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### Abstract

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 . These three exponents correspond to three displacement range zones of about , , and , 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 ) inverse power exponent because of the specific number (1, 2, or ) 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.

Human motion is a particular type of recurrent diffusion process where agents move from a location to another and return home after a given complex trajectory made of a certain number of displacement steps of given lengths. For a large set of agents this motion can be considered equivalent to a complex random walk described by a peculiar statistics. Understanding the statistical nature of human motion is the topic of the present work. In the literature, human displacement distributions are traditionally fit using an exponentially truncated Pareto or inverse power law distribution based on a single inverse power-law exponent. However, the physical meaning of this specific distribution function as well as of the measured inverse power law exponent (for example, β = 1.75) have never been explained. Herein, we show that the same human displacement distribution can be better fit with three inverse power law distributions described by simple integer exponents (β = 1, 2, 3, or larger). Each exponent characterizes a specific displacement length zone whose ranges are about , , and , respectively. This alternative fit methodology for interpreting human displacement distributions is crucial because it suggests a clear physical/geometrical generating mechanism. Essentially, people would divide the space into three regions (close, medium, and far distances), and they would assume that in each zone region the travel benefits are randomly/uniformly distributed mostly within the urban area of the cities and would associate to each zone region an increasing integer number of cost mechanisms required to cover the given distance. We suggest that the probability associated to a displacement length is a monotonic function of its

*cost*: less expensive (or shorter) displacements take place more likely than more expensive (or longer) displacements. Human mobility may be conditioned by multiple independent cost functions that work alone or may be statistically combined together according to the displacement length. For example, typical cost functions taken into account by people are the cost of the fuel, the time duration for the displacement, lodging costs that are required for very long trips, and others. The crucial fact is that on average these cost functions are likely directly proportional to the length of the displacement itself. For example, on average, the fuel cost for covering a distance

*L*is half the cost for covering a distance 2

*L*: the same is true for the time interval needed to cover a give distance. Thus, we can expect that the simplest strategy adopted by humans in optimizing their movements is that when, for example, fuel cost alone are taken into account by a person, displacements of length

*L*are twice more probable than displacements of length 2

*L*because of the homogeneous benefit distribution assumption within each urban zone. Thus, by assuming that people optimize their movement by uniformly distributing their travel resources (in time, energy, and money) within given range zones, we demonstrate by simple geometrical considerations based on area ratios that each displacement cost function yields a displacement probability distribution described by a basic inverse power law distribution

*P*(Δ

*r*) ∝ 1/Δ

*r*. When nearby cities are visited, a similar reasoning would yield an inverse power law with β = 2 because at least 2 cost basic mechanisms would be activated and because the desired benefit would still be mostly searched within an urban zone. If 3 or more displacement cost functions condition the decision of an agent, as for visits to far locations, then the combined probability distribution is the product of more basic probability distributions and would be characterized by inverse power law distributions with integer exponent β = 3 or more. We also show that a similar model predicts the statistical properties of human displacements below 1 km ranges by people who just walk, and whose walking decision could be determined by time and physical energy considerations. In the case of just walking people, the threshold between zone 1 and zone 2 may be a few hundred meters and, perhaps, it may have been evolutionary determined by the natural human high resolution visual range. In conclusion, the peculiarity of the three-scale pattern shown by the human displacement distribution suggests that on average humans take into account one, two, three (or more) alternative displacement cost mechanisms according to the length of the displacement itself and assume that the benefits are uniformly/randomly located only within specific areas of interests, which characterize zone 1. This property may be more general and may characterize other complex random walk phenomena that are commonly interpreted in the scientific literature with simple exponentially truncated inverse power law distributions based on a single exponent that may obscure, instead of clarifying, the physical mechanism behind a physical phenomenon.

