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Explaining the power-law distribution of human mobility through transportationmodality decomposition

Abstract

Human mobility has been empirically observed to exhibit Lévy flightcharacteristics and behaviour with power-law distributed jump size. The fundamentalmechanisms behind this behaviour has not yet been fully explained. In thispaper, we propose to explain the Lévy walk behaviour observed in humanmobility patterns by decomposing them into different classes according tothe different transportation modes, such as Walk/Run, Bike, Train/Subway orCar/Taxi/Bus. Our analysis is based on two real-life GPS datasets containingapproximately 10 and 20 million GPS samples with transportation mode information.We show that human mobility can be modelled as a mixture of different transportationmodes and that these single movement patterns can be approximated by a lognormaldistribution rather than a power-law distribution. Then, we demonstrate thatthe mixture of the decomposed lognormal flight distributions associated witheach modality is a power-law distribution, providing an explanation to theemergence of Lévy Walk patterns that characterize human mobility patterns.

Introduction

Understanding human mobility is crucial for epidemic control1,2,3,4,urban planning5,6, traffic forecasting systems7,8and, more recently, various mobile and network applications9,10,11,12,13.Previous research has shown that trajectories in human mobility have statisticallysimilar features as Lévy Walks by studying the traces of bank notes14, cell phone users' locations15 and GPS16,17,18,19. According to the this model, human movement containsmany short flights and some long flights and these flights follow a power-lawdistribution.

Intuitively, these long flights and short flights reflect different transportationmodalities. Figure 1 shows a person's one-day tripwith three transportation modalities in Beijing based on the Geolife dataset(Table 1)20,21,22. Starting fromthe bottom right corner of the figure, the person takes a taxi and then walksto the destination in the top left part. After two hours, the person takesthe subway to another location (bottom left) and spends five hours there.Then the journey continues and the person takes a taxi back to the originallocation (bottom right). The short flights are associated with walking andthe second short-distance taxi trip, whereas the long flights are associatedwith the subway and the initial taxi trip. Based on this simple example, weobserve that the flight distribution of each transportation mode is different.

Table 1 The Geolife and the NokiaMDC Human Mobility Datasets
Figure 1
figure 1

Illustration of a synthetic trail (taxi, walk, subway, walk, taxi,walk) for one day trip and their corresponding flights.

This figure shows that the flight distribution of each transportationmode (walk, taxi, subway) is very different.

In this paper, we study human mobility with two large GPS datasets, theGeolife and Nokia MDC datasets (approximately 20 million and 10 million GPSsamples respectively), both containing transportation mode information suchas Walk/Run, Bike, Train/Subway or Car/Taxi/Bus. The four transportation modes(Walk/Run, Bike, Train/Subway and Car/Taxi/Bus) cover the most frequentlyused human mobility cases. First, we simplify the trajectories obtained fromthe datasets using a rectangular model, from which we obtain the flight length16. Here a flight is the longest straight-line trip from one pointto another without change of direction16,19. One trail froman origin to a destination may include several different flights (Fig.1). Then, we determine the flight length distributions for differenttransportation modes. We fit the flight distribution of each transportationmode according to the Akaike information criteria23 in orderto find the best fit distribution.

We show that human movement exhibiting different transportation modalitiesis better fitted with the lognormal distribution rather than the power-lawdistribution. Finally, we demonstrate that the mixture of these transportationmode distributions is a power-law distribution based on two new findings:first, there is a significant positive correlation between consecutive flightsin the same transportation mode and second, the elapsed time in each transportationmode is exponentially distributed.

The contribution of this paper is twofold. First, we extract the distributionfunction of displacement with different transportation modes. This is importantfor many applications7,8,9,10,11. For example, a population-weightedopportunities (PWO) model24 has been developed to predict humanmobility patterns in cities. They find that there is a relatively high mobilityat the city scale due to highly developed traffic systems inside cities. Ourresults significantly deepen the understanding of urban human mobilities withdifferent transportation modes. Second, we demonstrate that the mixture ofdifferent transportations can be approximated as a truncated Lévy Walk.This result is a step towards explaining the emergence of Lévy Walkpatterns in human mobility.

