Understanding the complexity of the Lévy-walk nature of human mobility with a multi-scale cost/benefit model

Probability distributions of human displacements have been fit with exponentially truncated Lévy flights or fat tailed Pareto inverse power law probability distributions. Thus, people usually stay within a given location (for example, the city of residence), but with a non-vanishing frequency they visit nearby or far locations too. Herein, we show that an important empirical distribution of human displacements (range: from 1 to 1000 km) can be well fit by three consecutive Pareto distributions with simple integer exponents equal to 1, 2, and $(>≈) 3$⁠. These three exponents correspond to three displacement range zones of about $1 km ≲Δr ≲10 km$⁠, $10 km ≲Δr ≲300 km$⁠, and $300 km ≲Δr ≲1000 km$⁠, respectively. These three zones can be geographically and physically well determined as displacements within a city, visits to nearby cities that may occur within just one-day trips, and visit to far locations that may require multi-days trips. The incremental integer values of the three exponents can be easily explained with a three-scale mobility cost/benefit model for human displacements based on simple geometrical constrains. Essentially, people would divide the space into three major regions (close, medium, and far distances) and would assume that the travel benefits are randomly/uniformly distributed mostly only within specific urban-like areas. The three displacement distribution zones appear to be characterized by an integer (1, 2, or $>≈ 3$⁠) inverse power exponent because of the specific number (1, 2, or $>≈ 3$⁠) of cost mechanisms (each of which is proportional to the displacement length). The distributions in the first two zones would be associated to Pareto distributions with exponent β = 1 and β = 2 because of simple geometrical statistical considerations due to the a priori assumption that most benefits are searched in the urban area of the city of residence or in the urban area of specific nearby cities. We also show, by using independent records of human mobility, that the proposed model predicts the statistical properties of human mobility below 1 km ranges, where people just walk. In the latter case, the threshold between zone 1 and zone 2 may be around 100–200 m and, perhaps, may have been evolutionary determined by the natural human high resolution visual range, which characterizes an area of interest where the benefits are assumed to be randomly and uniformly distributed. This rich and suggestive interpretation of human mobility may characterize other complex random walk phenomena that may also be described by a N-piece fit Pareto distributions with increasing integer exponents. This study also suggests that distribution functions used to fit experimental probability distributions must be carefully chosen for not improperly obscuring the physics underlying a phenomenon.