Power law and small world properties in a comparison of traffic city networks
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1 Artcle Statstcal Physcs and Mathematcs for Complex Systems December 20 Vol.56 No.34: do: 0.007/s SPECIAL TOPICS: Power law and small world propertes n a comparson of traffc cty networks MA Ke 2* WANG ZhongWen 3 JIANG Jan 4 ZHU GuangX 2 & LI We 5 Department of Physcs College of Maths and Physcs Chna Unversty of Geoscences Wuhan Chna; 2 Department of Electroncs and Informaton Engneerng Huazhong Unversty of Scence and Technology Wuhan Chna; 3 Department of Mathematcs Kashgar Teachers College Kashgar Chna; 4 Insttut Supereur des Materaux du Mans 44 Avenue F.A. Barthold Le Mans France; 5 Insttute of Partcle Physcs Central Chna Normal Unversty Wuhan Chna Receved May 30 20; accepted July 6 20 We analyze the statstcal propertes of the urban publc bus networks of two ctes (Bejng and Chengdu) n Chna. To ths end we present a comprehensve survey of the degree dstrbuton average path length and clusterng of both networks. It s shown that both networks exhbt small world behavor and are herarchcally organzed. We also dscuss the dfferences between the statstcal propertes dsplayed by the two networks. In addton we propose a weght dstrbuton approach to study the passenger flow through the publc bus networks we consdered. A herarchcal structure s observed here also. publc transport network complex network power-law dstrbuton small-world Ctaton: Ma K Wang Z W Jang J et al. Power law and small world propertes n a comparson of traffc cty networks. Chnese Sc Bull 20 56: do: 0.007/s Solvng the problem of traffc congeston s of great mportance for the safety and convenence of modern socety [ 6]. Recently emprcal evdence has shown that many transportaton systems can be descrbed by complex networks characterzed by the small world [7] and/or scale-free propertes [8]. The small-world effect became so known as t provdes an elegant explanaton for Mlgram s experment of the sx degrees of separaton [9]. The frst sutable model capable of explanng the small-world effect was reported by Watts and Strogatz [7] whch has motvated several applcatons to real-world systems. In recent years several publc transport systems (PTS) have been nvestgated usng varous concepts of the statstcal physcs of complex networks [0 20]. Among them many studes have focused on the complexty of urban publc transport systems. Latora and Marchor [92] ntroduced the defnton of effcency coeffcent and appled t to a study of the Boston subway. Sen et al. [22] concluded *Correspondng author (emal: mark@opp.ccnu.edu.cn) that Inda s ralway network exhbted small world propertes. Smlar propertes were reported by Seaten and Hackett [23] n a study of ralway networks n Boston and Venna. Jang and Claramunt [24] found that the topologcal networks of streets exhbted small world propertes but were not scale-free. Gumera et al. [25] showed that the worldwde ar traffc network was a scale-free and small world network. Wang et al. [26] proposed a herarchcal geographcal model to mmc a real traffc system upon whch a random walk generates a power-law-lke travel dsplacement dstrbuton wth tunable exponent. L and Ca [6] showed that the topologcal structure of the ar traffc network of Chna (ANC) had two key characterstcs of smallworld networks a short average path-length and a hgh degree of clusterng. He et al. [2728] nvestgated the urban transport networks of four ctes n Chna and ntroduced a model whose numercal results can ft the emprcal data well. Wu et al. [29] concluded that the urban transt system n Bejng s a scale-free network. Zhao et al. [30] ntroduced three knds of models to study the propertes of the The Author(s) 20. Ths artcle s publshed wth open access at Sprngerlnk.com csb.scchna.com
