Understanding individual human mobility patterns

Understandin g indi vidual hu man mobility patter ns Marta C. Gonz ´ alez, 1, 2 C ´ esar A. Hidalgo, 1 and Albert-L ´ aszl ´ o Barab ´ asi 1, 2, 3 1 Center for Comple x Network Resear c h and D epartmen t of P hysics and Computer Science , Univer sity of N otr e Dame, Notr e Dame IN 46556. 2 Center for Complex Network Resear c h and Department of Physics, Biolog y and Compute r Science , Northeas tern University , Boston MA 0 2115. 3 Center for Ca ncer Systems Bio log y , Dana F arber Cancer Institu te, Boston, MA 02115. (Dated: Nove mber 26, 2024) Despite their importance f or urban planning [1], traffic for ecasting [2], and the spread of biological [3, 4, 5] and mobile viruses [6], our understanding of the basic laws gover n- ing human motion remains li mi ted thanks to the l ack of tools to monitor the ti me resolved location of individuals. Here we s tudy the trajectory of 100 , 000 anonymized mobile phone users whose position is tracke d f or a six month period. W e find that in contrast w i th the random trajectories predicted by the prev ailing L ´ evy flight and random wa l k models [7], human trajectories s how a high degree of temporal a nd spatial regu larity , each individual being characterized by a time indepe ndent characteristic length scale and a significant pr ob- ability to r etur n to a few highly fr equented locations. After correcting for differ ences in tra vel distances and the inherent anisotr opy of each trajectory , the individual tra vel p atterns collapse i nto a singl e spatial probability distribution, indicating that despite the diversity of their travel history , humans follow simple repr o ducible patterns. This inher ent simil arity in trav el pattern s could impact all phenomena driven by human mobil i ty , fr om epidemic pr ev ention to emergency re sponse, urban planning and agent based modeling. Giv en the many unknown f actors that i nfluence a po pulation’ s mobil ity patterns, ranging from means of transportation to job and family imposed restrictions and priori ties, human trajectories are often approxim ated wit h various random walk or diffusion m odels [7, 8]. Indeed, early mea- surements on albatros ses, b umblebees, deer and m onkeys [ 9, 10] and m ore recent ones on m arine predators [11] sugg ested t hat ani mal trajec tory is approxim ated by a L ´ evy fl ight [12, 13], a random walk whose step size ∆ r follows a powe r -law distrib ution P (∆ r ) ∼ ∆ r − (1+ β ) with β < 2 . While the L ´ evy statist ics for some anim als require furth er study [14], Brockmann et al . [7] generalized this finding to humans, docum enting that the distribution of di stances between consecutive sight- 2 ings of nearly half-million bank notes is fat tailed. Giv en th at money is carried by individuals, bank note d ispersal i s a proxy for human movement, suggest ing that human trajectories are best modeled as a continuo us tim e random walk wit h fat tailed displacements and waiting tim e dis- tributions [7]. A particle following a L ´ evy flight has a sign ificant probability t o travel very long distances in a si ngle step [12, 13], which appears to be c onsistent with human t ra vel patt erns: m ost of the tim e we trav el onl y over short dist ances, between hom e and work, while occasionally we take longer trips. Each consecutive sighti ngs of a bank not e reflects the com posite mo tion of two or more in di- viduals, who owned the bill between two reported sighti ngs. Thus it is not clear i f the observed distribution reflects t he mo tion of individual users, or so me hit ero u nknown con volution between population based heterogeneiti es and i ndividual human trajectories. Contrary to bank not es, mo- bile phones are carried by the same individual during his/her daily routine, of fering the best proxy to capture individual human trajectories [15, 16, 17, 18, 19]. W e used two d ata sets t o explore the mo bility pattern of individuals. The first ( D 1 ) cons ists of the mobili ty patterns recorded over a six month period for 100 , 000 individuals selected randomly from a sam ple of over 6 m illion anonymized m obile phone us ers. Each