Exploring Human Mobility Patterns Based on Location Information of US Flights. Bin Jiang and Tao Jia

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1 Eplorg Huma Moblty Pattrs Basd o Locato Iformato of US Flghts B Jag ad Tao Ja Dpartmt of Tchology ad Bult Evromt, Dvso of Gomatcs Uvrsty of Gävl, SE Gävl, Swd Emal: b.jag@hg.s, jatao8@6.com (Draft: Aprl 0, Rvso: August 0) Abstract A rag of arly studs hav b coductd to llustrat huma moblty pattrs usg dffrt trackg data, such as dollar ots, cll phos ad tacabs. Hr, w plor huma moblty pattrs basd o massv trackg data of US flghts. Both topologcal ad gomtrc proprts ar ad dtal. W foud that topologcal proprts, such as traffc volum (btw arports) ad dgr of coctvty (of dvdual arports), cludg both - ad outdgrs, follow a powr law dstrbuto but ot a gomtrc proprty lk travl lgths. Th travl lgths hbt a potal dstrbuto rathr tha a powr law wth a potal cutoff as prvous studs llustratd. W furthr smulatd huma moblty o th stablshd topologs of arports wth varous movg bhavors ad foud that th moblty pattrs ar maly attrbutd to th udrlyg bary topology of arports ad hav lttl to do wth othr factors, such as movg bhavors ad gomtrc dstacs. Apart from th abov fdgs, ths study adopts th had/tal dvso rul, whch s rgularty bhd ay havy-tald dstrbuto for tractg dvdual arports. Th adopto of ths rul for data procssg costtuts aothr major cotrbuto of ths papr. Kywords: scalg of gographc spac, had/tal dvso rul, powr law, gographc formato, agt-basd smulatos Itroducto Huma moblty s a rsarch topc of prmary trst to may dscpls, such as gography, urba plag, pdmology ad v tlcommucato (Hägrstrad 970, Hllr t al. 99, Hufagl, Brockma ad Gsl 00, L t al. 009). Rctly, wth th avalablty of massv amouts of varous trackg data, rsarchrs hav attmptd to vstgat th udrlyg rgularts ad mchasms of huma movmt pattrs (Brockma, Hufag ad Gsl 006, Gozalz, Hdalgo ad Barabás 008, Jag, Y ad Zhao 009, Sog t al. 00, Roth t al. 0). It was foud that huma moblty hbts a scalg proprty wth a hgh probablty of prdctg dvdual ad collctv movmt pattrs. Th trackg data rlatd to US dollar ots (Brockma, Hufag ad Gsl 006), cll phos (Gozalz, Hdalgo ad Barabás 008, Sog t al. 00), ad tacabs (Jag, Y ad Zhao 009) hav b usd. Th scalg proprty dcats that huma actvts gographc spac ar thr radom or v but dmostrat a vry hgh dgr of htrogty trms of dstacs ad prfrrd placs. I ths papr, w adopt locato formato of US flghts, capturd vry 5 uts by GPS rcvrs whl th plas ar srvc. I total, ovr 7 mllo locatos of varous US flghts wr usd to tract formato rlatd to both gomtrc ad topologcal proprts, such as travl lgths, traffc volum, ad arport dgrs of coctvty (by routs or by flghts) for furthr vstgato of huma moblty pattrs. W charactrz huma movmt pattrs by fv havy-tald dstrbutos,.., powr law, powr law wth cutoff, potal, strtchd potal ad logormal dstrbutos (Claust, Shalz ad Nwma 009). I cotrast to arlr studs that sought powr law-lk dstrbutos oly, w blv that all th fv modls ca charactrz huma movmt pattrs. Thus, w mak a par-ws comparso amog th fv modls ad choos o bst ft to charactrz huma movmt pattrs. Aothr dstgushd fatur of ths study s that w placd cosdrabl ffort to data procssg to tract th gomtrc ad topologcal proprts for plorg huma moblty pattrs. Th basc

2 prcpl of th data procssg s th had/tal dvso rul that was formulatd for mprcal data that hav a havy-tald dstrbuto (Jag ad Lu 0). W procssd ovr mllo capturd locatos of US domstc flghts wth a -day tm prod. From th massv amout of locatos or pots (, y, z, t), w tractd ovr,000 routs ad 00,000 flghts ad stablshd a topologcal rlatoshp for ovr 700 arports across th coutry ad wth th spcfd tm prod (s th t scto o data ad data procssg). W dsgatd th flght movmt as a proy for huma movmt at th coutry lvl ad ad th pattrs from both gomtrc ad topologcal prspctvs. To furthr plor huma moblty pattrs, w stup varous agt-basd smulatos to ucovr th udrlyg mchasms.. Data ad data procssg W obtad, total, 7,685,98 flght locatos or pots wth days btw 8 ad 8 August 00,.., (, y, z, t ) whr 7,685, 98. Th startg tm o th 8 th was :6AM, whl th dg tm o th 8 th was 8:9AM. Th locatos or pots ar rfrcd to th World Godtc Systm, 98. Apart from th locato formato, thr ar som othr attrbuts, such as arl cod, ad flght umbr ad spd; s Tabl for a small sampl. Tabl : A sampl of orgal flght data capturd by GPS rcvrs Arl Flght Spd Alttud Lattud Logtud Tmstamp MES 9 NULL NULL :0:9 COM :9:6 MES 9 NULL NULL :0:9 COM :5:6 MES NULL NULL :0: COM :59:6 W had to procss th data, tally a massv st of pots (500MB), to dtr th dvdual arports ad how th arports ar lkd or rlatd by flghts or routs. W tractd two typs of graphs: rout graphs ad flght graphs. Th dffrc btw a rout graph ad a flght graph s that th formr bars a bary rlatoshp whl th lattr s wghtd by th umbr of flghts. I othr words, a rout has at last o flght; thr would b o rout wthout a sgl flght. To buld up th two graphs, frst w tractd th dvdual flghts. Ths procss s show Fgur ad s dscrbd dtals as follows. Fgur : Flowchart of th data procssg to tract 05,66 vald flghts from 7,685,98 flght locatos