I. INTRODUCTION

II. THE TRADITIONAL FIT METHODOLOGY: RANGE FROM 1 KM TO 1000 KM

III. AN ALTERNATIVE FIT METHODOLOGY

IV. A SIMPLE MULTI-SCALE DIFFUSIONMODEL OF HUMAN DISPLACEMENTS AND ITS STATISTICAL/PHYSICAL INTERPRETATION

V. DISPLACEMENT DISTRIBUTIONS OF WALKING PEOPLE: Range Below 1 km

VI. CONCLUSION

### Key Topics

- Probability theory
- 29.0
- Statistical properties
- 14.0
- Motor vehicles
- 9.0
- Topology
- 7.0
- Cumulative distribution functions
- 5.0

## Figures

(Color online) Distribution of human displacements (from record D1 and Figure 1(c) in Gonzalez *et al.* (12)). The length Δ*r* is the travel distance. The distribution is fit with Eq. (2).

(Color online) Distribution of human displacements (from record D1 and Figure 1(c) in Gonzalez *et al.* (12)). The length Δ*r* is the travel distance. The distribution is fit with Eq. (2).

(Color online) Distribution of human displacements (from record D1 and Figure 1(C) in Gonzalez *et al.* (12)). The length Δ*r* is the travel distance. The distribution is approximately reproduced within three consecutive ranges with incremental inverse power law functions proportional to 1/Δ*r* ^{β}. We get: 1) β ≈ 1 for ; 2) β ≈ 2 for ; 3) β ≈ 3 for .

(Color online) Distribution of human displacements (from record D1 and Figure 1(C) in Gonzalez *et al.* (12)). The length Δ*r* is the travel distance. The distribution is approximately reproduced within three consecutive ranges with incremental inverse power law functions proportional to 1/Δ*r* ^{β}. We get: 1) β ≈ 1 for ; 2) β ≈ 2 for ; 3) β ≈ 3 for .

(Color online) Schematic representation of human motion with Lévy flights in three different displacement zone ranges. The bottom left corner represents the residence location where the walker returns. The regions with a higher density of displacements represent cities or towns (zone 1) where the walker stays longer. The distance among the cities classifies zone 2 or zone 3 according to whether the journey is medium or long. The shadowed areas represent the urban area where the benefits are assumed to be randomly/uniformly distributed. Displacements within these restricted areas yield to Pareto distribution with exponent β = 1 according to Eq. (5). Displacements between close areas of interest yield to Pareto distribution with exponent β = 2 according to Eq. (8).

(Color online) Schematic representation of human motion with Lévy flights in three different displacement zone ranges. The bottom left corner represents the residence location where the walker returns. The regions with a higher density of displacements represent cities or towns (zone 1) where the walker stays longer. The distance among the cities classifies zone 2 or zone 3 according to whether the journey is medium or long. The shadowed areas represent the urban area where the benefits are assumed to be randomly/uniformly distributed. Displacements within these restricted areas yield to Pareto distribution with exponent β = 1 according to Eq. (5). Displacements between close areas of interest yield to Pareto distribution with exponent β = 2 according to Eq. (8).

(Color online) Reproduction of Figure 6 in Rhee *et al.* (18) which refers to a distribution of displacements at Disney World (Orlando, FL) by a sample of traces obtained from 18 volunteers that mainly walked in the park during holidays. The figure shows in the wide lighter (green) curve the authors’ original fit obtained with a truncated Pareto distribution that gives an exponent β = 1.86 for the entire range. The two narrow straight (red and blue) lines, instead, were added by me and highlight that the smallest displacement range up to 100–200 m is compatible with a Pareto distribution with β = 1, while the longer scale is compatible with a Pareto distribution with β = 2. Note that the statistics for displacements above approximately 1000 m is visibly poorer.

(Color online) Reproduction of Figure 6 in Rhee *et al.* (18) which refers to a distribution of displacements at Disney World (Orlando, FL) by a sample of traces obtained from 18 volunteers that mainly walked in the park during holidays. The figure shows in the wide lighter (green) curve the authors’ original fit obtained with a truncated Pareto distribution that gives an exponent β = 1.86 for the entire range. The two narrow straight (red and blue) lines, instead, were added by me and highlight that the smallest displacement range up to 100–200 m is compatible with a Pareto distribution with β = 1, while the longer scale is compatible with a Pareto distribution with β = 2. Note that the statistics for displacements above approximately 1000 m is visibly poorer.

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