Statistical Analysis of the Metropolitan Seoul Subway System: Network Structure and Passenger Flows
aa r X i v : . [ phy s i c s . s o c - ph ] M a y Statistical Analysis of the Metropolitan SeoulSubway System: Network Structure andPassenger Flows
Keumsook Lee a , b Woo-Sung Jung c Jong Soo Park d M. Y. Choi c , e , ∗ a Department of Geography, Sungshin Women’s University, Seoul 136-742,Republic of Korea b Center for Transportation Studies, Boston University, Boston, MA 02215, USA c Center for Polymer Studies and Department of Physics, Boston University,Boston, MA 02215, USA d School of Computer Science and Engineering, Sungshin Women’s University,Seoul 136-742, Republic of Korea e Department of Physics and Astronomy, Seoul National University, Seoul 151-747,Republic of Korea
Abstract
The Metropolitan Seoul Subway system, consisting of 380 stations, provides themajor transportation mode in the metropolitan Seoul area. Focusing on the net-work structure, we analyze statistical properties and topological consequences ofthe subway system. We further study the passenger flows on the system, and findthat the flow weight distribution exhibits a power-law behavior. In addition, thedegree distribution of the spanning tree of the flows also follows a power law.
Key words: passenger flow, transportation, subway, power law
PACS:
Complex networks have been an active research topic in the physics commu-nity since models for complex networks were announced [1,2]. Subsequently,numerous real networks observed in biological and social systems as well asphysical ones have been studied [3,4,5,6,7,8,9,10,11,12,13,14,15]. Those studiedalso include transportation systems such as airline networks, subway networks, ∗ Permanent address: Department of Physics and Astronomy, Seoul National Uni-versity, Republic of Korea
Preprint submitted to Physica A 23 October 2018 nd highway systems. For example, studies of world-wide airport networks aswell as Indian and Chinese airport networks have disclosed small-world be-haviors and truncated power-law distributions [6,7,8,9,10]. For the subwaysystems in Boston and Vienna, various network properties, such as the clus-tering coefficient and network size, have been reported [11,12]. Further, sta-tistical properties of the Polish public transport network have been examined[13] and the Korean highway system has been analyzed with respect to thegravity model [14].In this manuscript, we consider the Metropolitan Seoul Subway (MSS) sys-tem, which consists of N = 380 stations, and serves as the major public trans-portation mode in the greater Seoul area, Republic of Korea. The system in anearlier phase was analyzed with regard to the accessibility measurement [16].In the present phase, the maximum distance between a pair of stations in thesystem is 126 km while the minimum value is 238 m. When we construct thesubway network with N nodes, each corresponding to a station, the numberof links connecting two nearest nodes turns out to be 424. The characteristicpath length L is defined in terms of the network distance n ij , which representsthe shortest path length between nodes i and j . Usually, the clustering coeffi-cient C also provides an important measure for a complex network. However,since a few nodes of the subway system have only one nearest neighbor, theclustering coefficient C is not well defined. Accordingly, we define the clus-tering coefficient C ∗ of the subway system, excluding the nodes which haveone neighbor. The eccentricity of node i corresponds to the greatest distancebetween i and other node. The radius R and the diameter D of a network arethen defined to be the minimum eccentricity and the maximum eccentricity,respectively, among all nodes. We further define the efficiency according to ǫ ≡ N ( N − X i = j n ij . (1)In the ideal case for the efficiency ǫ , all nodes are connected to each other, sothat the network of N nodes has N ( N − / network efficiency E , which takes avalue between zero and unity: 0 ≤ E ≤ N stations should be farless than N ( N − E . On the other hand, it is desirable to build the network in such a way thatthe physical distance measured along the links between a pair of stations isas short as possible, compared with the actual distance (along the straightline) between the two. Accordingly, it is more appropriate to use the physical2istance d ij rather than the network distance n ij in measuring the networkefficiency E . Whereas the value of E in the network distance represents theefficiency with respect to the ideal but unrealistic network (with all-to-allconnections), the value in the physical distance measures the efficiency withrespect to the optimal network in reality. In terms of the physical distance,the characteristic path length, diameter, and radius are given by L = 27 . D = 139 km, and R = 69 . w ij of a link between stations i and j is taken to be the sum ofpassenger flows in both directions on the link, i.e., i → j and j → i . Thestrength s i of station i is then defined to be the sum: s i ≡ P Nj =1 w ij . Displayedin Fig. 2 are the obtained distributions of (a) weights and (b) strengths forthe MSS network. It is observed that the weight distribution P ( w ) apparentlyexhibits power-law behavior with an exponent around 0 .