3 Bcaus w wr trstd US domstc flghts, w kpt,999,57 domstc pots wth th tag 0 ad fltrd out tratoal flghts wth th tag. Thr wr thus 75,599,999,57,8, 658 vald pots. A pot s vald f thr s th sam pot at th sam stat tm or f o of th four dmsos s st to NULL. Th,8,659 vald pots blog to,0 vald routs that ca b dtrd by a uqu combato of arl cod ad flght umbr. Th,0 routs ar furthr dvdd to two typs: rgular ad rrgular =.58 L (Pr(X )) 0 - =.56 Tragl sz Powr law ft Powr law ft L () Fgur : (Color ol) Bpartt powr law dstrbuto of tragl szs of th TIN To dstgush btw rgular ad rrgular routs, w plord th tm dffrc t t btw two coscutv tms ( t, t ). For ay rrgular rout that appars oly oc wth th days, dos ot chag much. It fluctuats at appromatly 5 uts, whch s th tm trval to captur flght locatos. Bcaus all th pots wr capturd whl th flghts wr srvc, for ay rgular rout, fluctuatd appromatly 5 uts whl srvc but aroud a largr valu (.g., s hours btw ths flght ad th t o) whl ot srvc. Gv th dffrc, th arthmtc ma of for ay rrgular rout ( ) s gratr tha th stadard dvato ( ),.,, whl th arthmtc ma of for ay rgular rout ( ) s far lss tha th stadard dvato ( ),..,. Usg ths statstcal proprty,.., th dffrc btw th arthmtc ma ad stadard dvato, w obtad 7,7 rgular routs ad,60 rrgular routs. Cosqutly, th 7,7 rgular routs cotad,70,55 rgular pots, whl th,60 rrgular routs cotad 8,0 rrgular pots. Thus, th prctags of rgular pots, rgular routs ad rgular flghts ar prtty hgh rspctvly 98%, 86% ad 98% as dcatd Fgur. Not that o rrgular rout s cosdrd to b o rrgular flght,.., th,60 rrgular routs qual th sam umbr of flghts. Howvr, o rgular rout cotas multpl flghts (Fgur a).th procss of drvg multpl flghts from a rout s achvd through th arthmtc ma of ad th stadard dvato wth ach rout. Thos pots wth lss tha th sum of th ma ad th stadard dvato blog to a complt flght wth th startg pot as th org ad dg pot as th dstato (c.f., Fgur b for a llustrato). Th 7,7 rgular routs fact cotad 0,0 rpatd flghts. Evtually, w obtad 05,66 vald flghts by addg th,60 rrgular flghts to th 0,0 rgular flghts.

4 0 0 L ( ) 0 Rgular rout Irrgular rout μ μ :59 08 : 08 : 5 08 : : Dat (a) Day Day Flght Flght (b) Fgur : Drvato of multpl flghts from o rout: (a) O rrgular rout (rd) ad o rgular rout cludg 9 rgular flghts (blu), ad (b) a fctv ampl of daly flghts wth th tm trvals bg prcsly 5 uts whl srvc (Not: W adopt a fctv ampl to llustrat how to drv multpl flghts from a rgular rout,., how to drv thos flght locatos as complt flghts btw th org ad th dstato. For th sak of smplcty, w assum a daly flght btw two arports lastg o hour. Evry day thr would b tm trvals (, whr,,... ), 5 for, but 60 for whch prod a flght s ot srvc. Th arthmtc ma of all tm trvals s ( 5 60) / 0.8 uts, ad th stadard dvato s 8.. Gv th fact that th tm trvals ar ot prcsly 5 uts, w ca us th sum of th arthmtc ma ad th stadard dvato ( ) = 9. to obta th cosctv pots as a complt flght joury.) Wth th 05,66 vald flghts (or mor spcfcally, th corrspodg rgular ad rrgular pots as show Fgur ), w could tract th dvdual arports for sttg up th rout graphs ad flght graphs. Evry flght cotas a org (O) ad a dstato (D), so thory thr ar, 05,66 O/D locatos. Howvr, w rtrvd oly 0,56 O/D locatos, slghtly lss tha th thortcal stmat, bcaus som O/D locatos ar shard. From th 0,56 pots, w gratd four tragulatd rrgular tworks (TIN) for four aras: th Malad, Alaska, Hawa, ad Purto Rco ad th Vrg slads. Th TIN tragl szs follow a bpartt powr law dstrbuto (Fgur ). Cosqutly, accordg to th had/tal dvso rul (Jag ad Lu 0) ad usg th arthmtc ma of th TIN tragl szs, w placd th tragls to two catgors: thos smallr tha th ma ad thos gratr tha th ma. W dsgatd thos smallr tha th ma as th arports, sc arports ar hghly crowdd wth O/D pots. Evtually, w gratd 7 arports ad dtrd th arport topologs trms of both flghts ad routs for furthr aalyss (Fgur ).