Results

Power-law fit for overall flight

First, we fit the flight length distribution of the Geolife and Nokia MDCdatasets regardless of transportation modes (see Methods section). We fittruncated power-law, lognormal, power-law and exponential distribution (see Supplementary Table S1). We find that the overall flight length(l) distributions fit a truncated power-law P(l) lαeγlwith exponent α as 1.57 in the Geolife dataset (γ =0.00025) and 1.39 in the Nokia MDC dataset (γ = 0.00016) (Fig. 2), better than other alternatives such as power-law,lognormal or exponential. Figure 2 illustrates the PDFsand their best fitted distributions according to Akaike weights. The bestfitted distribution (truncated power-law) is represented as a solid line andthe rest are dotted lines. We use logarithm bins to remove tail noises16,25. Our result is consistent with previous research (Refs. 14–19) and the exponent α is close totheir results.

Figure 2
figure 2

Power-law fit for overall flight.

(a–b) Power-law fitting of all flights regardless of transportationmodes in the Geolife and the Nokia MDC dataset. The green points refer tothe flight length samples obtained from the GeoLife and the Nokia MDC dataset,while the solid red line represents the best fitted distribution accordingto Akaike weights. The overall flight length distribution regardless of transportationmodes is well fitted with a truncated power-law distribution.

We show the Akaike weights for all fitted distributions in the SupplementaryTable S2. The Akaike weight is a value between 0 and 1. The larger itis, the better the distribution is fitted23,25. The Akaikeweights of the power-law distributions regardless of transportation modesare 1.0000 in both datasets. The p-value is less than 0.01 in all our tests,which means that our results are very strong in terms of statistical significance.Note that here the differences between fitted distributions are not remarkableas shown in the Fig. 2, especially between the truncatedpower-law and the lognormal distribution. We use the loglikelihood ratio tofurther compare these two candidate distributions. The loglikelihood ratiois positive if the data is more likely in the power-law distribution, andnegative if the data is more likely in the lognormal distribution. The loglikelihoodratio is 1279.98 and 3279.82 (with the significance value p < 0.01)in the Geolife and the NokiaMDC datasets respectively, indicating that thedata is much better fitted with the truncated power-law distribution.

Lognormal fit for single transportation mode

However, the distribution of flight lengths in each single transportationmode is not well fitted with the power-law distribution. Instead, they arebetter fitted with the lognormal distribution (see SupplementaryTable S2). All the segments of each transportation flight length arebest approximated by the lognormal distribution with different parameters.In Fig. 3 and Supplementary Fig. S1,we represent the flight length distributions of Walk/Run, Bike, Subway/Trainand Car/Taxi/Bus in the Geolife and the Nokia MDC dataset correspondingly.The best fitted distribution (lognormal) is represented as a solid line andthe rest are dotted lines.

Figure 3
figure 3

Lognormal fit for single transportation mode in the Geolife dataset.

(a–d) Flight distribution of all transportation modes (Car/Taxi/Bus,Walk/Run, Subway/Train, Bike). The green points refer to the flight lengthsamples obtained from the GeoLife, while the solid blue line represents thebest fitted distribution according to Akaike weights. The flight length distributionin each transportation mode is well fitted with a lognormal distribution.

Table 2 shows the fitted parameter for all thedistributions (α in the truncated power-law, μ and σin the lognormal). We can easily find that the μ is increasingover these transportation modes (Walk/Run, Bike, Car/Taxi/Bus and Subway/Train),identifying an increasing average distance. Compared to Walk/Run, Bike orCar/Taxi/Bus, the flight distribution in Subway/Train mode is more right-skewed,which means that people usually travel to a more distant location by Subway/Train.

Table 2 Parameters of fitteddistributions in the GeoLife and in the Nokia MDC datasets. The p-value isless than 0.01 in all the fitted distributions, identifying a strong statisticalsignificance

It must be noted that our findings for the Car/Taxi/Bus mode are differentfrom these recent research results26,27, which also investigatedthe case of a single transportation mode and found that the scaling of humanmobility is exponential by examining taxi GPS datasets. The differences aremainly because few people tend to travel a long distance by taxi due to economicconsiderations. So the displacements in their results decay faster than thosemeasured in our Car/Taxi/Bus mode cases.

