The dynamic spatial redistribution of individuals is a key driving force of various spatiotemporal phenomena on geographical scales. It can synchronize populations of interacting species, stabilize them, and diversify gene pools [1] [2] [3] . Human travel, for example, is responsible for the geographical spread of human infectious disease [4] [5] [6] [7] [8] [9] . In the light of increasing international trade, intensified human mobility and the imminent threat of an influenza A epidemic 10 , the knowledge of dynamical and statistical properties of human travel is of fundamental importance. Despite its crucial role, a quantitative assessment of these properties on geographical scales remains elusive, and the assumption that humans disperse diffusively still prevails in models. Here we report on a solid and quantitative assessment of human travelling statistics by analysing the circulation of bank notes in the United States. Using a comprehensive data set of over a million individual displacements, we find that dispersal is anomalous in two ways. First, the distribution of travelling distances decays as a power law, indicating that trajectories of bank notes are reminiscent of scalefree random walks known as Lévy flights. Second, the probability of remaining in a small, spatially confined region for a time T is dominated by algebraically long tails that attenuate the superdiffusive spread. We show that human travelling behaviour can be described mathematically on many spatiotemporal scales by a two-parameter continuous-time random walk model to a surprising accuracy, and conclude that human travel on geographical scales is an ambivalent and effectively superdiffusive process. Quantitative aspects of dispersal in ecology are based on the dispersal curve, which quantifies the relative frequency of travel distances of individuals as a function of geographical distance 11 . A large class of dispersal curves (for example, exponential, gaussian, stretched exponential) permits the identification of a typical length scale by the variance of the displacement length or equivalent quantities. When interpreted as the probability P(r) of finding a displacement of length r in a short time dt, the existence of a typical length scale often justifies the description of dispersal in terms of diffusion equations on large spatiotemporal scales. If, however, P(r) lacks a typical length scale, that is P(r) , r 2(1þb) with b , 2, the diffusion approximation fails. In physics, random processes with such a single-step distribution are known as Lévy flights [12] [13] [14] [15] [16] (see Supplementary Information) . Recently, the related notion of longdistance-dispersal (LDD) has been established in dispersal ecology 17 , taking into account the observation that dispersal curves of a number of species show power-law tails owing to long-range movements [18] [19] [20] [21] . (In ecological literature, the term 'dispersal' is commonly used in the context of the spatial displacement of individuals of a species between their geographical origin of birth and the location of their first breeding place. Here we use the term dispersal to refer to geographical displacements that occur on much shorter timescales, that is, due to travel by various means of transportation.) Nowadays, humans travel on many spatial scales, ranging from a few to thousands of kilometres over short periods of time. The direct quantitative assessment of human movements, however, is difficult, and a statistically reliable estimate of human dispersal comprising all spatial scales does not exist. The central aim of this work is to use data collected at online bill-tracking websites (which monitor the worldwide dispersal of large numbers of individual bank notes) to infer the statistical properties of human dispersal with very high spatiotemporal precision. Our analysis of human movement is based on the trajectories of 464,670 dollar bills obtained from the bill-tracking system www.wheresgeorge.com. We analysed the dispersal of bank notes in the United States, excluding Alaska and Hawaii. The core data consists of 1,033,095 reports to the bill-tracking website. From these reports we calculated the geographical displacements r ¼ jx 2 2 x 1 j between a first (x 1 ) and secondary (x 2 ) report location of a bank note and the elapsed time T between successive reports. In order to illustrate qualitative features of bank note trajectories, Fig. 1b depicts short-time trajectories (T , 14 days) originating from three major US cities: Seattle, New York and Jacksonville. After their initial entry into the tracking system, most bank notes are next reported in the vicinity of the initial entry location, that is jx 2 2 x 1 j # 10 km (Seattle, 52.7%; New York, 57.7%; Jacksonville, 71.4%). However, a small but considerable fraction is reported beyond a distance of 800 km (Seattle, 7.8%; New York, 7.4%; Jacksonville, 2.9%). From a total of 20,540 short-time trajectories originating across the United States, we measured the probability P(r) of traversing a distance r in a time interval dT of 1-4 days (Fig. 1c) . A total of 14,730 (that is, a fraction Q ¼ 0.71) secondary reports occurred outside a short range radius L min ¼ 10 km. Between L min and the approximate average East-West extension of the United States, L max < 3,200 km, the kernel shows power-law behaviour P(r) , r 2(1þb) with an exponent b ¼ 0.59^0.02. For r , L min , P(r) increases linearly with r, which implies that displacements are distributed uniformly inside the disk jx 2 2 x 1 j # L min . We measured P(r) for three classes of initial entry locations: highly populated metropolitan areas (191 sites, local population N loc . 120,000), cities of intermediate size (1,544 sites, local population 120,000 . N loc . 22,000) and small towns (23,640 sites, local population N loc , 22,000), comprising 35.7%, 29.1% and 25.2% of the entire population of the United States, respectively. The inset in Fig. 1c shows P(r) for these classes. Despite systematic deviations for short distances, all distributions show an algebraic tail with the same exponent b < 0.6, which confirms that the observed power-law is an intrinsic and universal property of dispersal. However, the situation is more complex. If we assume that the dispersal of bank notes can be described by a Lévy flight with a shorttime probability distribution P(r), we can estimate the time T eq for an initially localized ensemble of bank notes to reach the stationary distribution 22 (maps in Fig. 1a) , obtaining a value of T eq < 68 days (see Supplementary Information) . Thus, after 2-3 months, bank notes should have reached an equilibrium distribution. Surprisingly, the long-time dispersal data does not reflect a relaxation within this LETTERS
Recently, Brockmann et al. REF analyzed human traveling patterns from the circulation patterns of bank notes, in the scale of several hundred to thousand kilometers, and proved that human long-distance traveling patterns at a macro scale show Levy-walk patterns.
The scaling laws of human travel
Nature
Nature
Geography Physics Medicine