# Origin of the Scaling Law in Human Mobility: Hierarchical Organization of Traffic Systems

###### Abstract

Uncovering the mechanism leading to the scaling law in human trajectories is of fundamental importance in understanding many spatiotemporal phenomena. We propose a hierarchical geographical model to mimic the real traffic system, upon which a random walker will generate a power-law travel displacement distribution with exponent -2. When considering the inhomogeneities of cities’ locations and attractions, this model reproduces a power-law displacement distribution with an exponential cutoff, as well as a scaling behavior in the probability density of having traveled a certain distance at a certain time. Our results agree very well with the empirical observations reported in [D. Brockmann et al., Nature 439, 462 (2006)].

###### pacs:

89.75.Fb, 05.40.Fb, 89.75.DaStudies on the non-Poisson statistics of human behaviors have recently attracted much attention bara (); vqz1 (); zhou (). Besides the inter-event or waiting time distribution, the spatial movements of human also exhibit non-Poisson statistics. Brockmann et al. bro () investigated the bank note dispersal, as a proxy for human movements, and revealed indirectly a power-law distribution of human travel displacements. Gonzalez et al. gon () studied the human travel patterns by measuring the distance of mobile phone users’ movements in different stations, and observed a similar scaling law. Actually, the mobility patterns of many animals also show power-law-like displacement distributions bart1 (); ramo (); sims (). The ubiquity of such kind of distributions attracts scientists to dig into the underlying mechanism. Some interpretations, such as optimal search strategy visw (); bart2 (), olfactory-driven foraging reyn () and deterministic walks santo (), have already been raised for the power-law displacement distribution in animals mobility patterns, however, they are based on the prey processes and thus cannot be used to explain the observed scaling law in human trajectories, which is still an open problem. In this paper, we propose a model to mimic the human travel pattern, where the hierarchical organization of the real human traffic systems is taken into account. Our model can reproduce the power-law displacement distributions, as well as the scaling behavior in probability density of having traveled a certain distance at a certain time, agreeing very well with the empirical results reported in Ref. bro ().

Let’s think about the real human traffic systems. Generally speaking, a district (e.g., a province or a state) usually has a core city, like its capital; around this core city, there are several big cities as the secondary centers (e.g., municipalities); then, each of these centers is rounded by some counties; and towns and villages will surround each of the counties. A hierarchical traffic system is built accordingly. Imaging people traveling from a town, , subordinating to the central city, , to another town that is subordinated to the central city . There is usually no direct way connecting and , and the typical route is . This kind of hierarchical organization is not just inside a country or a district, but across the whole world. For example, if one wants to travel from the University of Science and Technology of China to the University of Fribourg, there is no direct way connecting Hefei and Fribourg, instead, one has to follow the route HefeiShanghaiZürichFribourg although it is much longer than the geographical distance between Hefei and Fribourg. Such a hierarchical organization as well as the resulting scale invariance in road networks have already been demonstrated recently Kalapala2006 ().

For simplicity, we call all the units cities. In our model, cities are organized in layers. A uniform 3-layer system is shown in Fig. 1, in which, 81 cities locate on the centers of a lattice. The most central city is put in the first layer, and the whole region is divided into 9 sub-regions, each contains a lattice. Except the middle sub-region, all other eight sub-regions have their own central cities, namely the second layer cities, which locate at the centers of those sub-regions. Meanwhile, there are eight third layer cities around each of the second layer cities as well as the first layer city. An illustration is shown in Fig. 1. Denote the number of layers and the number of first layer cities. We assign sub-regions to each of the 1st-layer cities, with the 1st-layer city locating in the center, and 2nd-layer cities are respectively put in the remain sub-regions. Each of the sub-regions is further divided into sub-sub-regions, with the 1st- or 2nd-layer cities locating in the center and newly generated 3rd-layer cities put in the remain sub-sub-regions. Repeating this process until the th-layer cities are generated. For , there are th-layer cities. Note that, in this model, to make sure the lattice is fulfilled with cities, must be equal to where is a certain integer larger than 1.

The first layer cities are fully connected with each other, each of which is connected with the nearest th-layer cities and the nearest second layer cities. Each of the second layer cities is connected with the nearest th-layer cities, the nearest third layer cities, as well as the other second layer cities belonging to the same first layer city. Actually, for , each of the th-layer cities is connected with the nearest th-layer cities, the nearest th-layer cities, as well as the other th-layer cities belong to the same th-layer city. Note that, all the connections are symmetry, and the direct move between two cities is allowed only if they are connected. Figure 1 illustrates an example with , and . The modeled system can be viewed as a hierarchical network. Different from the real hierarchical networks rav2 () or the mathematical models an (); ran (), it is hierarchical but not scale-free.

We consider the simplest case where a random walker is consequently moving from one city to a random neighboring city (two cities are said to be neighboring if they are connected). Figure 1(d) shows a typical trajectory of a walker moving from a lower-layer city to a higher-layer city in other sub-region. The displacement of the walker in one step is defined as the geometric distance , and the distribution of is what we mainly concern in this paper. As shown in Fig. 2, burstiness of long-range travels is clearly observed and the distribution of travel displacement, , approximately obeys a power-law form with exponent -2.