5 Fgur : (Color ol) Illustrato of data procssg from a pot cloud to 7 atural arports. Mathmatcal charactrzato of flghts or huma moblty pattrs Th ctral thm of our study s to dtr whch havy tald dstrbuto charactrzs huma moblty pattrs. Th havy tald dstrbutos rfr oft to thr basc olar rlatoshps btw quatty ad ts probablty: powr law, logormal ad potal, ad thr dgrat vrsos: powr law wth a potal cutoff ad strtchd potal (Claust, Shalz ad Nwma 009). Th focus of th work by Claust ad hs collagus was to dtct a powr law 5

6 dstrbuto ad dffrtat t from th four altratvs basd o som statstcal tsts. Thr mthods ca b summarzd by thr stps: () comput ad for a powr law ft, () coduct Kolmogorov-Smrov (KS) tst ad calculat th p valu to assss th goodss of ft, ad () comput th lklhood rato (LR) valus to compar wth altratvs cas th ft caot pass th KS tst. I ths papr, w adopt smlar mthods but do so to dtr ay havy tald dstrbuto ad dffrtat t from th altratvs (s Appd A). Lt us frst troduc th fv dstrbutos. Th powr law dstrbuto dcats a olar rlatoshp btw a quatty () ad ts probablty (y), gv by y ts smplst form, whr s usually btw ad. Ths s a dalzd prsso. I ral world data, th powr law rlatoshp appars oly for th tal whl s gratr tha a mum. Th powr law s prssd as follows: y C, [] whr C s ( ) [] Th powr law has a dgrat form,.., a powr law wth a potal cutoff. I othr words, th tal s ot a prfct powr law dstrbuto but rathr a mtur of powr law ad potal. y C, [] whr C s (, ) [] I ts smplst format, a potal rlatoshp btw ad y s prssd by y. Takg th logarthm of both sds of th prsso, w obta l y, whch mpls that th rlatoshp btw ad l y s lar. Mor grally, a potal dstrbuto s prssd as follows: y C, [5] whr C s [6] A dgrat vrso of th potal fucto s th strtchd potal,.., y C, [7] whr C s [8] Th last havy tald dstrbuto s logormal. Ltrally, logormal mpls that th logarthm of a quatty follows a ormal or Gaussa dstrbuto,.., (l ) 5 y C, [9] whr C 5 s l rfc [0] 6

7 As show th abov formulas, vry o of th havy tald dstrbutos cossts of two parts: th ma fucto that charactrzs th dstrbuto ad th costat C. Thr s a srs of paramtrs that dscrb th fuctos ad costats,..,,,,,,. Th objctv of Claust, Shalz ad Nwma (009) was to dtct a powr law dstrbuto th form of quato [] ad to dtr how to dffrtat t from altratvs th form of quatos [], [5], [7] ad [9]. Ths papr, howvr, sks a bst ft fucto for charactrzg varous proprts of huma moblty pattrs. Th mthods for dtfyg th powr law dstrbutos ca b tdd for dtfyg ay havy tald dstrbuto. Th mthods ca b usd ot oly () to fd th bst ft modls but also () to valuat how good th ft s. For task (), all of th paramtrs for th abov modls ca b computd (s Appd A). For task (), thr ar two typs of mthods for th goodss-of-ft tst. Th frst mthod s th KS tst. For a ral world datast, thr s a hypothszd modl for. Th mamum dffrc (δ) btw th cumulatv dstrbuto fucto (CDF) of th datast ad th CDF of th hypothszd modl dcats th goodss-of-ft or how clos th modl fts th ral world datast: ma f ( ) g( ), [] whr f () s th CDF of th datast wth a valu of at last ad g() s th CDF for th hypothszd modl that bst fts th datast for. Howvr, to quatfy th ft, w gratd 000 sythtc datasts accordg to th hypothszd modl. Th 000 sythszd datasts had 000 corrspodg hypothszd modls for. Followg th sam da of th mamum dffrc btw th modl ad th data, w would hav 000, dcatg th dffrcs btw th hypothszd modls ad th sythtc data. Evtually a goodss-of-ft d p was dfd as a rato of th umbr of whos valus ar gratr tha δ to 000. If th p valu s gratr tha a gv thrshold (.g., 0.0), th th hypothszd modl s accptabl, mplyg that 0 amog th 000 hav a mamum dstac gratr tha δ. For modls that could ot pass th abov KS tst, w dd altratv mthods to compar th two comptg modls. O of th most frqutly usd mthods s to calculat th lklhood rato of two modls. Bcaus ths study, th KS tst was suffct to dtty th pottal modls, w wll ot troduc th mthods; trstd radrs ca rfr to Claust, Shalz ad Nwma (009) for mor dtals.. Smulatos of huma moblty comparso wth th obsrvd W smulatd huma movmt usg movg bhavors as show Fgur 5 to ucovr th udrlyg mchasm of huma moblty pattrs. O of th mchasms s basd o gomtrc dstac (G), thr of th mchasms ar basd o topologcal proprts (T, T, ad T), ad a fal o s a prfrtal rtur (PR) suggstd by Sog t al. (00). For ach scaro or mchasm, w stup 500 movg agts, movg a rout graph or flght graph from od to aothr for about 000 tms utl th vstd tms ar hghly corrlatd to th dgr of od coctvty (.g. R squar > 0.9). To ths pot, w thk th smulato s saturatd. Through th smulatos, w obtad th smulatd travl lgths (stad of traffc flow) ad compard thm wth th obsrvd os. Not that th smulatos ar basd o th topology of th 7 arports, both th rout graphs ad th flght graphs. Th smulatd travl lgths wr was compard to th obsrvd os btw th 7 arports. Th comparso was basd o a smlar da to th o llustratd quato [],.., th 7