2 3732 Ma K et al. Chnese Sc Bull December (20) Vol.56 No.34 publc transport n Bejng. Zhu et al. [3] nvestgated the scalng of drected dynamcal small-world networks wth random responses. Some other noteworthy results can be seen n [32 40]. In ths paper we construct the publc bus network based on the data for two Chnese ctes Chengdu (N = 220 E =2280 where N s the number of bus stops and E s the number of bus lnes between two stops cn) and Bejng (N=498 E= Three most robust measures of network topology: degree dstrbuton characterstc path length and clusterng coeffcent have been checked as well as the passenger flow analyss. Despte a large dfference n sze of the consdered networks they share several unversal features. Comparson of fundamental characterstcs of networks The degree dstrbuton s one of the three most robust measures of network topology. The degree of a node s the number of edges connected wth ths node. The degree k of node can be defned as k a j (a j s the adjacency G of connecton among nodes). The average degree s k k / N 2 K / N (K s the total number of lnks n G the network and N s the total number of nodes). The way the degree s dstrbuted among the nodes s an mportant property of a network that can be nvestgated by calculatng the degree dstrbuton P(k).e. the probablty of fndng nodes wth k lnks. The degree dstrbuton s defned as P(k)= N(k)/N where N(k) s the number of nodes wth k lnks. The majorty of papers about complex networks have shown that n most of the real systems the degree dstrbuton follows a power law for large k: Pk ( )~ Nk ( )~ k wth the exponent between 2 and 3 [4 43]. Networks wth such a degree dstrbuton are called scale-free [43]. The results found are n contrast wth what are expected for random graphs [44]. In fact a random graph wth N nodes and K edges (an average of k per node).e. a graph obtaned randomly selectng the K couples of nodes to be connected exhbts a Posson degree dstrbuton centered at k. The comparson of degree dstrbuton of the publc bus networks of Bejng and Chengdu s presented n Fgure. The power-law-tal behavor ndcates that both networks have scale-free topology. The consequence of heavy tals s that the average behavor of the system s not typcal [45]. There s overwhelmng evdence that human actvtes ncludng traffc networks are characterzed by heavy-taled statstcs [46 49]. The exponent of degree dstrbuton for Bejng s 3.7 much smaller than that for Chengdu. Ths result shows there are comparatvely more nodes wth hgh degrees n Bejng than n Chengdu where t s 6.. The Fgure Comparson of the degree dstrbutons of the publc bus networks of ctes Bejng (a) and Chengdu (b) n Chna. P(k) s the degree dstrbuton functon and k s the degree of each node. A power-law dstrbuton s obvously shown n the log-log plot and the fttng curve wth black sold lne matches the data well. reason s that more hub bus statons exst n the Bejng publc bus network. Another robust measure of network topology s characterstc path length. Analyss of many networks has shown smlar propertes: n most real-world networks t s possble to reach any node from another one gong through a number of edges that s small compared wth the total number of exstng nodes n the system [9]. The typcal separaton between two generc nodes n a graph G can be measured by the characterstc path length L defned as L d () j N( N ) j G j where d j s the length of the shortest path between nodes and j.e. the mnmum number of edges covered to go from to j. Clusterng s the thrd robust measure of network topology. In many real-world networks for nstance n socal systems there s a hgh probablty that two ndvduals lnked by an acquantance have a thrd acquantance n common. Such tendency can be measured by the clusterng coeffcent C. For each node of G we consder the subgraph G of frst neghbors that s obtaned n two steps: () extractng and ts frst neghbors from G; (2) removng the node and all the ncdent edges. If node has k neghbors then G wll have k nodes and at most k (k )/2 edges. C s proportonal to the fracton of these edges that really exst and measures the local group cohesveness of vertex. C s the average of C calculated over all nodes: CG ( ) C C (2) where N G 2e a a a C () k k j j m m j ( ) jm k( k ) where e s the number of edges n G. By defnton C takes values n the nterval [0 ]. Notce that C s related to the (3)
