Graph-theoretic evaluation support tool for fixed-route transportdevelopment in metropolitan areas

Theodore Tsekeris a,n, Anastasia-Zoi Souliotou b

a Centre of Planning and Economic Research (KEPE), Amerikis 11, 10672 Athens, Greeceb EDESTA-INREV, Department of Arts and Image Technologies, University of Paris VIII, France

a r t i c l e i n f o

Available online 1 February 2014

Keywords:Public transportNetwork designEvaluationGraph theoryUrban planning

a b s t r a c t

This paper presents a graph-theoretic analysis for supporting the evaluation of alternative fixed-routepublic transport development plans in metropolitan areas. Several indicators grounded on the theory ofgraphs and network science are suggested and calculated for evaluating prospective developments of thefixed-route transport system in the Athens metropolitan area, which includes the metro, tram andsuburban railway. The comparative static analyses of past and scheduled line extensions and planned lineconstructions generally show the tendency of the system towards small-world networking with scale-free characteristics, which implies increasing scale economies and reliance on a few large transferstations. The results suggest that policy-makers can choose the option of constructing a semi-circumferential line in the middle (compared to the end) of the system development process, in orderto trade investment cost for increased levels of service and robustness.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Fixed-route public transport systems, particularly subway ormetro systems, are widely recognized as a type of transportinfrastructure that can sustainably accommodate urban growthneeds and address the increasing problems of traffic congestionand environmental degradation. Nonetheless, due to the consider-ably high cost of constructing, operating and maintaining thespecific infrastructure, special attention should be given to thelocation of passenger stations and their interconnections withinthe network at different stages of the system development.Traditional approaches in handling these issues rely on (static)optimization procedures, within the framework of transit networkdesign (van Nes, 2002; Guihaire and Hao, 2008) which primarilyaims at minimizing travel time and, possibly, some measure of itsvariance. However, it has been recognized that, in reality, thestructure of urban public transport networks is rather counter-intuitive, due to its intrinsic topological characteristics and otherfactors such as local development patterns, area coverage andoperating costs (von Ferber et al., 2009; Roth et al., 2011). Inaddition, they evolve and organize themselves through usuallyuncoordinated decisions taking place over time. The proper timingand coordination of investment decisions may considerablyenhance the anticipated benefits of public transport development

projects into the overall urban economy. Especially, the fixed-routepublic transport systems are of crucial importance for the resi-lience (Cox et al., 2011; Zemp et al., 2011) and the scalability(Bornholdt and Schuster, 2002) of urban operations, in the sensethat they help cities to ‘work on all scales’ and (re)build afractalistic character so as to attain long-term strategic goals.

The use of computational tools and metrics from the graphtheory and network science provides a sound methodologicalframework for investigating the self-organizing features anddynamic properties of fixed-route public transport networks. Suchrelevant measures as average path length, network diameter anddensity, clustering coefficient and robustness (see Section 2) arecritical for both the sustainable and resilient planning of publictransport systems. Their consideration typically corresponds to thefinal states of systems that have reached a sufficient level ofmaturity, in terms of the connectivity between stations and lines.

From the methodological point of view, the main purpose of thepaper is to demonstrate how a set of simple graph-theoreticmeasures can be used for the plausible quick-response evaluationof alternative network design options, rather than to suggest acomplete framework of a cost-benefit type of analysis. In contrastwith other studies in the existing literature (see Section 2), itperforms a comparative static analysis to examine the implicationsof these measures for the continuous planning and on-goingevaluation of a large fixed-route public transport system, that ofthe Greater Area of Athens in Greece. From the policy perspective,the present analysis can offer a decision-support mechanism forprioritizing and scheduling public transport investments according

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Transport Policy

0967-070X/$ - see front matter & 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.tranpol.2014.01.005

n Corresponding author. Tel.: þ30 210 3676371.E-mail address: tsek@kepe.gr (T. Tsekeris).

Transport Policy 32 (2014) 88–95

to a set of policy objectives. A principal question that is addressedhere concerns the potential benefits of constructing a semi-circumferential line at the mid-stage of the system developmentprocess, compared to the end stage, as has been originally con-sidered in the existing urban master plan.