tim e a user initiates or recei ves a call or SMS, the location of the t ower r outing the communi cation is recorded, allowing us t o reconstruct the user’ s time resolved trajectory (Figs. 1a and b). The time between consecutive calls follows a b ursty pattern [20] (see Fig. S1 in the SM), in dicating that whi le most consecutive calls are placed soon after a pre vious call, occasionally there are long periods without an y call activity . T o make sure t hat the obtained results are not affec ted by the irregular call patt ern, we also study a data set ( D 2 ) that captures the location of 206 mobile phone users, re corded e very two hours for an ent ire week. In b oth datasets the spatial resolution is determi ned by the local dens ity of the m ore than 10 4 mobile towers, registering movement o nly when the user moves between areas serviced by dif ferent to wers. T he a verage service area of each tower is a pproximately 3 km 2 and over 30% of the t ower s cov er an are a of 1 km 2 or less. T o explore the stat istical properties of the pop ulation’ s mo bility patterns we measured the dis- tance bet ween user’ s p ositions at consecutive calls, captu ring 16 , 264 , 308 disp lacements for the D 1 and 10 , 407 displ acements for t he D 2 datasets. W e find t hat the dist ribution of displacements over all users is well approximated by a truncated po wer -law P (∆ r ) = (∆ r + ∆ r 0 ) − β exp ( − ∆ r /κ ) , (1) 3 with β = 1 . 75 ± 0 . 15 , ∆ r 0 = 1 . 5 km and cutof f values κ | D 1 = 400 km, and κ | D 2 = 80 km (Fig. 1c, see the SM for statis tical validation). Note th at the o bserved scalin g exponent is not far from β B = 1 . 59 observed in Ref. [7] for bank note dispersal, suggesting that the two distrib utions may capture the same fundamental mechanism dri ving human mobility patterns. Equation (1) suggests that human motion follows a truncated L ´ evy flight [7]. Y et, the observed shape o f P (∆ r ) could b e explained by three distinct hy potheses: A. Each individual follows a L ´ evy trajectory wi th jump size dis tribution giv en by (1). B. The observed d istribution captures a population based heterogeneity , c orresponding to the inherent dif ferences between individuals. C. A pop ulation based h eterogeneity coexists w ith i ndividual L ´ evy trajectories, hence (1) represents a con volution of hypothesis A and B . T o disting uish between hypot heses A, B and C we calcul ated the radius of gyrati on for each user (see Methods), interpreted as t he typical distance trav eled by user a when observed up to time t (Fig. 1b). Next, we determined t he radius of gyration distribution P ( r g ) by calculati ng r g for all users in samples D 1 and D 2 , finding that they also can b e approximat ed with a truncated power -law P ( r g ) = ( r g + r 0 g ) − β r exp ( − r g /κ ) , (2) with r 0 g = 5 . 8 km , β r = 1 . 65 ± 0 . 1 5 and κ = 350 km (Fig. 1d, see SM for statis tical validation). L ´ evy flights are characterized b y a high degree of intrins ic heterogeneity , raising the possibi lity that (2) could emerge from an ensemble o f ident ical agents, each following a L ´ evy trajectory . Therefore, we d etermined P ( r g ) for an ensemble of agents following a Random W alk ( RW ), L ´ evy-Flight ( LF ) or T runcated L ´ evy-Flight ( T LF ) (Figu re 1d) [8, 12, 13]. W e find th at an en- semble o f L ´ evy agents d isplay a s ignificant d egree of heterogeneit y in r g , yet is not sufficient to explain the truncated power l aw dist ribution P ( r g ) exhibited by the mobile phone us ers. T aken together , Figs. 1c and d suggest that the dif ference in the range of typical mobility patterns of indi- viduals ( r g ) h as a s trong impact on the t runcated L ´ evy behavior seen in (1), ruling out hypothesi s A. If ind ividual trajectories are described by a LF or T LF , then the radius of gyration should increase i n t ime as r g ( t ) ∼ t 3 / (2+ β ) [21, 22] while for a RW r g ( t ) ∼ t 1 / 2 . That is, th e l onger we observe a user , the higher t he chances that she/ he will trav el to areas not vi sited before. T o check the validity of these predicti ons we measured the ti me dependence of the radius o f g yration