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The Scales of Human Mobility

There is a contradiction at the heart of our current understanding of individual and collective mobility patterns. On one hand, a highly influential stream of literature on human mobility driven by analyses of massive empirical datasets finds that human movements show no evidence of characteristic spatial scales. There, human mobility is described as scale-free. On the other hand, in geography, the concept of scale, referring to meaningful levels of description from individual buildings through neighborhoods, cities, regions, and countries, is central for the description of various aspects of human behavior such as socio-economic interactions, or political and cultural dynamics. Here, we resolve this apparent paradox by showing that day-to-day human mobility does indeed contain meaningful scales, corresponding to spatial containers restricting mobility behavior. The scale-free results arise from aggregating displacements across containers. We present a simple model, which given a person's trajectory, infers their neighborhoods, cities, and so on, as well as the sizes of these geographical containers. We find that the containers characterizing the trajectories of more than 700,000 individuals do indeed have typical sizes. We show that our model generates highly realistic trajectories without overfitting and provides a new lens through which to understand the differences in mobility behaviour across countries, gender groups, and urban-rural areas.

trajectory, infers their neighborhoods, cities, and so on, as well as the sizes of these geographical containers. We find that the containers characterizing the trajectories of more than 700 000 individuals do indeed have typical sizes. We show that our model generates highly realistic trajectories without overfitting and provides a new lens through which to understand the differences in mobility behaviour across countries, gender groups, and urban-rural areas.
It is nearly impossible to underestimate the importance of establishing a quantitative foundation for our understanding of how individuals move from place to place in their everyday lives. Hundreds of millions of individuals spend billions of collective hours commuting every day. 6 Goods and food are transported through a global network using shared infrastructure. 7 Understanding mobility patterns helps us mitigate epidemic spreading, 8 assist in crisis management, 9 prepare for dramatic shifts in modes of transportation, 10 and many other cases. 6 For this reason, understanding the origin of scale-free distributions of displacements in empirical mobility traces is crucial, as this issue currently separates the large-scale data-driven human mobility research 11 from the community of human geography 4, 5 and transportation research. 12 Our mental representation of physical space has a hierarchical structure. 13 We describe space referring to places, 4 meaningful spatial entities with associated typical size, or scale, from rooms and buildings -via neighborhoods, cities, and states -to nations and continents that are organized in a nested structure. 4,[14][15][16][17] Geographical borders confine residential mobility 18 and collective mobility fluxes. 19 Commuting is characterized by a typical travel-time budget, and, as a consequence, there exist characteristic spatial scales that have evolved in connection to the progress of transportation. 20 Further, it has been conjectured that there are fundamental differences between forms of moving at different scales, from moving within a building to traveling across the globe. 6,12,16 However, recent empirical research in the field of Human Mobility, 11 has found no evidence for characteristic spatial scales in how people travel. [1][2][3]21 On the contrary, studies have shown that the distribution of displacement lengths ∆r travelled by an individual has a power law tail P(∆r) ∼ ∆r −β over several orders of magnitude, where typically 1 ≤ β ≤ 2. 22 Power law distributions are also called scale free, because they are the only mathematical distribution to have no associated typical scale 23 (see Supplemen-tary Note 1).
Nested scales generate power laws. So the question becomes: How is it possible that our intuitive conception of space is clearly hierarchical and characterized by typical scales, when a broad range of empirical datasets, ranging from displacements of dollar bills 1 or cell-tower data 2 to public transportation systems, 24 and GPS data 25, 26 all suggest that human mobility is scale free?
To explain this apparent contradiction, we propose that each typical scale of human mobility corresponds to a container of a certain mobility behavior. These containers (rooms, buildings, neighborhoods, cities, countries, and so on) have typical sizes, see Figure 1a, and roughly correspond to the notion of places in geography. 4 The observed power law arises when we aggregate mobility behavior within containers and mobility that transports a person between containers. Specifically, it is well known that mixtures of normal (or lognormal) distributions with different variances can generate power laws 27 (see Figure 1d). More specifically, we assume that for each individual, physical space is organized as a nested structure of containers. This structure relates, in part, to the organization of the transportation system 20 and to the concrete structure of our built environment, 16 see Figure 1a.