8 mamum dstac btw two CDF of th smulatd ad obsrvd travl lgths. W ra ach scaro 00 tms ad avragd th travl lgths for comparso to th obsrvd os. L L L L L L L L L L L L L L L (a) G (b) T (c) T 5 (d) T 7 P P 6 0 P 6 P 0 P 6 () PR Fgur 5: (Color ol) Fv scaros of smulatos dcatd by (a) G = gomtry (rout graph), (b) T = topology (rout graph), (c) T = topology (flght graph), (d) T = topology (rout graph), ad () PR = prfrtal rtur (rout graph) (NOTE: A movg agt s currtly at th blu od, ad th agt wats to mak th t mov wth a dffrt probablty as dcatd. Th gomtrc scaro s basd o gomtrc dstac,.., th closr, th bttr th probablty (pal a). Th t thr bhavors ar basd o som topologcal proprts: T s smply basd o lkag (ys/o whthr or ot thr s a flght) (pal b), T cosdrs th actual wght (how may actual flghts) (pal c), whl T cosdr how attractv ts ghbors ar (pal d). Prfrtal rtur s basd o Barabas s prfrtal attachmt (Sog t al. 00),.., thr s a hghr chac to rtur to th ods that wr prvously vstd. I pal (), all th lkd ods ar placd to catgors: th vstd ods to th rght (th tms ar dcatd by th umbrs), ad th uvstd ods to th lft (gray ods). Th frst probablty (P) s accordg to th prctag of th two catgory ods, whl th scod probablty (P) rlats to a prfrc to rtur to thos prvously vstd ods.) 5. Rsults ad dscusso W foud that huma moblty pattrs ar maly shapd by th udrlg topology of arports ad hav lttl to do wth huma movmt bhavors ad dstacs. Ths fdg s cosstt wth th rct studs (.g., Jag ad Ja 0, Ha t al. 0), but th prmtal sttgs wr rathr dffrt. For ampl, th two prvous studs dalt wth smulatos: th formr for strtcostrad movmt, th lattr work for smulatg movmt som dalzd topology or tworks. Th currt study s cosdrd to b comprhsv bcaus t covrs () massv data procssg, () mathmatcal charactrzato of th obsrvd moblty pattrs, ad () agt-basd smulatos of dffrt movmt to compar wth th obsrvd flow or pattrs. I what follows, w rport th rsults rlatd to thos thr ma topcs. 8