The essential physics of this model is a random-walk process in a geographical network where edges are of different geometric distances. For a random walker in a connected symmetry (undirected) network, in the long time limit, each edge has the same chance to be visited (this proposition is hold even for a very heterogenous network, since for an arbitrary node, the number of times being visited is proportional to its degree while the probability that a specific adjacent edge of this node is consecutively visited is inversely proportional to the degree. Details can be found in Ref. Lovasz1996 ()). Therefore, the displacement distribution of a random walker is equivalent to the distribution of edges’ geometric distances. Let denote the average geometric distance of edges connecting two th-layer cities (they must belong to the same th-layer city) and an th-layer city and an th-layer city (the former belongs to the latter), and denote the total number of edges contributed to . Obviously, for , . , and can be considered as constants, and thus we have . On the other hand, for , where . That is, . Roughly speaking, and play the roles of geometric distance and the number of edges associated with such a distance. As we have already obtained the scaling , we can deductive that the displacement distribution for a system with sufficiently large and in the long time limit obeys the scaling , which is in accordance with the simulation result shown in Fig. 2(c). The observed result implies that the scaling law in human trajectories may results from the inherent hierarchical organization in traffic systems.

Although the present model can reproduce the power-law displacement distribution, the absolute value of exponent is higher than the empirical ones bro (); gon (), and the model is obviously oversimplified. Firstly, real cities are not located in a completely uniform matter, but with some irregularity. Secondly, the model assumes that each city has the same attraction for the walker, however, in the real world, a central city is generally much more attractive than a small town. We next propose a modified model taking into consideration the inhomogeneous locations and attractions of cities. In this inhomogeneous model, all cities are randomly distributed in an square (we keep the number of cities in each layer the same as the original model), each of the th-layer cities is connected to the nearest higher-layer city, and two th-layer cities are connected if they are connected to the same higher-layer city. All the 1st-layer cities are fully connected to each other.

As mentioned above, the center city should have greater attraction, which is represented by a layer-dependent weight, , where denotes the layer and is a free parameter. The probability that the walker will move along an edge is proportional to its weight (A similar weighted random walk model has previously been proposed to explain the nonlinear dependence of the airport throughput on the connectivity Ou2007 ()). Clearly, the larger indicates higher heterogeneity. In Fig. 3, we show an illustration of a typical trajectory in a 3-layer inhomogeneous model, and in Fig. 4 we report the trajectories for and in a 5-layer inhomogeneous model. As shown in Fig. 5, for the inhomogeneous model, the displacement distribution, , is still heavy-tailed and can be well fitted by a power-law function with an exponential cutoff, as . In addition, as shown in the inset of Fig. 5, when increases from 1.0 to 2.0, the power-law exponent, , monotonously decreases from 2.70 to 1.51, covering the range of empirical observations bro (); gon (). This result suggests that the inhomogeneity in cities’ attractions may be the reason why the absolute value of power-law exponent in the real human displacement distribution is lower than that predicted by the homogeneous model (i.e., 2.0), while the inhomogeneity of cities’ locations enlarges the absolute value of such exponent.

Finally, we check whether our model can reproduce the spatiotemporal statistics of real human mobility. Providing the trajectory of a random walker, one can obtain the probability of having traveled a distance at time (the same technique has been adopted in preparing Fig. 2a in Ref. bro (), please see details there). As shown in Fig. 6, a scaling behavior with is clearly observed, which agrees well with the empirical result, , reported in Ref. bro (). Similar scaling behavior can also be observed for , however, the exponent, , is far less than the empirical value. In addition, providing the travel displacement distribution, this scaling behavior with around 1.0 can not be reproduced by a Lévy flight bro ().

Uncovering the human traveling pattern is of fundamental importance in the understanding of various spatiotemporal phenomena bro (); gon (), and may finds applications in the design of traffic systems Barthelemy2006 (), the control of human infectious disease Hufnagel2004 (), the military service planning Zhao2008 (), and so on. Although empirical results about the scaling law of long-range human travels have been reported for years, it lacks the understanding of the underlying mechanism. This work gives raise to a very possible reason causing the heavy-tailed displacement distribution, , that is, the hierarchical organization of traffic systems. The secondary ingredient, also playing appreciable role in determining the traveling pattern, is the inhomogeneities of the locations and attractions of cities: The former enlarges the exponent , while the latter depresses it (essentially, the inhomogeneity of attractions results from the inhomogeneous population density and economic development). Actually, as shown in Fig. 5, with tunable strength of inhomogeneity, the exponent is also tunable. When , meaning the topper-layer cities having greater attractions, the statistical features produced by our model are very close to the empirical ones reported in Ref. bro (), not only the displacement distribution, but also the spatiotemporal statistics of mobility.

We acknowledge Changsong Zhou and Aaron Clauset for valuable suggestions. This work is supported by 973 program (2006CB705500), and the National Natural Science Foundation of China (10532060, 10635040 and 70871082). Authors are ordered in alphabet.

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