Despite their importance for urban planning, traffic forecasting and the spread of biological and mobile viruses, our understanding of the basic laws governing human motion remains limited owing to the lack of tools to monitor the time-resolved location of individuals. Here we study the trajectory of 100,000 anonymized mobile phone users whose position is tracked for a six-month period. We find that, in contrast with the random trajectories predicted by the prevailing Lévy flight and random walk models, human trajectories show a high degree of temporal and spatial regularity, each individual being characterized by a time-independent characteristic travel distance and a significant probability to return to a few highly frequented locations. After correcting for differences in travel distances and the inherent anisotropy of each trajectory, the individual travel patterns collapse into a single spatial probability distribution, indicating that, despite the diversity of their travel history, humans follow simple reproducible patterns. This inherent similarity in travel patterns could impact all phenomena driven by human mobility, from epidemic prevention to emergency response, urban planning and agent-based modelling.

Despite their importance for urban planning [1], traffic forecasting [2], and the spread of biological [3, 4, 5] and mobile viruses [6], our understanding of the basic laws governing human motion remains limited thanks to the lack of tools to monitor the time resolved location of individuals. Here we study the trajectory of 100, 000 anonymized mobile phone users whose position is tracked for a six month period. We find that in contrast with the random trajectories predicted by the prevailing Lévy flight and random walk models [7], human trajectories show a high degree of temporal and spatial regularity, each individual being characterized by a time independent characteristic length scale and a significant probability to return to a few highly frequented locations. After correcting for differences in travel distances and the inherent anisotropy of each trajectory, the individual travel patterns collapse into a single spatial probability distribution, indicating that despite the diversity of their travel history, humans follow simple reproducible patterns. This inherent similarity in travel patterns could impact all phenomena driven by human mobility, from epidemic prevention to emergency response, urban planning and agent based modeling.Given the many unknown factors that influence a population's mobility patterns, ranging from means of transportation to job and family imposed restrictions and priorities, human trajectories are often approximated with various random walk or diffusion models [7,8]. Indeed, early measurements on albatrosses, bumblebees, deer and monkeys [9, 10] and more recent ones on marine predators [11] suggested that animal trajectory is approximated by a Lévy flight [12,13], a random walk whose step size ∆r follows a power-law distribution P (∆r) ∼ ∆r −(1+β) with β < 2. While the Lévy statistics for some animals require further study [14], Brockmann et al. [7] generalized this finding to humans, documenting that the distribution of distances between consecutive sight-

In analogy with Newton's law of gravity, the gravity law assumes that the number of individuals T ij that move between locations i and j per unit time is proportional to some power of the population of the source (m i ) and destination (n j ) locations, and decays with the distance r ij between them as ,where ! and ! are adjustable exponents and the deterrence function is chosen to fit the empirical data. Occasionally T ij is interpreted as the probability rate of individuals !of traveling from i to j, or an effective coupling between the two locations 24 . Despite its widespread use, the gravity law has notable limitations:i) We lack a rigorous derivation of (1). While entropy maximization 25 leads to (1) with ! = " = 1 , it fails to offer the functional form of f(r).ii) Lacking theoretical guidance, practitioners use a range of deterrence functions (power law or exponential) and up to nine parameters to fit the empirical data 5,7,8,11,14 .iii) As (1) requires previous traffic data to fit the parameters [ ], it is unable to predict mobility in regions where we lack systematic traffic data, areas of major interest in modeling of infectious diseases.iv)The gravity law has systematic predictive discrepancies. Indeed, in Fig. 1a we highlight two pairs of counties with similar origin and destination populations and comparable distance, so according to (1) the flux between them should be the same. Yet, the US census (see SI) documents an order of magnitude difference between the two fluxes: only 6 individuals commute between the two Alabama counties, while 44 in Utah.v) Equation (1) predicts that the number of commuters increases without limit as we increase the destination population n j , yet the number of commuters cannot exceed the source population m i , highlighting the gravity law's analytical inconsistency (see SI, Sect. 4).vi) Being deterministic, the gravity law cannot account for fluctuations in the number of travelers between two locations.Motivated by these known limitations, alternative approaches like the intervening opportunity model 26 or the random utility model 27 (SI, Sect. 7) have been proposed.While derived from first principles, these models continue to contain context specific tunable parameters, and their predictive power is at best comparable to the gravity law 28 .Here we introduce a modelling framework that relies on first principles and overcomes the problems (i) -(vi) of the gravity law. While commuting is a daily process, its source and destination is determined by job selection, a decision made over longer timescales. Using the natural partition of a country into counties (for which commuting data are collected), we assume that job selection consists of two steps ( Fig. 1 b, c):An individual seeks job offers from all counties, including his/her home county.The number of employment opportunities in each county is proportional to the resident population, n, assuming that there is one job opening for every n jobs individuals. We capture the benefits of a potential employment opportunity...

Novel aspects of human dynamics and social interactions are investigated by means of mobile phone data. Using extensive phone records resolved in both time and space, we study the mean collective behavior at large scales and focus on the occurrence of anomalous events. We discuss how these spatiotemporal anomalies can be described using standard percolation theory tools. We also investigate patterns of calling activity at the individual level and show that the interevent time of consecutive calls is heavy-tailed. This finding, which has implications for dynamics of spreading phenomena in social networks, agrees with results previously reported on other human activities.