9 Th dtald data procssg s dscrbd dtals th abov Scto. A ot-worthy rsult of th data procssg s that th had/tal dvso rul (Jag ad Lu 0) s applcabl for dlatg arports. Th had/tal dvso rul stats that for gv a varabl X, f ts valus follow a havytald dstrbuto, th th ma (m) of th valus ca dvd all th valus to two parts: a hgh prctag th tal ad a low prctag th had. It should b otd that th havy-tald dstrbuto s obtad by plottg rakg a dcrasg ordr as th -as, ad th corrspodg valus as th y-as, so-calld rak-sz plot or Zpf plot. Th two parts, or both th had ad th tal, corrspod wll to huma cogto about ral world phoma, such as rch ad poor, rcsso ad prosprty, urba ad rural, just to am a fw ampls. Ths rul was tally usd to dlat cty boudars wh t was frst formulatd. Hr, th applcato of th rul provds furthr vdc for ts usfulss. Not that th arports ar aturally dtfd dpdg o th dsty of O/D pots so-calld atural arports. Th atural arport boudars may dvat from th ral os, but th boudars ar suffctly accurat to st up arport rlatoshps or topology L (Pr(X )) Traffc volum Powr law ft L () Fgur 6: (Color ol) Powr law dstrbuto of traffc volum W foud that topologcal proprts, cludg traffc volum (flght umbrs btw arports) ad dgr of coctvty (umbr of flghts arrvg or dpartg from th dvdual arports), show a powr law dstrbuto. Fgur 6 llustrats th powr law dstrbuto a doubl logarthm plot:.56 y 8* ( 9) wth th KS tst d p = 0.. To furthr llustrat th hrarchy of traffc volum, w rducd t to dffrt lvls of dtal accordg to th had/tal dvso rul (c.f., Fgur 7). Both th dgr ad outdgr hbt a strkg powr law dstrbuto wth slghtly dffrt.7 paramtr sttgs: y 0.68* ( 06) wth th KS tst d p = 0.0, ad.67 y.0* ( 8) wth th KS tst d p = 0.05 (Fgur 8). 9

10 Fgur 7: (Color ol) Four lvls of hrarchy of traffc volum (NOTE: Th frst lvl rprsts th most dtald lvl, whch all th routs wth at last o flght ar show togthr wth th rlatd arports or ods. Th scod lvl s obtad from th frst lvl by slctg thos lks wth mor tha th ma umbr (9) of flghts. Ths sam procdur cotus rcursvly for drvg th thrd ad fourth lvls. Th szs of th ods dcat th magtud of dgr, whch s hghly corrlatd wth that of th outdgr.) 0

11 0 0 L (Pr(X )) Idgr Powr law ft Outdgr Powr law ft L () Fgur 8: (Color ol) Powr law dstrbutos of arport dgrs W foud that huma travl lgths dmostrat a potal dstrbuto wth a powr law cutoff,.., ( 9000 km) p 0. y ( 9000 km) p 0.5 Ths rsult dcats that th dstrbuto s potal for travl lgths lss tha 9000 km, whras t s a powr law for travl lgths gratr tha 9000 km (Fgur 9). It should b otd that th powr law part cotas oly 8 flghts (whch ca b gord du to suffct statstcs), whl th potal part cotas 05,5 flghts. Ths rsult s vry dffrt from th arly fdg, whch clams that huma travl lgths dmostrat a powr law wth a potal cutoff (Brockma, Hufag ad Gsl 006, Gozalz, Hdalgo ad Barabás 008, Jag, Y ad Zhao 009). I othr words, for thos short lgths lss tha th thrshold, thr s a powr law dstrbuto, whl for log os, thr s a potal dstrbuto. Ths cocluso s oppost from our fdg. 0 0 L (Pr(X )) L (Pr(X )) L (Pr(X )) Flght lgth 9000Km Epotal ft Flght lgth 9000Km Powr law ft L () L () Fgur 9: (Color ol) Epotal dstrbuto wth a powr law cutoff for travl lgth

12 Not that th abov fdgs ar basd o a flght graph whos lks ar wghtd by th umbr of flghts. Howvr, arlr studs ar maly basd o a rout graph that s a bary graph th rlato btw arports s thr or 0 (.g., Gumrà ad Amaral 00, Gumrà t al. 005). I othr words, thr was o traffc volum formato volvd th arlr studs. Th arport topology usd th prvous studs s purly topologcal,.., ay two arports wth a lk as log as thr s o flght rgardlss of th umbr of flghts. Our arport topology s wghtd,.., th lk s wghtd by th umbr of flghts. A bary rout graph caot fully charactrz huma movmt pattrs or flght pattrs. Nvrthlss, w rducd th flght graph to a rout graph ad ad both th topologcal ad gomtrc proprts. W foud that wth th bary rout graph both dgr ad btwss ctralts hbt a powr law dstrbuto wth a potal cutoff. Ths rsult rgardg th bary rout graph s cosstt wth arlr studs (.g., Gumrà ad Amaral 00, Gumrà t al. 005). Th rout lgths dmostrat a potal dstrbuto,.., 0.00 y 0.00 ( 768) wth th KS tst d p = 0.8 (Fgur 0). If w tak th rout lgths as a proy for travl lgths, th th rsult s also dffrt from what was rportd th ltratur that clams a powr law or powr law wth a potal cutoff (Brockma, Hufag ad Gsl 006, Gozalz, Hdalgo ad Barabás 008, Jag, Y ad Zhao 009). Aothr trstg fdg s th rch club phomo obsrvd wth th bary rout graph,.., th top 50 arports costtut a arly complt graph wth at last routs to othr arports L (Pr(X )) Rout lgth Epotal ft Fgur 0: (Color ol) Epotal dstrbuto of rout lgths Through agt-basd smulatos of huma moblty (c.f., Scto ad Fgur 5), w foud that th travl lgths wth scaro T, basd o th rout graph ad radom movg bhavor, bst match th obsrvd travl lgths. T s squtally followd by T, PR, T ad G trms of matchg dgr. Ths rsult s coutr-tutv bcaus w avly td to blv that th othr two topologcally basd scaros T ad T would hav a bttr ffct rplcatg th obsrvd travl lgths. Scaro T s basd o th d facto flght graph, whl scaro T cosdrs how attractv th ghbors ar. Th scaro basd o prfrtal rtur s good but ot th bst. Ths fdg llustrats th udrlyg topologcal structur (th bary o rathr tha th wghtd o), ad radom moto ca dscrb th movmt pattrs wll. I othr words, huma movmt pattrs hav lttl to do wth gomtrc factors or movg bhavor. 6. Cocluso Massv trackg data of flghts provd a w mas of studyg huma moblty pattrs. I ths papr, w assum US flghts to b a proy for huma movmt at th coutry lvl. From th massv locato formato of US flghts wth a -day prod, w tractd dvdual routs, flghts ad arports ad stup both flght graphs ad rout graphs for plorg th moblty pattrs. Th rsultg data ar publshd togthr wth th papr. It s foud that topologcal proprts, such as