for users whose gyration radi us would be considered sm all ( r g ( T ) ≤ 3 km ), mediu m ( 20 < r g ( T ) ≤ 30 km) or lar ge ( r g ( T ) > 100 km) at the end of our observation period ( T = 6 mo nths). The 4 results indicate that t he t ime dependence of t he av erage radius of gyration of mobile phone users is bett er approxi mated by a logarithmi c increase, not only a manifestly slower d ependence th an the one predicted by a power law , but one that m ay appear s imilar to a s aturation process (Fig. 2a and Fig. S4). In Fig. 2 b, we ha ve chosen us ers with similar as ymptotic r g ( T ) after T = 6 months , and measured the jump size distrib ution P (∆ r | r g ) for each group. As the inset of Fig. 2b sho ws, users with small r g tra vel most ly over small distances, whereas thos e wit h large r g tend to display a combination of m any sm all and a few larger jump sizes. Once we rescale the distributions wit h r g (Fig. 2b), we find that the data collapses into a si ngle curve, sugges ting that a single jump size distribution characterizes al l users, independent of their r g . This indicates t hat P (∆ r | r g ) ∼ r − α g F (∆ r /r g ) , where α ≈ 1 . 2 ± 0 . 1 and F ( x ) is an r g independent function with asym ptotic beha vior F ( x < 1) ∼ x − α and rapidly decreasing for x ≫ 1 . Therefore the trav el patterns of i ndividual users m ay be approxi mated by a L ´ evy flight up to a di stance characterized by r g . Most important, h owe ver , is the fact that the ind ividual trajectori es are bou nded beyond r g , thus lar ge displacements which are the source of t he distinct and anom alous nature of L ´ evy flights, are statisti cally absent. T o understand the relations hip between the diffe rent exponents, we note that the measured probability distributions are related by P (∆ r ) = R ∞ 0 P (∆ r | r g ) P ( r g ) dr g , which suggests (see SM) that up to the leading order we hav e β = β r + α − 1 , c onsistent, within error bar s, with the m easured exponents. Thi s in dicates that t he observed jump s ize dist ribution P (∆ r ) i s in fact the con volution between the statistics of indi vidual trajectories P (∆ r g | r g ) and the population heterogeneity P ( r g ) , consistent with hypothesis C. T o uncov er the mechanism stabil izing r g we measured the return probabili ty for each indi- vidual F pt ( t ) [22], defined as the probability that a user returns to the position w here it was first observed after t hours (Fig. 2c). For a two dim ensional random walk F pt ( t ) should foll ow ∼ 1 / ( t ln( t ) 2 ) [22]. In contrast, we find that the return probability is characterized by s e veral peaks at 24 h, 48 h , and 72 h, capturing a stron g tendency of h umans to return to lo cations they vi sited before, describing the recurrence and temporal periodicity inherent to human mobility [23, 24]. T o explore if ind ividuals return to t he same location over and over , we ranked each l ocation based on the num ber of tim es an in dividual was recorded in its v icinity , such that a location with L = 3 represents the t hird most visited location for the selected individual. W e find that the probability of findi ng a user at a location with a gi ven rank L i s well approximat ed by P ( L ) ∼ 1 /L , independent of the number of locations visited by the user (Fig. 2d). Therefore people dev ote most 5 of their time to a fe w locations, while spending their r emaining time in 5 to 50 places, visited with diminish ed re gularity . Therefore, the observ ed logarithmic saturation of r g ( t ) is rooted in the high degree o f regularity in their daily tra vel patterns, captured by the high re turn probabilities (Fig. 2b) to a few highly frequented locations (Fig. 2d). An important quantity for modeling human mobility patterns is the probability Φ a ( x, y ) to find an individual a in a given position ( x, y ). As it is evident from Fig. 1b, in dividuals li ve and tra vel in different regions, yet each u ser can be ass igned to a well defined area, defined b y home and workplace, where she or he can be found most of the time. W e can compare the trajectories o f diffe rent users by diagonalizing each trajectory’ s inerti a