Mechanisms behind the power-law pattern

We characterize the mechanism of the power-law pattern with Lévyflights by mixing the lognormal distributions of the transportation modes.Previous research has shown that a mixture of lognormal distributions basedon an exponential distribution is a power-law distribution30,31,32,33.Based on their findings, we demonstrate that the reason that human movementfollows the Lévy Walk pattern is due to the mixture of the transportationmodes they take.

We demonstrate that the mixture of the lognormal distributions of differenttransportation modes (Walk/Run, Bike, Train/Subway or Car/Taxi/Bus) is a power-lawdistribution given two new findings: first, we define the change rate as therelative change of length between two consecutive flights with the same transportmode. The change rate in the same transportation mode is small over time.Second, the elapsed time between different transportation modes is exponentiallydistributed.

Lognormal in the same transportation mode

Let us consider a generic flight lt. The flight lengthat next interval of time lt+1, given the changerate ct+1, is

It has been found that the change rate ct in the sametransportation mode is small over time9,22. The change rate ctreflects the correlation between two consecutive displacements in one trip.To obtain the pattern of correlation between consecutive displacements ineach transportation mode, we plot the flight length point (lt, lt+1)from the GeoLife dataset (Fig. 4). Here ltrepresents the t-th flight length and lt+1represents the t + 1-th flight length in a consecutive trajectory inone transportation mode34. Figure 4 showsthe density of flight lengths correlation in Car/Taxi/Bus, Walk/Run, Subway/Trainand Bike correspondingly. (lt, lt+1)are posited near the diagonal line, which identifies a clear positive correlation.Similar results are also found in the Nokia MDC dataset (see SupplementaryFig. S2).

Figure 4
figure 4

Flight length correlation for each transportation mode.

(a–d) Consecutive Flight length correlation of all transportationmodes (Car/Taxi/Bus, Walk/Run, Subway/Train, Bike) in the GeoLife dataset.A high density of points are near diagonal line lt = lt+ 1, identifying a small difference lt+1ltin the same transportation mode between two time steps.

We use the Pearson correlation coefficient to quantify the strength ofthe correlation between two consecutive flights in one transportation mode35. The value of Pearson correlation coefficient r is shownin the Supplementary Table S3. The p value is lessthan 0.01 in all the cases, identifying very strong statistical significances. ris positive in each transportation mode and ranges from 0.3640 to 0.6445,which means that there is a significant positive correlation between consecutiveflights in the same transportation mode and the change rate ctin the same transportation mode between two time steps is small.

The difference ct in the same transportation mode betweentwo time steps is small due to a small difference lt+1ltin consecutive flights. We sum all the contributions as follows:

We plot the change rate samples ct of the Car/Taxi/Busmode from the Geolife dataset as an example in SupplementaryFig. S3. We observe that the change rate ct fluctuatesin an uncorrelated fashion from one time interval to the other in one transportationmode due to the unpredictable character of the change rate. The Pearson correlationcoefficient accepts the findings at the 0.03–0.13 level with p-valueless than 0.05 (see Supplementary Table S4). By the CentralLimit Theorem, the sum of the change rate ct is normallydistributed with the mean μT and the variance σ2T,where μ and σ2 are the mean and varianceof the change rate ct and T is the elapsed time.Then we can assert that for every time step t, the logarithm of lis also normally distributed with a mean μt and variance σ2t36. Note here that lT is the length of the flightat the time T after T intervals of elapsed time. In the sametransportation mode, the distribution of the flight length with the same changerate mean is lognormal, its density is given by

whichcorresponds to our findings that in each single transportation mode the flightlength is lognormal distributed.