13 traffc volum ad arport dgrs, ar dd powr-law dstrbutd. Travl lgths hbt a potal dstrbuto wth a powr law cutoff rathr tha a powr law dstrbuto wth a potal cutoff as prvous studs llustratd. To ucovr th udrlyg mchasms, w costructd varous smulato scaros basd o both flght graphs ad rout graphs ad foud that rout graphs wth a radom-moto bhavor rplcat th obsrvd travl lgths pattr. Ths fdg dcats that huma movmt pattrs at a collctv lvl ar maly shapd by th udrlyg structur th smplst topology rprstd by th bary rout graph. Ackowldgmts Th flght data was collctd by Flytcomm Ic., ad w ar gratful for thr prmsso to us t ths papr. Rfrcs: Brockma D., Hufag L., ad Gsl T. (006), Th scalg laws of huma travl, Natur, 9, Claust A., Shalz C. R., ad Nwma M. E. J. (009), Powr-law dstrbutos mprcal data, SIAM Rvw, 5, Gozalz M., Hdalgo C. A., ad Barabás A.-L. (008), Udrstadg dvdual huma moblty pattrs, Natur, 5, Gumrà R. ad Amaral L. A. N. (00), Modlg th world-wd arport twork, Th Europa Physcal Joural B, 8(), Gumrà R., Mossa S., Turtsch A. ad Amaral L. A. N. (005), Th worldwd ar trasportato twork: Aomalous ctralty, commuty structur, ad cts' global rols, Procdgs of th Natoal Acadmy of Sccs of th Utd Stats of Amrca, 0(), Hägrstrad T. (970), What about popl rgoal scc? Paprs of th Rgoal Scc Assocato,, 7. Ha X., Hao Q., Wag B. ad Zhou T. (0), Org of th scalg law huma moblty: Hrarchy of traffc systms, Physcal Rvw E, 8, 067. Hllr B., P A., Haso J., Grajwsk T. ad Xu J. (99), Natural movmt: cofgurato ad attracto urba pdstra movmt, Evromt ad Plag B: Plag ad Dsg, 0, Hufagl L., Brockma D. ad Gsl T. (00), Forcast ad cotrol of pdmcs a globalzd world, Procdgs of th Natoal Acadmy of Sccs, 0(), Jag B. ad Ja T. (0), Agt-basd smulato of huma movmt shapd by th udrlyg strt structur, Itratoal Joural of Gographcal Iformato Scc, 5(), 5 6. Jag B. ad Lu X. (0), Scalg of gographc spac from th prspctv of cty ad fld blocks ad usg volutrd gographc formato, Itratoal Joural of Gographcal Iformato Scc,, -, Prprt, arv.org/abs/ Jag B., Y J. ad Zhao S. (009), Charactrzg huma moblty pattrs a larg strt twork, Physcal Rvw E, 80, 06. L K., Hog S., Km S. J., Rh I. ad Chog S. (009), SLAW: A moblty modl for huma walks, IEEE INFOCOM 009, IEEE. Roth C., Kag S. M., Batty M, ad Barthélmy M. (0), Structur of urba movmts: Polyctrc actvty ad tagld hrarchcal flows, PLoS ONE, 6(), 59. Sog C., Kor T., Wag P. ad Barabás A. (00), Modllg th scalg proprts of huma moblty, Natur Physcs, DOI: 0.08/NPHYS760. Ypma T. J. (995), Hstorcal dvlopmt of th Nwto-Raphso mthod, SIAM Rvw, 7(), 5 55.