We propose that these nested containers affect how individuals move, and therefore can be inferred from the raw mobility data. Specifically, the amount of time spent within a container can depend on its hierarchical level. The connection between hierarchical level and mobility is supported by the literature, which shows that e.g. transitions between regions are more frequent than transitions between countries. 28 A simple model identifies containers. We now describe the associated container model of mobility, a model which estimates a person's containers from their empirical mobility patterns, see Figure 1c. For each individual, we model physical space as a hierarchy of L levels, ordered from the smallest to largest (e.g. individual locations to countries). At any level l, space is partitioned into topologically compact containers, with a characteristic size. For l < L, a container is fully included within a single parent container (for example each neighbourhood is part of a single city). Hence, each geographical location k can be identified as a sequence of containers, k = (k 1 , ..., k l , ..., k L ), where container k l  is included in k l+1 .
Next, consistent with most models of human mobility, 11,29 each container k l is characterized by its probability to be selected within its parent container, its attractiveness a(k l ). We define the level-distance d(j, k) between locations j and k as the highest index at which the two sequences of containers describing j and k differ. 30 We model traces individually; each trace results in a unique hierarchical structure.
Based on the assumption that the amount of time spent in a container depends upon its place in the hierarchy, we design a model of trajectories, where the probability of transitioning from location j to location k depends on the level-distance between them.
For an agent located in j, we model the probability of moving to k as the product of two factors: see also Methods, section 'Model description'. The first factor, p d(j,k),d(j,h) , represents the probability of traveling at level-distance d(j, k), given that the current location j is at level-distance d(j, h) from the individual home-location, h. This probability follows a multinomial distribution, which must depend on level-distance from home to account for the fact that higher-level transitions are more likely when individuals are not in the home container; for example, one is typically more likely to transition at the country scale, when not in the home country. The second factor ∏ l≤d(j,k) a(k l ) is the probability of choosing a specific location k at that level-distance, where a(k l ) is the attractiveness of a container at level l including location k.  Figure S5). In both datasets of GPS traces, our model finds characteristic sizes of containers. The sizes of containers -defined as the maximum distance between two locations in a container at a given level -are not broad, but well described by lognormal distribution across the population. Our results are robust across datasets (see Extended Data Table S1).

Results
We argue that the characteristic sizes of containers are precisely the 'scales' of human mobility.
These typical sizes of containers can be characterized by the median value e µ l , of the lognormal distributions Lognormal(µ l , σ 2 l ), 32 for each hierarchical level l. We find e µ 2 = 3.089 ± 0.006 km, e µ 3 = 27.064 ± 0.006 km, e µ 4 = 88.442 ± 0.022 km and e µ 5 = 161.634 ± 0.049 km, see Figure 1b and Extended Data Table S3. The coefficient of variation C l = e σ 2 l − 1, 33 characterizing the relative dispersion of the lognormal distribution, is included in the range [2.721,3.042] for l in the range [2,5].
The median time spent within the same container at a given level is also well described by a lognormal distribution (see Figure 1c and Extended Data Table S2), implying that there are characteristic temporal scales associated to spatial scales.
Having shown that we can infer information on geographical scales directly from the raw data, we now demonstrate the usefulness of this novel description of mobility patterns. We approach this task in two steps. First, we argue that the hierarchical description generated by the container model generalizes to unseen data without overfitting, while providing a more expressive and nuanced description of mobility relative to stateof-the-art models according to unbiased performance estimates. Secondly, drawing on demographic and environmental data, we show that the container model produces results that converge with existing literature on gender differences, urban/rural divides, and walkability scores.   In panels a,c,d, the filled areas for synthetic traces include two standard deviations around the mean computed across 1000 simulations for each user. In panel b filled areas include the interquartile range.
For each individual, we fitted the EPR and container models considering a training period of 1 year.
The data used here for validation corresponds to the 50 individual displacements following the training period. Results are shown for a random sample of 9000 individuals.
We focus on four key properties of mobility in the generated data: Distribution of dis-placements, evolution of radius of gyration, time allocation among locations, and entropy.