Transportation mode elapsed time

We define elapsed time as the time spent in a particular transportationmode; we found that it is exponentially distributed. For example, the trajectorysamples shown in Fig. 1 contain six trajectories withthree different transportation modes, (taxi, walk, subway, walk, taxi, walk).Thus the elapsed time also consists of six samples (ttaxi1, twalk1, tsubway1, twalk2, ttaxi2, twalk3).The elapsed time t is weighted exponentially between the differenttransportation modes (see Supplementary Fig. S4). Similarresults are also reported in Refs. 26, 37. The exponentially weighted time interval is mainlydue to a large portion of Walk/Run flight intervals. Walk/Run is usually aconnecting mode between different transportation modes (e.g., the trajectorysamples shown in Fig. 1) and Walk/Run usually takesmuch shorter time than any other modes. Thus the elapsed time decays exponentially.For example, 87.93% of the walk flight distance connecting other transportationmodes is within 500 meters and the travelling time is within 5 minutes inthe Geolife dataset.

Mixture of the transportation modes

Given these lognormal distributions Psinglemode(l)in each transportation mode and the exponential elapsed time t betweendifferent modes, we make use of mixtures of distributions. We obtain the overallhuman mobility probability by considering that the distribution of flightlength is determined by the time t, the transportation mode changerate ct mean μ and variance σ2.We obtain the distribution of single transportation mode distribution withthe time t, the change rate mean μ and variance σ2fixed. We then compute the mixture over the distribution of t since tis exponentially distributed over different transportation modes with an exponentialparameter λ. If the distribution of l, p(l, t),depends on the parameter t. t is also distributed accordingto its own distribution r(t). Then the distribution of l, p(l)is given by . Here the t in p(l, t)is the same as the t in the r(t). r(t)is the exponential distribution of elapsed time t with an exponentialparameter λ.

So the mixture (overall flight length Poverall(l))of these lognormal distributions in one transportation mode given an exponentialelapsed time (with an exponent λ) between each transportation modeis

which can be calculated to give

where the power law exponent α′ is determined by 31,32,33. The calculation to obtain α′is given in Supplementary Note 1. If we substitute theparameters presented in Table 2, we will get the α′= 1.55 in the Geolife dataset, which is close to the original parameter α= 1.57 and α′ = 1.40 in the Nokia MDC dataset, which isclose to the original parameter α = 1.39. The result verifies thatthe mixture of these correlated lognormal distributed flights in one transportationmode given an exponential elapsed time between different modes is a truncatedpower-law distribution.

Discussion

Previous research suggests that it might be the underlying road networkthat governs the Lévy flight human mobility, by exploring the humanmobility and examining taxi traces in one city in Sweden19.To verify their hypothesis, we use a road network dataset of Beijing containing433,391 roads with 171,504 conjunctions and plot the road length distribution16,28,29. As shown in Supplementary Fig. S5,the road length distribution is very different to our power-law fit in flightsdistribution regardless of transportation modes. The α in roadlength distribution is 3.4, much larger than our previous findings α= 1.57 in the Geolife and α = 1.39 in the Nokia MDC. Thus the underlyingstreet network cannot fully explain the Lévy flight in human mobility.This is mainly due to the fact that it does not consider many long flightscaused by metro or train and people do not always turn even if they arriveat a conjunction of a road. Thus the flight length tails in the human mobilityshould be much larger than those in the road networks.

Methods

Data Sets

We use two large real-life GPS trajectory datasets in our work, the Geolifedataset20 and the Nokia MDC dataset38. The keyinformation provided by these two datasets is summarized in Table1. We extract the following information from the dataset: flight lengthsand their corresponding transportation modes.

Geolife20,21,22 is a public dataset with 182 users'GPS trajectory over five years (from April 2007 to August 2012) gathered mainlyin Beijing, China. This dataset contains over 24 million GPS samples witha total distance of 1,292,951 kilometers and a total of 50,176 hours.It includes not only daily life routines such as going to work and back homein Beijing, but also some leisure and sports activities, such as sightseeing,and walking in other cities. The transportation mode information in this datasetis manually logged by the participants.

The Nokia MDC dataset38 is a public dataset from Nokia ResearchSwitzerland that aims to study smartphone user behaviour. The dataset containsextensive smartphone data of two hundred volunteers in the Lake Geneva regionover one and a half years (from September 2009 to April 2011). This datasetcontains 11 million data points and the corresponding transportation modes.