3 Ma K et al. Chnese Sc Bull December (20) Vol.56 No number of trangles present on the network. A detaled descrpton of the network can be obtaned by the dstrbuton of C among the nodes of the network. For nstance mportant nformaton can be extracted by consderng C(k) the average clusterng coeffcent restrcted to classes of vertces of degree k. In many cases C(k) exhbts a power law decay as a functon of k.e. a herarchy wth low degree vertces belongng to well nterconnected communtes and hubs connectng many vertces not drectly connected to each other. Random graphs have a characterstc path length L that grows only logarthmcally wth N and a clusterng coeffcent C= k =N gong to zero for large N. Conversely a regular lattce has a fnte clusterng coeffcent and a characterstc path length L whch grows lnearly wth N. Watts and Strogatz have shown that many real-world networks have propertes ntermedate between random graphs and regular lattces. All such networks that have been named small worlds have at the same tme: () a small characterstc path length as n random graphs; and (2) a large clusterng coeffcent typcal of regular lattces. To check whether a network s a small world t has been proposed to compare the value of L and C wth those obtaned for the randomzed verson of the network.e. for a network wth the same N and K and n whch the edges are randomly dstrbuted wth a unform probablty among all the nodes. In a small-world network L~L rand and C >>C rand. Therefore we calculate the average path length and the clusterng coeffcent of the publc bus networks of Bejng and Chengdu. The results n Table are compared wth those obtaned from random graphs wth the same number of nodes and lnks. Though both networks have a small average path length Chengdu s publc bus network has the much smaller average path length n that any two bus stops can be connected n just two steps. In addton the two networks have C >>C rand. The smaller C of Chengdu s publc bus network compared wth Bejng s publc bus network s because there are fewer bus hub statons n Chengdu than there are n Bejng. The correlaton between the cluster coeffcent C(k) and the degree k of two publc bus networks s also nvestgated (Fgure 2). A scale-free behavor s clearly emergng n Fgure 2: () the power law ft yelds exponents 0.65 and 0.75 for Bejng and Chengdu respectvely; (2) both exponents are sgnfcantly larger for large N than the random network predcton C(k)~k [4450]. Moreover the negatve correlatons account for the herarchy of the publc bus Table The coeffcent of the publc bus networks of Bejng and Chengdu n Chna C L C rand L rand K Chengdu Bejng Fgure 2 Comparson of the correlaton between the cluster coeffcent and degree of two publc bus networks of ctes Bejng (a) and Chengdu (b) n Chna. C(k) s the cluster coeffcent and k s the degree. From the straght lnes we fnd that there s negatve correlaton between the two quanttes n each panel. As the slopes are 0.65 and 0.75 respectvely the degrees of correlaton n the two panels are almost equal. networks we consdered. Many real-lfe networks ncludng the Internet World Wde Web and the actor network are characterzed by the exstence of a herarchcal structure [5 53] whch can usually be detected by the negatve correlaton between the clusterng coeffcent and the degree. All the general small-world-propertes mentoned above are smultaneously present n our publc bus networks. As an obvous human actvty the passenger flow s the man purpose of the PTS and naturally reflects ts sgnfcant features. We nvestgate the passenger flow on our publc bus networks. The weght w j of a lnk between statons and j s taken to be the sum of passenger flows n both drectons on the lnk.e. j and j. The strength s N of staton s then defned to be the sum s w. j j Fgure 3 shows the dstrbutons of passenger flow n the two publc bus networks. It s observed that the weght dstrbuton P(S) exhbts power-law behavor wth exponents of 2.2 and 2. respectvely. Despte a sgnfcant dfference n passenger flow sze such nearly smlar passenger flow dstrbuton exponents account for the unform herarchcal structure as prevously found and dscussed of passenger flows n both Bejng and Chengdu s publc bus network. That s not unexpected because as the metropolses of Chna Bejng and Chengdu s bus stops are located n