Considering the distribution of displacement lengths between consecutive locations, a widely studied property of mobility traces, 22 the likelihood ratio test 34 shows that the container model provides a significantly better description of the data than the EPR model 3 and its variations (see Figure 2a and Extended Data Figure S4 and Table S4, Next, the radius of gyration 2 (see Methods section 'Metrics'), quantifies the spatial extent of an individual's mobility. Here we find that while the evolution of individuals' radius of gyration over time is well described by a logarithmic growth in all cases: real, 2 EPR, 3  The final property of synthetic traces is the individual difference between the uncorrelated entropy S unc , which characterizes the heterogeneity of visitation patterns, and the temporal entropy, S temp , which depends not only on the frequency of visitation, but also the order in which locations were visited 35  and walkability score 38 in the area surrounding one's home-location. First, we find that gender differences can partly explain the observed heterogeneity, in line with previous findings, 36 although not in all of the countries under study (see Figure 3a, and Supplementary Table 1). A novel finding concerns the fact that in 21 out of 53 countries, females are characterized by a significantly larger number of hierarchical levels than males, while the opposite is not the case for any country (see Supplementary Table 2).
As a key observation inviting further research, we find that the difference between genders across countries, measured as the Kullback-Leibler divergence between the distributions of number of levels, is strongly correlated with the Gender Inequality Index (GII). 39 GII measures the percentage of potential human development loss due to gender inequality (Spearman correlation ρ = 0.69, p < 10 −6 , see Figure 3b). Turning next to the size of containers, we find that in 48 out of 53 countries, the containers characterizing the mobility of females are smaller compared to males (see Supplementary Note 2), in line with previous research. 36 Secondly, we find that the urban/rural divide partly explains differences in mobility patterns, in line with the fact that rural areas are characterized by limited accessibility. 16,40 Individuals living in rural areas (see Methods section 'Other data' for definitions) have significantly larger containers compared to urban individuals, and this difference is more pronounced at the lowest hierarchical levels (see Figure 3c, and Supplementary Table 3).
Finally, we find that the walkability score around an individual's home location correlates negatively with the size of containers at the lowest hierarchical level (Spearman correlation ρ = −0.96, p < 10 −9 , see Figure 3d), in line with the finding that improved walkability increases accessibility to goods, services, and activities. 41 The correlation between walkability and container-size is significant up to the third level of description (see Supplementary Note 2).
Discussion. The paradigm of power law descriptions does not stand entirely unchallenged within the quantitative analysis literature. For example, it has been argued that exponential or lognormal functions may be more suitable to describe the distributions of displacements within cities, 22 hinting that human mobility may not be completely free of scales. Until now, however, the nature of the probability distribution of displacements has been unclear. 22,26 For example, it has been suggested that scaling laws could be the signature of Lévy flights, a type of random walk with scale-free step-size attributed to animal foraging, 42 but Lévy flights do not reproduce all statistical properties of human trajectories. 3 It has also been proposed that the structure of the transportation system, 26,43,44 where each mode of transportation corresponds to a typical distance traveled, could explain the observed scaling laws. Intra-city displacements considering all transportation modes, however, are not scale-free distributed, as this theory would suggest. 21,24 Because of the lack of agreement on the functional form of distribution of displacements, many state-of-the-art agent-based models of individual mobility focus on temporal aspects, 11 including the interplay between exploration and exploitation, 3, 45 recency and memory effects, 25,31,46 weekly and circadian rhythms. 47 With few exceptions, 43 these models do not account for effects due to the spatial distribution of locations.
In this paper, we have proposed a model in which human mobility is organized according to a hierarchical structure of spatial containers, corresponding to the notion of places in geography (see Equation 1). Under this model, the observed power law data arises by merging mobility within containers with mobility that transports a person between containers. The container model focuses on a specific aspect of mobility, and neglects other important features, including temporal visitation patterns, exploration, 3 and the structural connectedness of geographical spaces (e.g. through transportation networks). [48][49][50] These could be incorporated in future versions of the model. Fitting the model to trajectories collected in two distinct datasets, consisting of ∼ 700 000 GPS traces of individuals distributed across the world, we found that -across individuals -the containers have typical sizes, representing the 'scales' of human mobility. We showed that our model allows for better understanding of mobility behavior and improves on the stateof-the-art in modeling. The D2 data was collected as part of an experiment that took place between September 2013

Methods
and September 2015. 51 The experiment involved 851 Technical University of Denmark students (∼ 22% female and ∼ 78% male), typically aged between 19 and 21 years old. Participants' position over time was estimated from a combination of GPS and WiFi information, resulting in samples every 1 − 2 minutes. The location estimation error was below 50 m in 95% of the cases.