tensor , providing the probability of finding a user in a giv en position (see Fig. 3a) in the us er’ s intrinsic reference frame (see SM for the details). A st riking feature of Φ( x, y ) is i ts prominent spati al anisotropy in this int rinsic reference frame (note the different scales in Fig 3a), and we find that t he larger an individual’ s r g the more pronounced is this ani sotropy . T o quantify t his effec t we defined t he anisot ropy ratio S ≡ σ y /σ x , where σ x and σ y represent the standard deviation of the trajectory measured in the u ser’ s i ntrinsic reference frame (see SM). W e find that S decreases monoton ically with r g (Fig. 3c), bein g well approximated with S ∼ r − η g , for η ≈ 0 . 12 . Gi ven the small value o f the scaling exponent, other functional forms may of fer an equally good fit, thus mechanistic models are required to identify if this represents a true scaling law , or only a reasonable approximation to the data. T o compare t he t rajectories o f dif ferent users we remove t he in dividual anis otropies, rescal- ing each user trajectory with it s respectiv e σ x and σ y . The rescaled ˜ Φ( x/σ x , y /σ y ) di stribution (Fig. 3b) is similar for groups of users with considerably differ ent r g , i.e. , after the anisotropy and the r g dependence is removed all individuals appear to follo w the same uni versal ˜ Φ( ˜ x, ˜ y ) prob- ability distribution. This is particularly e vi dent in Fig. 3d, where we s how the cross section of ˜ Φ( x/σ x , 0) for th e t hree g roups of us ers, finding t hat apart from th e n oise in the data the curves are indisti nguishable. T aken together , our results suggest that the L ´ evy statistics observed in bank note measurements capture a con volution of the population heterogeneity (2) a nd the motion of indi vidual users. Indi - viduals display sig nificant regularity , as the y return to a fe w highly frequented locations, l ike home or work. This regularity do es not apply to the bank notes: a bill always follows the trajectory of its current owner , i.e. dollar bills diff use, b ut humans do not. The fact that individual trajectories are characterized by the sam e r g -independent t wo d imen- sional p robability di stribution ˜ Φ( x/σ x , y /σ y ) suggests that key s tatistical characteristics of indi- 6 vidual trajectories are lar gely indistinguishable after re scaling. Therefore, our results establish the basic ingredients of realistic agent based models, requiring us t o place users in nu mber propor- tional with the p opulation density of a gi ven region and assig n each user an r g taken from t he observed P ( r g ) dis tribution. Using the predicted anisotropic rescaling, combined with the density function ˜ Φ( x, y ) , whose shape is provided as T able 1 in the SM, we can obtain the li kelihood of finding a user in any l ocation. Gi ven the known correlations between spatial proximity and social links, our results could help quantify the role of space in network de velopment and ev olu- tion [25, 26, 27, 28, 29] and improve our understanding of dif fusion processes [8, 30]. W e thank D. Brockmann, T . Geisel, J. Park, S. Redner , Z. T oroczkai and P . W ang for discus- sions and com ments on th e m anuscript. This work was supported by the James S. McDonnell Foundation 21st Century Initi ativ e in Studyi ng Complex System s, th e Nat ional Science Founda- tion w ithin the DDD AS (CNS-0540348), ITR (DMR-0426737 ) and IIS-0513650 programs, and the U.S. Office of Na v al Research A w ard N0001 4-07-C. Data analysis was performed on the Notre Dame Biocompl exity Cluster supported in part by NSF MRI Grant No. DBI-0420980. C.A. Hi - dalgo ackno wledges support from the K ellogg Institute at Notre Dame. Supplemen tary Inf ormation is linked to the online version of the paper at www .nature.com/ nature. 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Physical Revie w Letter s 96 , 088702 (2006). [30] Cecconi, F ., Marsili, M., Banav ar , J.R. & Maritan, A. Dif fusio n, pee r p ressu re, and tailed distrib utions. 9 Physical Revie w Letter s 89 , 088102 (2002). 