4 3734 Ma K et al. Chnese Sc Bull December (20) Vol.56 No.34 Fgure 3 Comparson of the dstrbuton of weghts for the publc bus networks of ctes Bejng (a) and Chengdu (b) n Chna. P(S) s the weght functon and S s the weght. (a) There s a power law dstrbuton n most of the observed range of weght and the slope of fttng straght lne s 2.2; (b) a power law dstrbuton s also observed and the slope s 2.. a smlar way. For example most facltes are located near statons so each staton naturally has an abundance of passengers. 2 Conclusons We have constructed the publc bus network based on the data of two Chnese ctes (Bejng and Chengdu). The publc bus networks under consderaton appear to be strongly correlated small-world and herarchcal structures wth powerlaw dstrbuton of degree hgh values of clusterng coeffcents (especally n L and less n C-spaces) and comparatvely low average shortest path values. It s found that the clusterng coeffcent has negatve correlaton wth the numbers of the bus lnes that pass that stop. In partcular we have consdered the passenger flow on two ctes publc bus networks by the weght denotng the number of passengers through a lnk between statons. The weghts characterzng the passenger flow also exhbt a herarchcal structure feature wth power-law dstrbuton behavor. The comparson of statstcal propertes between Bejng and Chengdu n detal shows more bus hub statons n Bejng than that n Chengdu therefore a hgher effcency publc bus network of Bejng. These results may contrbute to the descrpton of the herarches and organzatonal prncples for the publc transport networks. Ths work was supported by the Natonal Natural Scence Foundaton of Chna ( ). Nagel K Schreckenberg M. A cellular automaton model for freeway traffc. J Phys I France 992 2: Helbng D Huberman B A. Coherent movng states n hghway traffc. Nature : Kerner B S. The Physcs of Traffc. Berln: Sprnger Bham O Mddleton A Levne D. Self-organzaton and a dynamcal transton n traffc-flow models. Phys Rev A : Chung K H Hu P M Gu G Q. Two-dmensonal traffc flow problems wth faulty traffc lghts. Phys Rev E 995 5: Angel O Holroyd A E Martn J B. The jammed phase of the Bham-Mddleton-Levne traffc model. Elec Comm In Prob : Watts D J Strogatz S H. Collectve dynamcs of small-world networks. Nature : Albert R Jeong H Barabas A L. Internet: Dameter of the worldwde web. Nature : Mlgram S. The small world problem. Psychology Today 967 2: Matheus P V Lucano F C. How many nodes are effectvely accessed n complex networks? Phys Lett A : Barthelemy M. Spatal networks. Phys Rep : 0 2 Scellato S Fortuna L Frasca M et al. Traffc optmzaton n transport networks based on local routng. Eur Phys J B : L G Res S D Morera A A et al. Towards desgn prncples for optmal transport networks. Phys Rev Lett : Bagler G. Analyss of the arport network of Inda as a complex weghted network. Physca A : Albert R Albert I Nakarado G L. Structural vulnerablty of the North Amercan power grd. Phys Rev Letts : L W Ca X. Statstcal analyss of arport network of chna. Phys Rev E : Barrat A Barthelemy M Pastor-Satorras R et al. The archtecture of complex weghted networks. Proc Natl Acad Sc USA : Strogatz S H. Explorng complex networks. Nature : Latora V Marchor M. Effcent behavor of small-world networks. Phys Rev Lett : 4 20 Amaral L A Scala A Barthelemy M et al. Classes of small-world networks. Proc Natl Acad Sc USA : Latora V Marchor M. Is the Boston subway a small-world network? Physca A : Sen P Dasgupta S Chatterjee A. Small-world propertes of the Indan ralway network. Phys Rev E : Seaton K A Hackett L. Statons trans and small-world networks. Physca A : Jang B Claramunt C. Topologcal analyss of urban street networks. Envron Plan B : Gumera R Mossa S Turtsch A et al. The worldwde ar transportaton network: Anomalous centralty communty structure and ctes global roles. Proc Natl Acad Sc USA : Han X P Hao Q Wang B H et al. Orgn of the scalng law n human moblty: Herarchy of traffc systems. Phys Rev E 20 83: Chen Y Z Fu C Chang H et al. Connectvty correlatons n three topologcal spaces of urban bus-transport networks n Chna. Chn Phys B : Chen Y Z L N He D. A study on some urban bus transport networks. Physca A : Wu J J Gao Z Y Sun H J. Urban transt system as a scale-free network. Mod Phys Letts B : Zhao J S D Z R Wang D H. Complexty and effcency of Bejng transt network. Comp Syst Comp Sc : Zhu C Xong S Tan Y et al. Scalng of drected dynamcal small-world networks wth random responses. Phys Rev Letts : Da J C L X. Mult-agent systems for smulatng traffc behavors. Chnese Sc Bull : Sun X Y Wang B H Yang H X et al. Effects of nformaton feedback on an asymmetrcal two-route scenaro. Chnese Sc Bull : Zang C X Huang B C L Z Y et al. Magnetc propertes of hghroad-sde pne tree leaves n Bejng and ther envronmental sgnfcance. Chnese Sc Bull :
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