14 Appd A: Estmato of th paramtrs for som havy tald dstrbutos basd o th mamum lklhood mthod Basd o th mthods suggstd by Claust, Shalz ad Nwma (009) for dtctg a powr law dstrbuto, ths appd, w prst varous procdurs for stmatg th paramtrs of th othr four havy tald, cotuous dstrbutos. Th prmary mthod usd s th mamum lklhood mthod.. Mamum lklhood stmato for th cotuous potal dstrbuto. Th gral form of th cotuous potal dstrbuto ca b dscrbd by y C [.] whr λ s th rat paramtr ( 0 ) ad C s th ormalzg costat.. Calculatg th ormalzg costat C It s obvous that y dvrgs as 0 ; thus, th abov quato dos ot hold for all valus of. Thr must b a lowr boud for th potal dstrbuto. Supposg that ad λ ar kow, w ca asly drv th ormalzg costat C. To fd th ormalzg costat, w us th fact that Thus, Fally, C d [.] C C C C d( ) [.] [.] [.5] [.6]. Estmatg th rat paramtr Now suppos, gv a mprcal datast cotag obsrvatos (,,..., ), w wat to kow th probablty that ths datast s draw from th cotuous potal dstrbuto modl. Ths probablty s also kow as th lklhood, ad t s dscrbd by Thus, ths cas, L y ( ) ( ) [.7] L. [.8] W ow wsh to fd th valu of that mamzs L (). Mathmatcally, ths valu ca b obtad by sttg th drvatv dl ( ) d qual to 0 ad solvg for. Howvr, t s tdous to fd th drvatv of L () bcaus t s a product of fuctos. Hc, w adopt th atural logarthm form of L () bcaus t s asr to fd th valu of that mamzs L [ L( )]. W hav

15 L[ L( )] L. [.9] Th drvatv of L [ L( )] wth rspct to s dl[ L( )] d. [.0] Thus, th valu of that mamzs L [ L( )] s th soluto of th quato 0. [.] Solvg, w obta th stmats ˆ as ˆ. [.]. Mamum lklhood stmato for th cotuous strtchd potal dstrbuto. Th gral form of th cotuous strtchd potal dstrbuto ca b dscrbd as: y C [.] whr s th rat paramtr ( 0 ), s th strtchg pot ( 0 ) ad C s th ormalzg costat. For, t dgrats to a potal dstrbuto as dscussd abov.. Calculat th ormalzg costat C To calculat th ormalzg costat C, w mpos a lowr boud for th strtchd potal dstrbuto. Lt us assum that, ad ar kow; w ca asly drv th ormalzg costat C. To fd th ormalzg costat, w us th fact that Thus, Fally, C C C C d d( ), [.] [.] [.], [.5] C, [.6]. Estmatg th rat paramtr ad strtchg pot 5

16 Now suppos, gv a mprcal datast cotag obsrvatos (,,..., ), w wat to kow th lklhood that ths datast s draw from th cotuous strtchd potal dstrbuto modl. I ths cas, th lklhood s gv by Furthrmor, ( ) L(, ), [.7] L L[ L(, )] L L ( ). [.8] Th mamum lklhood stmators of ad ar th valus that mamz L [ L(, )]. Takg drvatvs wth rspct to both ad, w obta ad L( L(, ))) ( L( L(, ))) ( L L L [.9]. [.0] Sttg th abov two drvatvs qual to 0, w obta th followg systm of quatos, L 0 L L 0. [.] Solvg th frst quato, w obta ˆ ˆ ˆ [ ]. [.] Substtutg ˆ for th scod quato, w obta ˆ ˆ ˆ ˆ L [ ] ( ) 0 L L ˆ. [.] Notc that th lft sd of th abov quato s kow as f (ˆ ), ad our goal s to fd th valu of ˆ that maks ths fucto qual to 0. Howvr, t s dffcult to solv t drctly to obta th valu of ˆ bcaus t s dffcult to drv th vrs fucto of f (ˆ ) as ˆ f (, ). Hc, w us th smpl Nwto Raphso mthod (Ypma, 995) to drv th soluto to f (ˆ ). Oc th valu of ˆ s dtrd, th valu of ˆ s also dtrd. Th Nwto Raphso mthod s usd for fdg succssvly bttr appromatos to th roots of a fucto f. Itally, w gv a stmato of th root 0 that s rasoably clos to th tru root. Th, w draw a tagt l of th fucto f at ths stmato pot, ad th trscto pot btw ths tagt l ad th -as s dtrd as th scod stmato of th root, whch s cosdrd to b a bttr appromato. Aftr svral tratos, f th currt stmato satsfs our accuracy rqurmt, th ths procss s tratd; othrws, ths procss s rpatd utl a 6

17 rasoabl accuracy s accptd. Not that th drvatv of th fucto f should b calculatd drctly bcaus th tagt l must b costructd at th stmato pot. Th followg fgur llustrats ths mthod. Fgur A: (Color ol) Illustrato of th Nwto Raphso mthod Basd o ths mthod, frstly, w st th tal valu of ˆ as th pot valu of a pur powr law dstrbuto fttd by th obsrvd datast. Th, w obta th drvatv of ths fucto, ' [ f ( ˆ) ˆ ˆ L ˆ L ]( ˆ ( ˆ ˆ ) ) [ ˆ L [.] Nt, followg th stps of Nwto Raphso mthod, w ca calculat th t bttr stmato valu of ˆ wth th formula ˆ ˆ f ( ˆ). [.5] ' f ( ˆ) Fally, ths procdur wll b tratd f f ˆ ), [.6] ( whr s th accptabl rror. Othrws, ths procdur must b tratd utl ths codto s satsfd. ˆ L ]. Mamum lklhood stmato for th cotuous logormal dstrbuto. Th gral form of th cotuous logormal dstrbuto ca b dscrbd by y ( C /( ))p( ( L ) / ), [.] 5 whr s th ma valu, s th stadard dvato valu ad C 5 s th ormalzg costat. 7

