the paper. C.R. and G.B.W. contributed equally as senior authors. All authors discussed the results and reviewed the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 The spatio-temporal structure of movement in cities.
Panels show visitor influx maps for Greater Boston for different parameters (r, f). The colour of each grid cell (500 m × 500 m) indicates the value of the spectral flow ρ. Remarkably, visitor influx maps for the same quantity v = rf are nearly identical, as is clear from viewing along the diagonals indicated by the coloured arrows in the figure. Hence, doubling the visitation frequency f (from top row to bottom row) results in the same quantitative decrease of the influx as doubling the travel distance r (from left column to right column).
Extended Data Fig. 2 Empirical power-law exponents of the distance–frequency distribution.
a, Histogram of the exponents for all locations in the Greater Boston area. The values were determined using ordinary least squares minimization to a linear relation of the logarithmically transformed variables. The red line shows η = 2, consistent with our theoretical argument. b, Corresponding histogram of the R2 values.
Extended Data Fig. 3 Universality of the scaling relation ρ ∝ (rf)−2 across Greater Boston.
The panels depict the data for individual locations (500 m × 500 m grid cells), ranked according to the total number of visitors from neighbouring cells. Shown are locations of rank 1–30 (from top left to bottom right). The geographic coordinates of each location (latitude and longitude of the centre point of the grid cell) are indicated. The straight lines denote the inverse square of rf (slope = −2), consistent with our theoretical argument.
Extended Data Fig. 4 Universality of the scaling relation ρ ∝ (rf)−2 across Portugal.
The panels depict the data for individual locations (1 km × 1 km grid cells), ranked according to the total number of visitors from neighbouring cells. Shown are locations of rank 1–30 (from top left to bottom right). The geographic coordinates of each location (latitude and longitude of the centre point of the grid cell) are indicated. The straight lines denote the inverse square of rf (slope = −2), consistent with our theoretical argument.
Extended Data Fig. 5 Universality of the scaling relation ρ ∝ (rf)−2 across Dakar.
The panels depict the data for individual locations (1 km × 1 km grid cells), ranked according to the total number of visitors from neighbouring cells. Shown are locations of rank 1–30 (from top left to bottom right). The geographic coordinates of each location (latitude and longitude of the centre point of the grid cell) are indicated. The straight lines denote the inverse square of rf (slope = −2), consistent with our theoretical argument.
Extended Data Fig. 6 Universality of the scaling relation ρ ∝ (rf)−2 across Singapore.
The panels depict the data for individual locations (500 m × 500 m grid cells), ranked according to the total number of visitors from neighbouring cells. Shown are locations of rank 1–30 (from top left to bottom right). The geographic coordinates of each location (latitude and longitude of the centre point of the grid cell) are indicated. The straight lines denote the inverse square of rf (slope = −2), consistent with our theoretical argument.
Extended Data Fig. 7 Simulation results of the EPR model.
a, b, Generated number of visits (a) and attractiveness values μi (b). c, d, The EPR model generates the rf scaling of the population flows with a scaling exponent that is in remarkable agreement with the data. The generated visitor counts, Ni(r, f), are shown in c, and the resulting rf scaling of the spectral flows, ρi(r, f), is shown in d. The generated attractiveness values μi are rather homogeneous and uniform across space, which is in contrast to the empirical data (b). Model parameters are taken from Song et al.32 (Methods).
Extended Data Fig. 8 Estimation of the magnitude of flows from population density ρpop.
The schematic shows a zoom-in on the immediate vicinity of a destination location j (small values of r), where it is reasonable to assume that ρpop(j) ≈ constant. Hence, the local population density imposes an upper bound on the influx, ∫ρjdf ≤ ρpop(j). A simple boundary condition of the continuous model then dictates that the minimum visiting frequency of all individuals living directly on the boundary (each being assigned to a point at r = rj) assumes the minimum frequency with which the individuals living inside the attracting location return home, fmin ≈ fhome. The minimum distance rmin for locations from which individuals visit with minimum frequency fmin < fhome increases with decreasing value of fmin.
Extended Data Fig. 9 CPT and radius of attraction.
a, Schematics of CPT, showing the spatial arrangement of three tiers of centres (see Supplementary Information for details). This hierarchical arrangement of central places results in the most efficient transport network. b–f, Average travel distance per visit ⟨r⟩f to perform activities with fixed visiting frequency f across all locations in Greater Boston (b), Singapore (c), Dakar (d), Abidjan (e) and Lisbon (f). We find a clear inverse relation, ⟨r⟩f ∝ 1/f. The quantity ⟨r⟩f can be interpreted as the characteristic distance associated with the level of specialization of the functions provided by the locations.
Extended Data Fig. 10 Fermat–Torricelli–Weber (FTW) efficiency of collective human movements.
a, The schematic shows how the FTW efficiency is computed (see Supplementary Information). The effective distance travelled by the visitors of a specific location (cell) can be minimized by moving it on the grid. The efficiency is \(\Delta {\mathcal{D}}/{\mathcal{D}}\), which is the ratio between the reduction of the effective travel distance of all visitors when moving the cell from its actual location to the optimum FTW point (\(\Delta {\mathcal{D}}\)) and the actual effective travel distance of all visitors to that cell (\({\mathcal{D}}\)). b, Density plots representing the number of cells with a given number of visits and FTW efficiency for the Greater Boston area (for the month of August 2009). The FTW efficiency is computed for each cell based on visits made by visitors who live at distances larger than a given threshold value rthr. For rthr = 0 (top left), the density of locations is particularly high where the FTW efficiency is very high. As the number of visits is increased, the distribution becomes narrower and the FTW efficiency increases. This pattern is generally also valid for larger values of rthr but becomes weaker.
Supplementary information
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Schläpfer, M., Dong, L., O’Keeffe, K. et al. The universal visitation law of human mobility. Nature 593, 522–527 (2021). https://doi.org/10.1038/s41586-021-03480-9
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DOI: https://doi.org/10.1038/s41586-021-03480-9
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