The power law in human mobility is a mixture artifact: evidence from a pandemic natural experiment

Abstract

For nearly two decades, human mobility has been read as scale free. Displacement distributions follow heavy tails that look like truncated power laws, traced to individual Lévy flights. A rival account holds that movement within each spatial container is lognormal, and the aggregate power law is an artifact of mixing containers of different sizes. The two fit the same aggregate data, so the debate has been hard to settle. We use the COVID-19 lockdowns as a natural experiment that removes long-distance travel and leaves local travel intact. We analyze 2.1 billion displacements from 4.4 million mobile-phone users across three distinct periods. The aggregate exponent increases under lockdown, from 1.66 to 1.74. Resampling the pre-lockdown traveling population to match the lockdown population reproduces that shift on its own, so we must look at the individual level to decides the question. Lognormal classification is a stable attractor (81\% retained) while the power-law classification is fragile (32\% retained), and the users who switch are the ones whose travel range collapsed the most. Matching the sample size we find that single users' tails reject the power law and pooled mixtures of equal size pass, so the heavy tail behavior originates in the aggregate behavior and not on the individual. Tail tests find no power-law threshold at full sample size, and the apparent power law disappears above ten thousand points. A level mixture rebuilds the aggregate ($R^2$ up to 0.98), the radius-of-gyration collapse fails and worsens under lockdown (CV $= 0.62$ to $0.76$), and the steepening concentrates in wide-ranging users ($P = 0.0003$). The power law of human travel is a feature of aggregation, not of individual movement.