A stochastic model of randomly accelerated walkers for human mobility

ARTICLE Received 20 Jun 2016 | Accepted 15 Jul 2016 | Published 30 Aug 2016 DOI: 10.1038/ncomms12600 OPEN A stochastic model of randomly accelerated walkers for human mobility Riccardo Gallotti1, Armando Bazzani2,3, Sandro Rambaldi2,3 & Marc Barthelemy1,4 Recent studies of human mobility largely focus on displacements patterns and power law fits of empirical long-tailed distributions of distances are usually associated to scale-free superdiffusive random walks called Lévy flights. However, drawing conclusions about a complex system from a fit, without any further knowledge of the underlying dynamics, might lead to erroneous interpretations. Here we show, on the basis of a data set describing the trajectories of 780,000 private vehicles in Italy, that the Lévy flight model cannot explain the behaviour of travel times and speeds. We therefore introduce a class of accelerated random walks, validated by empirical observations, where the velocity changes due to acceleration kicks at random times. Combining this mechanism with an exponentially decaying distribution of travel times leads to a short-tailed distribution of distances which could indeed be mistaken with a truncated power law. These results illustrate the limits of purely descriptive models and provide a mechanistic view of mobility. 1 Institut de Physique Théorique, CEA, CNRS-URA 2306, F-91191 Gif-sur-Yvette, France. 2 Department of Physics and Astronomy, University of Bologna, Viale Berti Pichat 6/2, 40126 Bologna, Italy. 3 INFN Bologna Section, 40126 Bologna, Italy. 4 CAMS (CNRS/EHESS) 190-198, avenue de France, 75244 Paris, France. Correspondence and requests for materials should be addressed to M.B. (email: ). NATURE COMMUNICATIONS | 7:12600 | DOI: 10.1038/ncomms12600 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12600 U nderstanding individual mobility has important implications for traffic forecasting1, epidemics spreading2,3 or the evolution of cities4–6. With the development of Information and Communication Technologies7, the investigations’ focus shifted from the traditional travel diary surveys8–10 to several new data sources. In particular, it became possible to follow individual trajectories from mobile phone calls11–13, location-sharing services14–16 and microblogging17, or directly extracted from public transport ticketing system10,18, global positioning system (GPS) tracks of taxis10,19–23, private cars24–26 or single individuals27,28. For most data sources, the spatial position r is the most reliable quantity. This information can be used for studying two different aspects of human mobility: how far and where we are moving. The second question is far more complex than the first and can be approached with several different tools, from aggregated origin–destination matrices29 for mobility prediction30 or land use analysis31 to individual mobility networks and patterns suitable for describing the natural tendency to return frequently to a few locations (such as homes, offices and so on)11,12,24–26,32. On the other hand, the first question, generally characterized by the distribution P(Dr) of the individuals’ displacements Dr across all users, although apparently simple is still far from being completely understood. Indeed, even if the study of the distribution P(Dr) has become a trademark for recent works on human mobility, there are still no consensus about the functional form of this distribution. At a large scale (national or inter-urban), one may observe a long tail behaviour9,11,12,14,15,17,20,22,33 characterized by a power law decay for long displacements. At a smaller (urban) scale, the distribution seems to have an exponential tail10,13,19,21,22,24,25. The unclear nature of this probability distribution makes its interpretation difficult and dependent on the data set used, the scale and possible empirical and fitting errors34. It is therefore necessary to obtain data as clean as possible and to propose a model that can be tested against empirical results. So far, essentially power law fits were used and led the authors to draw conclusions about the nature and mechanisms of the mobility, but this way of proceeding could actually lead to erroneous conclusions. Similar unresolved controversy also exists in the study of animal’s foraging movements: relying only on a fit of the empirical data, the same distribution can be understood in different ways, leading to contrasting conclusions on the nature of the underlying process35–38. Remarkably enough, what appears to be under-evaluated in the study of human mobility is the relevance of travel itself. Human travelling behaviour can in general be described as a sequence of rest times of duration t and jumps Dr in space12. These two processes need to be separated for modelling human mobility, since costs are in general associated to trips while a positive utility can be associated to activities performed during stops1. However, proposed models usually neglect the role of travel time and the moving velocity and assume instantaneous jumps. This is essentially a consequence of the limitations inherent to data sources: phone calls or social networks capture the spatial character of individuals’ movements39, but are limited by sampling rates or by the bursty nature of human communications40,41 and are thus not suitable for an exhaustive temporal description of human mobility. In this paper, we show that the observed truncated power laws in the jump size distribution can be the consequence of simple processes such as random walks with random velocities42. We test this model over a large GPS database describing the mobility of 780,000 private vehicles in Italy, where travels and pauses can be easily separated, as the transition is identified by the moment when the engine is turned on or off (but we introduce a lower threshold of 5 min in the elapsed time, to distinguish real stops from accidentally switched off of the engine during a trip). This allows us to evaluate accurately not only the displacements Dr, but also travel times t, speeds v and rest times t. Results The current empirical view. Several studies suggested that the displacements’ distribution P(Dr) has a fat tail, and power law fits display a wide range of exponent values depending on the data set and the fitting form used (see Table 1). We note that, strictly speaking, the distribution cannot be scale-free since displacements are always limited in space43. Thus, human movements could possibly be identified as truncated Lévy flights only11,44. In contrast, displacements at the urban scale consistently display an exponential tail19,24,25. Short tails also emerge when studying the distribution P(t) of travel times t of individual trajectories originating in different cities. For private cars’ mobility, as in the data set studied here, we observe that P(t) is indeed characterized by an exponential decay Pðt Þ / e  t=t as in refs 9,19,28,45 (see Table (...truncated)