18 . Calculatg th ormalzg costat C 5 To calculat th ormalzg costat C 5, w mpos a lowr boud o th logormal dstrbuto. Lt us assum that, ad ar kow. Th, w ca asly drv th ormalzg costat C 5. To fd th ormalzg costat, w us th fact that ( C /( ))p( ( L ) / ) d. [.] 5 Thus, C 5 p( ( ) / ) ( ) L L d C5 L rfc( ) C5 L rfc( ) [.] [.]. [.5] Fally, L 5 [ ( )] C rfc, [.6] whr rfc( ) t dt. [.7]. Estmatg th ma paramtr ad th stadard dvato paramtr pot Suppos that, gv a mprcal datast cotag obsrvatos (,,..., ), w wat to kow th lklhood that ths datast s draw from th cotuous logormal dstrbuto modl. I ths cas, th lklhood s gv by L ( L ) t, ) p( )[ L dt], [.8] ( Furthrmor, L [ L(, )] ( L ) L L L( L t dt) Th mamum lklhood stmators of ad ar th valus that mamz L [ L(, )]. Hr, w adopt th tchqu provdd by th R packag to obta th stmato valu of ad bcaus t s ot covt to us th sam stratgy as th cas of th strtchd potal dstrbuto to hadl th abov complcatd log lklhood fucto. [.9] 8

19 R s a collcto of powrful packags that s usd for data mapulato, calculato ad graphcal dsplay. It ca b usd as a statstcal systm that mplmts a rch st of classcal ad modr statstcal tchqus. Th problm hr of obtag a stmat for th valu of ad to mamz L [ L(, )] ca b tratd as a problm of olar optmzato. I R, th fucto lm() s usd to carry out a mzato of th objctv fucto f basd o a Nwto-typ algorthm. Thrfor, our problm ca b solvd as a mzato of th gatv L [ L(, )], ad th procdur s dmostratd wth th followg psudo-cod. Fucto LogormalFt (, thrshold) St = ( thrshold) Guss th tal valu of both ad usg Ma( L( )) St stmato = lm ( f L[ L(, )], p (, )) Rtur stmato Ed Fucto ad Std( L( )) whr th varabl stmato rturs th bst stmato valu for both ad.. Mamum lklhood stmato for th cotuous powr law dstrbuto wth potal cutoff. Th gral form of th cotuous logormal dstrbuto ca b dscrbd by y C, [.] whr s th powr law pot paramtr, s th potal rat paramtr ad C s th ormalzg costat.. Calculatg th ormalzg costat C To calculat th ormalzg costat C, w mpos a lowr boud o th powr law dstrbuto wth a potal cutoff. Lt us assum that, ad ar kow. Th, w ca asly drv th costat C. To fd th ormalzg costat, w us th fact that Thus, C d C ( ) p( ) d( ). [.]. [.] Substtutg t for, Substtutg C t p( t) dt. [.] for a, a a C t p( t) dt. [.5] Th, a C ( a) ( a) ( a, )]. [.6] [ 9

20 Aga, substtutg a for, ad fally, whr c ( )[ (, )], [.7] t t 0 ( ) dt, [.8] ad t t dt 0 (, ). [.9] ( ). Estmatg th powr law pot paramtr ad th potal rat paramtr Suppos that, gv a mprcal datast cotag obsrvatos (,,..., ), w wat to kow th lklhood that ths datast s draw from th cotuous powr law dstrbuto wth a potal cutoff modl. I ths cas, th lklhood s gv by Furthrmor, L(, ). [.0] ( )[ (, )] L L [ L (, )] ( ) L L [ ( )[ (, )]] [.] Th mamum lklhood stmators of ad ar th valus that mamz L [ L(, )]. Howvr, bcaus of th complty of th abov log lklhood fucto, w adopt th sam stratgy as th cas of th logormal dstrbuto. O small dffrc s that hr w us th fucto CostrOptm() R to carry out a mzato of th objctv fucto f wth th costrats o th rag of paramtrs. Thrfor, our problm ca b solvd as a mzato of th gatv L [ L(, )] wth paramtr costrats, ad th procdur s dmostratd wth th followg psudo-cod. Fucto PowrlawEpFt (, thrshold) St = ( thrshold) Guss th tal valu of by fttg to a pur powr law dstrbuto Guss th tal valu of by fttg to a pur potal dstrbuto St th costrats as ad 0 St stmato = CostrOpt m( f L[ L(, )], p (, ), costrats) Rtur stmato Ed Fucto Whr varabl stmato rturs th bst stmato valu for both ad. 0