Data collection was approved by the Danish Data Protection Agency. All participants provided informed consent. Data to produce Figure 1a is the location trajectory of one of the authors. We pre-processed all trajectories to obtain stop-locations using the Infostop algorithm. 52   where it is located. At each level in the hierarchy, containers have comparable size. In the simplest form, this structure is a nested grid (see Supplementary Note 3).
• The collection of these containers' attractivenesses, a. The attractiveness a(k l ) is the probability of visiting container k l among all containers encapsulated within k l+1 . Accordingly, ∑ k l ∈k l+1 a(k l ) = 1.
• The L × L matrix, p characterizing the probability of travelling at a certain level-distance.
Each row in p is a probability vector that describes the probabilities p(d, d h ) of traveling at level-distance d when the level distance from home is d h . Here, home is defined as the location with largest attractiveness at all levels. By level-distance we mean the socalled cophenetic distance: 30 the highest level in the hierarchy one travels to reach the destination. It is necessary to maintain separate probability distributions for each leveldistance from home. This is, for example, because traveling at the highest-level distance (e.g. intercontinentally) is unlikely when one is near home, but comparatively likely when on a different continent.
Under the container model, each transition is the result of a two-stage decision process. First, the individual selects at which level-distance to travel. Then, she selects a specific destination based on container attractiveness. Specifically, an individual located in j, chooses destination location k with probability: The first factor, p d(j,k),d(j,h) , is the probability of traveling at level-distance d(j, k). The second factor a(k d(j,k) ) 1−a(j d(j,k) ) is the probability of choosing container k d(j,k) . Such a container is found at level d(j, k) in the hierarchy and has attractiveness a(k d(j,k) ). The renormalization 1 − a(j d(j,k) ) accounts for the fact that container j d(j,k) cannot be selected (this detail is not present in the main text for readability). The third factor ∏ d(j,k)−1 l=1 a(k l ) is the probability of picking all other containers k l that encapsulate location k, for any level in the hierarchy lower than d(j, k). Note that the way we model destination choice in a hierarchical fashion, connects to the class of choice models called nested logit models. 54 The nested structure of the physical space in the container model relates, in part, to the organization of the transportation system 20,55,56 and to the concrete structure of our built environment. 16,49 The importance of these contexts are also gradually being recognized in the Human Mobility literature, where early studies focused on large datasets, but did not consider the effect of contextual information, e.g. transportation type or other mobility characteristics, which can introduce heterogeneity. 21,36,44,[57][58][59][60][61][62] *Generating traces. We model transitions as a two-step decision process. Thus, we can simulate synthetic trajectories given a hierarchical description H, container attractivenesses a, and the probability matrix p, (either designed or obtained by fitting the container model to an empirical trace). We simulate the mobility of an agent by the following algorithm. To guide the reader we offer an Example at each step, describing an agent travelling across a hierarchy where levels correspond to countries, cities, neighbourhoods, buildings and locations.
1. Initialize the agent in a random location, j, at level-distance d(j, h) from the home location.
Example: the agent is initialized in location j located in a different country than her home country.
2. Select a level-distance l * that the agent should travel at, by drawing from the multinomial distribution, p d(j,h) . Example: the agent chooses to travel at the city-distance.
3. Select a destination, k: • At level l * , select a container k l * , by drawing from the attractiveness distribution over the containers encapsulated in j l * +1 . j l * cannot be selected in this process, so k l * = j l * .
Example: the agent chooses the destination city among other cities in the same country where she is currently located.
• At level (l * − 1), select a container k l * −1 encapsulated within k l * , by drawing from the attractiveness distribution over containers in k l * . Continue this process until level 1 is reached. Example: the agent picks a neighborhood, then a building and then a location encapsulated within the destination city chosen in the previous step.
where P H,a,p (k(i − 1) → k(i)) is the probability of a transition to occur.
Unlike spatial clustering methodologies, this method allows us to identify 'containers', structures that are compact in size, but also also contain mobility behavior. This optimization task, however, is computationally expensive, therefore we approach the problem according to the following heuristic.