56, albeit restricteddue to the finite system size; in contrast, the strength distribution follows alog-normal function with a peak at s ≈ × . Note that in the subwaynetwork the weight of a link connecting two stations represents the passengerflow between them and the strength of a station corresponds to the numberof passengers arriving at and departing from that station. Accordingly, whilepassenger flows do not have a characteristic size, numbers of passengers at sin-gle stations do have. In a metropolis, most facilities are located near stations,so that each station is naturally abundant in passengers, the number of whichreflects the capacity of facilities located near the station. The fact that a ma-jority of stations are used by a similar number of passengers, corresponding tothe peak of the strength distribution, thus indicates that with residential andcommercial facilities taken into account, places near most stations are alreadydeveloped fully to accommodate dense and compact location of facilities. Thepeak of the strength distribution thus gives a measure for the characteristiccapacity of the facilities near a station in the fully urbanized Seoul.Spanning trees are widely used to analyze a complex network [17]. Of partic-ular interest is the minimum spanning tree, which is constructed with weights3ot larger than those of other possible spanning trees. In the subway system,on the other hand, the link which has a larger passenger flow than others isimportant, demanding to consider the maximum spanning tree. Figure 3 showsthe maximum spanning tree of passenger flows in the MSS system. We thencompute the degree distribution P ( k ) of the maximum spanning tree, whichis shown in Fig. 4. It is observed to be consistent with a power-law behavior: P ( k ) ∼ k − γ with exponent γ ≈ .
7, obtained from the least-square fit.To summarize, we have analyzed the Metropolitan Seoul Subway system con-sisting of 380 stations, and obtained various network measurements includingthe path length, clustering coefficient, diameter, and radius as well as the ef-ficiency of the network. The path length, diameter, and radius have also beencomputed in terms of the physical distance between stations. We have furtherinvestigated the passenger flows in the system, and constructed the maximumspanning tree of the flows. It is found that the weight distribution displaysa power-law behavior whereas the strength distribution follows a log-normalone. Also revealed is the power-law behavior of the degree distribution of thespanning tree. The detailed analysis and implications are left for further study.
Acknowledgements
This work was supported by the Korea Research Foundation through GrantNo. KRF-2006-B00022 and by the Sungshin Women’s University ResearchGrant of 2008.
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31 69 . E . × −
20 40 60 n -5 -4 -3 -2 -1 P ( n ) d (km) -5 -4 -3 -2 -1 P ( d ) (a) (b) Fig. 1. Probability distribution of the shortest path length in terms of (a) the net-work distance n and (b) the physical distance d between stations on semi-log scales. w -4 -2 P ( w ) s P ( s ) (a) (b) γ=0.56 Fig. 2. Probability distribution of (a) weight w on the log-log scale and (b) strength s on the semi-log scale. Lines are guides to the eye. In (a) the line has the slope − .
56 whereas the line in (b) represents a log-normal distribution. ig. 3. Maximum spanning tree of passenger flows, consisting 380 stations in theMetropolitan Seoul Subway system. k -3 -2 -1 P ( k ) Fig. 4. Probability distribution of the degree k of the maximum spanning tree inFig. 3 on the log-log scale. The slope of the line, serving as a guide to the eye, isgiven by − .7.