10 FIG. 1: B asic human mobility patter ns . a, W eek-lon g trajecto ry of 40 mobile phone users indicate that most indi viduals trav el only ove r short distances , b ut a few regularl y mov e over hundreds of kilomete rs. Panel b, displays the detailed trajectory of a single user . The dif feren t phone towers are sho w n as green dots, and the V oronoi lattice in grey marks the approximate reception area of each tower . The datas et studie d by us records only the identity of the closest tower to a mobile user , thus we can not identify the positi on of a user within a V oronoi cell. The trajector y of the user sho wn in b is construc ted from 186 two hou rly reports, during w hich the user visited a total of 12 dif ferent locations (tower vic inities ). A mong these, the user is found 96 and 67 occasion s in the two m ost preferr ed locations , the frequenc y of visits for each location being sho wn as a vertical bar . The circle represents the radius of gyration centered in the trajectory’ s center of mass. c, P robabil ity density function P (∆ r ) of tra ve l distances obtained for the two studied datasets D 1 and D 2 . T he solid line indicat es a trun cated po wer law whose paramete rs are pro vided in the text (see E q. 1). d, T he distrib ution P ( r g ) of the radius of gyrati on measured for the users, where r g ( T ) was mea sured after T = 6 months of observ ation. The solid line represent a similar trun cated po wer law fit (see Eq. 2). T he d otted, da shed and dot-d ashed curv es show P ( r g ) o btaine d from th e standard null models ( RW , LF and T LF ), where for the T LF we used the same step size distrib ution as the one measured for the mobile phone users. 11 FIG. 2: Th e bound ed nature of h u man trajectories . a, Radius of gyration , h r g ( t ) i vs time for mobile phone users separated i n three groups acc ordin g to their final r g ( T ) , wher e T = 6 months. The black curves corres pond to the analytical prediction s fo r the random walk models , increasing in time as h r g ( t ) i| LF ,T LF ∼ t 3 / 2+ β (solid ), and h r g ( t ) i| RW ∼ t 0 . 5 (dotte d). T he dashed curves corresp ondin g to a loga rithmic fit of the form A + B ln( t ) , where A and B depe nd on r g . b, Probability density functio n o f indi vidual trav el distance s P (∆ r | r g ) for users with r g = 4 , 10 , 40 , 100 and 200 km. As the inse t sho ws, each group displays a quite dif ferent P (∆ r | r g ) dist rib ution. After rescaling the distan ce and the distrib ution w ith r g (main pane l), the dif ferent curves collaps e. The soli d line (po wer law) is sho wn as a guide to the eye . c, Return probability distrib ution, F pt ( t ) . The prominent peaks capture the tendenc y of humans to regular ly return t o the locat ions the y v isited before, in contrast with the smooth asymptotic behav ior ∼ 1 / ( t ln ( t ) 2 ) (solid li ne) predicted fo r random walks. d, A Zipf pl ot sho wing the frequenc y of vi siting diffe rent l ocatio ns. The symbols c orresp ond to users that ha ve been observed to visit n L = 5 , 10 , 30 , and 50 differe nt location s. Denoting with ( L ) the rank of the loc ation liste d in th e or der of the visit f requen cy , the data is well approximated by R ( L ) ∼ L − 1 . The inset is the s ame plot in line ar scale, illustrat ing that 40% of the time indi viduals are found at th eir first two pre ferred locations. 12 FIG. 3: Th e shape of human trajectori es. a, The probabi lity densit y functio n Φ( x, y ) of fi nding a mobile phone user in a locati on ( x, y ) in the user ’ s intrinsic referen ce frame (see SM for details). The three plots , from left to right, were generated for 10 , 000 users with: r g ≤ 3 , 20 < r g ≤ 30 and r g > 100 km. The traject ories become more anisotr opic as r g increa ses. b, After scalin g each position with σ x and σ y the resulti ng ˜ Φ( x/σ x , y /σ y ) has approx imately the same shape for each group. c, The change in the shape of Φ( x, y ) can be quantified calcula ting the isotrop y ratio S ≡ σ y /σ x as a function of r g , which decrease s as S ∼ r − 0 . 12 g (solid line). Error bars represent the standard error . d, ˜ Φ( x/σ x , 0) represent ing the x-axis cross sectio n of the rescal ed distrib ution ˜ Φ( x/σ x , y /σ y ) sho w n in b .