Obtaining Transportation Mode and The Corresponding Flight Length

We categorize human mobility into four different kinds of transportationmodality: Walk/Run, Car/Bus/Taxi, Subway/Train and Bike. The four transportationmodes cover the most frequently used human mobility cases. To the best ofour knowledge, this article is the first work that examines the flight distributionwith all kinds of transportation modes in both urban and intercity environments.In the Geolife dataset, users have labelled their trajectories with transportationmodes, such as driving, taking a bus or a train, riding a bike and walking.There is a label file storing the transportation mode labels in each user'sfolder, from which we can obtain the ground truth transportation mode eachuser is taking and the corresponding timestamps. Similar to the Geolife dataset,there is also a file storing the transportation mode with an activity ID inthe Nokia MDC dataset. We treat the transportation mode information in thesetwo datasets as the ground truth.

In order to obtain the flight distribution in each transportation mode,we need to extract the flights. We define a flight as the longest straight-linetrip from one point to another without change of direction16,19.One trail from an original to a destination may include several differentflights (Fig. 1). In order to mitigate GPS errors, werecompute a position by averaging samples (latitude, longitude) every minute.Since people do not necessarily move in perfect straight lines, we need toallow some margin of error in defining the ‘straight’ line. Weuse a rectangular model to simplify the trajectory and obtain the flight length:when we draw a straight line between the first point and the last point, thesampled positions between these two endpoints are at a distance less than10 meters from the line. The same trajectory simplification mechanism hasbeen used in other articles which investigates the Lévy walk natureof human mobility16. We map the flight length with transportationmodes according to timestamp in the Geolife dataset and activity ID in theNokia MDC dataset and obtain the final (transportation mode, flight length)patterns. We obtain 202,702 and 224,723 flights with transportation mode knowledgein the Geolife and Nokia MDC dataset, respectively.

Identifying the Scale Range

To fit a heavy tailed distribution such as a power-law distribution, weneed to determine what portion of the data to fit (xmin)and the scaling parameter (α). We use the methods from Refs. 30, 39 to determine xminand α. We create a power-law fit starting from each value in thedataset. Then we select the one that results in the minimal Kolmogorov-Smirnovdistance, between the data and the fit, as the optimal value of xmin.After that, the scaling parameter α in the power-law distributionis given by

where xiare the observed values of xi > xminand n is number of samples.

Akaike weights

We use Akaike weights to choose the best fitted distribution. An Akaikeweight is a normalized distribution selection criterion23.Its value is between 0 and 1. The larger the value is, the better the distributionis fitted.

Akaike's information criterion (AIC) is used in combination with Maximumlikelihood estimation (MLE). MLE finds an estimator of that maximizes the likelihood function of one distribution. AIC is used to describe the bestfitting one among all fitted distributions,

HereK is the number of estimable parameters in the approximating model.

After determining the AIC value of each fitted distribution, we normalizethese values as follows. First of all, we extract the difference between differentAIC values called Δi,

Then Akaike weights Wi are calculated as follows,

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Acknowledgements

The research at the University of Helsinki was supported by the DIGILEInternet of Things research program, ESENS project funded by Tekes and EITICT Labs. Mirco Musolesi would like to acknowledge the support of EPSRC throughthe grants “Trajectories of Depression” (P/L006340/1) and “TheUncertainty of Identity: Linking Spatiotemporal Information Between Virtualand Real Worlds” (EP/J005266/1). Weixiong Rao's work is partiallysupported by Shanghai Science and Technology Commission (15ZR1443000) andIBM Faculty award 2014. We thank Petteri Nurmi for discussions and feedbackon the manuscript.

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K.Z., M.M., P.H., W.R. and S.T. designed the research based on the initialidea by K.Z. and S.T. K.Z. executed the experiments guided by M.M., P.H.,W.R. and S.T. K.Z. and S.T. performed statistical analyses and prepared thefigures. K.Z., M.M., P.H., W.R. and S.T. wrote the manuscript. All authorsreviewed the manuscript.

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Zhao, K., Musolesi, M., Hui, P. et al. Explaining the power-law distribution of human mobility through transportationmodality decomposition. Sci Rep 5, 9136 (2015). https://doi.org/10.1038/srep09136

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