First, we note that, when n T is large and H is chosen, p and a are trivial to estimate. In fact, for In order to ensure that clusters are compact, we choose H among the solutions of the complete linkage hierarchical clustering algorithm. 63 First, we run the complete-linkage algorithm for the set of locations in sequence T. The algorithm initializes each location as a separate cluster. It then iteratively joins the two clusters whose union has the smallest diameter, defined as the maximum distance between two locations by computing the corresponding likelihoods. We find the cut C 2 of the dendogram resulting in the maximum likelihood L 2 . If L 1 is significantly larger than L 2 (tested by bootstrapping), we assign L * = L 1 , conclude that H * has only 1 level (individual locations), and stop the algorithm. Otherwise, we explore the set of partitions corresponding to two cuts of the dendogram (i.e. L = 3), where one of them is C 2 , and find the cut C 3 that yields the maximum likelihood L 3 .
We compare L 2 and L 3 , and stop the algorithm if L 2 is significantly larger than L 3 . We proceed for increasing values of L, until L = N or there is no significant improvement in likelihood.
In the worst-case scenario, we explore N! partitions. We validate the one-level-at-a-time algorithm against synthetic data (see Extended Data Figure S3 and Supplementary Note 3). We find that the algorithm recovers the correct number of hierarchical levels ∼ 95% of the times. The similarity between the correct and recovered hierarchical structure, measured as their cophenetic correlation 30 has median value 1 (the cophenetic correlation is the correlation between the cophenetic distance computed for all pairs of locations according to two different hierarchical descriptions, and thus is 1 for identical descriptions). The median absolute error relative to the estimation of the matrix of probabilities p is 0.03.

Model validation
Metrics. In Figure 2, we compare synthetic and real traces by computing quantities characterizing individual trajectories.
The radius of gyration for an individual u is defined as: where N is the total number of displacements (50 in our analysis), r u n is the position of u after n displacements, r u CM,n is its center of mass after n displacements, defined as: The uncorrelated entropy S unc is defined as: where P(i) is the probability of visiting location i, and N L is the total number of locations. The temporal entropy S temp , is defined as where P(T i ) is the probability of finding a particular time-ordered subsequence T i in the trajectory T i . We estimate S temp using the method described by Sekara

Code Availability.
Code is available at https://github.com/lalessan/scales human mobility/ An interactive illustration of the container model generative process can be found at https://observablehq.com/@ulfaslak/a-model-for-generating-multiscale-mobility-traces. acterizing human data -defined as the maximum distance between two locations in the same container at a given level -are not broad, but better described by lognormal distribution across the population under study. This is verified by considering a truncated power-law distribution: with parameters a and λ, and where Γ is the gamma function Γ(x) = (x − 1)!, in comparison with a lognormal distribution: with parameters σ and µ. We compare the goodness-of-fit of distributions f 1 and f 2 using the log-likelihood ratio test. 34 We find that the lognormal distribution are better suitable, compared to truncated power laws, to describe the distribution of container sizes at levels 2 to 5 at significance level α = 0.01 (see Extended Data Table 1 and Ex-tended Data Figure 2), for both datasets. The same result is verified for the distribution of time spent within the same container at a given level (see Extended Data Table 2).
Further, we find that the log-normal distribution is generally a better model of the containers size distribution in comparison to other log-well-behaved distributions: the loggamma, the log-logistic, the log-laplace and the log-Weibull distributions (see Extended Data Table 1). In Extended Data Table 3 Table 1.

Supplementary Note 2: Demographics and Built Environment
Gender differences revealed by the container model. We investigate gender differences with respect to mobility features revealed by the container model, by considering countries in our dataset that include more than 1000 individuals. First, we investigate the divergence between the number of hierarchical levels for males vs females. We find that the gender divide, measured as the Kullback-Leibler divergence between the distributions, is considerably more pronounced in certain countries compared to others, ranging from 0.09 in Saudi Arabia to 0.0036 in Germany (see Supplementary Table S1).
The gender divide in number of levels correlates with the gender inequality index (see main text, Figure 2B). In fact, under the Kolmogorov-Smirnov test, only in 21 countries out of 53 the difference is significant. We find that in 44 countries, females mobility has more hierarchical levels than males', under the Mann-Whitney U test (see Supplementary Table S1). The opposite is not verified in any country (see Supplementary Ta-ble S1). Then, we study differences in containers sizes. We find that in 48 countries out of 53, the size of level-2 containers is smaller for females compared to males, under the Mann-Whitney U test (see Supplementary Table S2). The Kullback-Leibler divergence between females and males distributions of level-2 containers sizes correlates positively with the Gender Inequality Index (ρ = 0.40, p < 10 −2 ).
