A power-law-like pattern in pooled human travel may be less a signature of how each person moves than a byproduct of combining people with different spatial ranges, according to a new analysis of Chilean mobile-phone data. During full lockdown, the pooled exponent rose from 1.657 to 1.737, while median displacement edged down from 1.98 to 1.90 kilometres and median radius of gyration—a measure of spatial range—fell from 7.31 to 2.29 kilometres.
Two explanations for one pattern
The study set out to distinguish between two explanations. H1 says the population-level power law reflects a scale-free individual process. H2 says the aggregate pattern emerges when lognormal movement is mixed across spatial scales.
The analysis covered 2.1 billion displacement records from 4.4 million anonymised mobile-phone users in Chile. The records were grouped into three periods: 464 million before lockdown, 1.3 billion during partial lockdown and 354 million during full lockdown.
The individual-level evidence moved in the same direction
Within-user comparisons pointed away from a stable individual power law. In a balanced panel of 1.52 million users, each with at least 50 displacements in every period, the share whose movements favoured a lognormal model rose from 49.5% before lockdown to 60.9% during full lockdown. The share favouring a truncated power law fell from 18.1% to 13.9%.
The classifications were not equally persistent. Among users initially classified as lognormal, 78.7% retained that classification, compared with 36.9% of users initially classified as power law. Switching from lognormal to power law occurred in 4.9% of cases, while switching in the opposite direction occurred in 30.9%.
The paired exponent change was 0.102, with a 95% confidence interval from 0.096 to 0.107. Its standardised size was modest, with Cohen’s d of 0.349, even though the paired test produced a very small P value of 8.8 × 10−251.
Pooling changed what the tail appeared to show
A separate reweighting test showed how much the mix of users could matter. When pre-lockdown users were reweighted to match the full-lockdown distribution of spatial ranges, the fitted exponent rose from 1.673 to 1.814. That composition-only increase of 0.141 was larger than the observed panel increase of 0.065. The authors describe this as a sufficiency result, not a differential test that settles every competing explanation.
The same issue appeared in a matched comparison of tails. Among 1,537 users with at least 300 pre-lockdown displacements, the top 20% of each person’s movements rejected a power law 52.9% of the time. Pooled samples of the same size rejected it 16.7% of the time, while their median tail fraction was 0.281, compared with 0.096 for individual tails.
A full scan across possible thresholds found rejection at 97% to 98% of thresholds in the two-sided test. In the extreme tail, there were only seven distinct displacement values during full lockdown, compared with 67 before lockdown, leaving no sustained power-law regime in that scan.
A layered mixture closely rebuilt the aggregate curve
The researchers also reconstructed the overall distribution by mixing movement patterns at different distance levels. The reconstruction closely matched the aggregate data, with R2 values of 0.977 before lockdown and 0.984 during full lockdown. At neighbourhood distances below 5 kilometres, lognormal fits performed better; the 5-to-50-kilometre range was mixed.
Another test rescaled each user’s movements by their radius of gyration. The reported collapse values rose from 0.616 before lockdown to 0.657 during partial lockdown and 0.762 during full lockdown, which the analysis treats as worsening collapse. The exponent increase was also larger among users with the widest pre-lockdown spatial ranges: 0.184 in the highest range quartile versus 0.051 in the lowest.
A comparison, not a perfect fit
The study’s strongest caveat is that neither candidate distribution was an adequate absolute description for most individual user-period datasets. The bootstrap check generated 500 synthetic datasets for each user-period and treated a model as plausible when its P value exceeded 0.05. Neither the lognormal nor truncated-power-law model passed for more than 95% of user-periods, so the authors frame the exercise as a comparison between models rather than proof that either one is complete.
The paper is an arXiv version 1 preprint dated 24 August 2026. Funding and conflicts of interest are not reported in the supplied text.
Paper data and sources
Original title: The power law in human mobility is a mixture artifact: evidence from a pandemic natural experiment
Authors: Leo Ferres, Bruno Gonçalves
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-24
DOI: Not available
Original paper · Full text