Urban/rural differences revealed by the container model. We investigate differences between users living in urban vs rural areas with respect to the size of containers. Data on the urbanisation level in the area surrounding individuals' home locations relies on the GHS Settlement Model grid. For the sake of our analysis, we merged towns and cities into the same category (urban). We find that, at all hierarchical levels, the size of containers is significantly larger for individuals living in rural areas compared to urban individuals (see Supplementary Table S3) under the Mann-Whitney U test. Differences between the distribution of container sizes, measured as Kullback-Leibler divergence, are more pronounced at low hierarchical levels (see Supplementary Table S3).

Differences reflecting the walkability around one's home location.
We investigate if the walkability score in the area surrounding an individual's home location affects the size of containers revealed by our model. We find that there is a negative correlation between the size of containers and the walkability score, for containers at level 2 (ρ = −0.98, p < 10 −9 , see main text, Figure 2D) and 3 (Spearman correlation ρ = 0.68, p = 0.001). The correlation is not significant at significance level α = 0.01 for higher hierarchical levels. These analyses are computed for the 11, 511 individuals living in New Zealand, Australia, Canada and the USA, for which WalkScore data was available (see Material and Methods, Data Description).

gorithm.
In this section, we evaluate the performance of the likelihood maximization algorithm for the container model (see main text, Methods, Model description). First we generate 5, 000 synthetic traces using the container model as follows: • We define a hierarchical partitioning H as a nested grid with L levels, where at each level i there are N i squared containers. L and all values of N i are random integers (from a uniform distribution). Each cell in the grid corresponds to a location.
• We fill the matrix p with values chosen from a uniform distribution, and we then normalize it appropriately (to ensure that values correspond to probabilities of travelling at given level-distance). Note that here, for simplicity, we consider that the probability of travelling at given level-distance does not depend on the distance from home.
• We assign to containers their attractiveness (stored in a) as follows. The distribution of attractiveness for the children of any given parent container is a power law (with exponent chosen from a uniform distribution), in agreement with the literature. 11 • Given the above choices of H, a and p, we simulate sequences of 2, 000 locations following the procedure described in the main text (Methods, Generation of traces).
We then estimate H, a and p using the one-level-at-a-time algorithm for maximum likelihood optimization described in the main text (Methods, Likelihood optimization). We evaluate the performance of the algorithm by computing: the absolute difference D between the original and recovered number of levels (we find that P(D = 0)=0.95, P(D ≤ 1)=0.99), the cophenetic correlation 30 between the original and recovered hierarchical structure (we find that the cophenetic correlation has median value 1), and the relative difference

Supplementary Note 4: Comparison across models.
We compare the ability to reproduce the statistical properties of human mobility of several individual-level models: the container model presented here, the Exploration and Preferential Return model (epr), 3 the memory EPR (m-epr), 25 Table 4). Using the log-likelihood ratio test, 34 we find that the container model provides a better description than the others across the measures considered at significance level α = 0.01. Further, we find that while the evolution of individuals' average radius of gyration over time is well described by a logarithmic growth in all cases, only the fit f(x) =a+b·log(x) for the container model traces is consistent with the real data within errors: (a data = −628 ± 138, b data = 1884 ± 44, a container = −523 ± 27, b container = 1844 ± 9), while the EPR model fit has markedly different parameters (a EPR = −5700 ± 92, b EPR = 6087 ± 36).
In panels b and f filled areas include the interquartile range. For each individual, we fitted the models considering a training period of 1 year. The data used here for validation corresponds to the 50 individual displacements following the training period. * p≤0.05, ** p≤0.01, *** p≤0.001 Extended Data Table S4: The container model describes unseen data better than other individual mobility models. The log-likelihood ratio R 34 comparing the likelihood of the container model to other models (one per row). When R is positive, the container model has higher likelihood compared to the alternative, and vice-versa. The table reports also the p-values associated with R (*p ≤ 0.05,**p ≤ 0.01,***p ≤ 0.001). Results are shown for the D1 and D2 datasets. We report the results obtained considering different properties of the trajectories: the probability of displacement length P(∆r), the probability of number of visits per location P( f l ), the probability of the difference between the uncorrelated and temporal entropy P(S unc − S temp ). and m-EPR (green line) models.