As far as the organization of the paper is concerned, Section 2reviews and describes the metrics of the network analysis of fixed-route transport systems, in accordance with the graph theory.Section 3 presents the different (current, scheduled and planned)stages of the development of the fixed-route public transportsystem in Athens. Section 4 reports and discusses the results of theempirical analysis of the study. Section 5 concludes and providespolicy implications of the study findings.

2. Network analysis of fixed-route transport systems

The increasing availability of geosimulation and network ana-lysis software tools has facilitated the development and applica-tion of various metrics for measuring connectivity properties andother topological aspects in transport systems. The graph-theoretic analysis of transport networks was originally implemen-ted for highways (Garrison, 1960). During the last two decades,numerous applications are encountered in various other types oftransport networks, including large urban arterial street (Jiang andClaramunt, 2004; Porta et al., 2006; Xie and Levinson, 2007),railway (Sen et al., 2003), maritime (Veenstra et al., 2005), airline(Xu and Harriss, 2008; Wang et al., 2011) and express-logistics(Yang et al., 2009) networks. These graph-theoretic studies oflarge-scale transport systems have contributed to the betterunderstanding of the complex interrelationships of their structuralcomponents and their evolution, and have enhanced the processesof planning, designing and operating them.

Especially regarding the metro and other fixed-route publictransport networks in large urban areas, graph-theoretic modelsare capable of parsimoniously representing the main structuraland functional properties of such systems. This paper employs awide range of indices which can depict these properties, includingthe following:

(i) Average Degree, which is defined by the sum of all the degreesdivided by the number of nodes (stations) in the network. Thetotal degrees encompass the in-degrees (or row degrees), that isthe number of links leading to a station from other stations, andthe out-degrees (or column degrees), that is the number of linksleading out of a station to other stations.

(ii) Network Diameter, that is the longest graph distancebetween any two nodes in the network; it indicates howfar apart the two most distant stations are and, hence, theease of access from one to the other.

(iii) Graph Density, which is defined by the ratio of the existingnumber of edges (links) to the total number of possibleedges, including all possible linkages between stations in thenetwork; it essentially measures how close the network is tocompletion (where it has all possible edges and the densityequals to one).

(iv) Modularity, which provides a measure of the decomposabilityof the network, namely, its ability to be decomposed orpartitioned into interconnected groups of nodes (neighboringstations). The community detection algorithm (Blondel et al.,2008) is implemented to calculate the modularity score.

(v) Average Path Length, that is the average network distancebetween all pairs of stations; its calculation relies on theresult of a shortest path-finding algorithm.

(vi) Clustering Coefficient, C, which is expressed here by the degreeof connectivity, C ¼ E=ð3V�6Þ, as originally introduced by

Kansky (1963), where E denotes the number of edges and V thenumber of vertices (nodes or stations). The latter measure offersan overall indication of the clustering or level of development ofthe system, compared to its potential development: moredeveloped transit networks tend to have shorter trip distances,i.e. higher clustering (and higher probability of a future linkagebetween two unconnected neighbors of a considered node). It isnoted that the measure of the clustering coefficient essentiallyrefers to the graph density for planar graphs, in contrast withthe general definition of graph density which holds for non-planar graphs and is calculated here for comparison purposesamong the different stages of the system development. In fact,the planar graph hypothesis is widely adopted for simplicitypurposes from the early studies of the graph-theoretic analysisof transport networks and it provides a sufficient representationof the graph density in metro and rail networks. For thisreason, the clustering coefficient is used here to denotethe planar property of the fixed-route transport network,demonstrate changes in the network connectivity and capturethe small-world properties of the system, in conjunction withthe average path length.

(vii) Robustness Index, r, which denotes the ability of the systemto successfully respond to failures/incidents and continue itsoperation to serve the demand between stations. Based onDerrible and Kennedy (2010), this index can be expressed asr¼ ðE�Vdþ1�EmÞ=V , where Vd is the number of diatonicstations, including all transfer stations and termini or end-stations, and Em is the number of diatonic (or multiple)edges, which connect the diatonic stations.

(viii) Ratio of transfer stations to total number of stations in thenetwork.

(ix) Scaling factor ε, which assigns the scale-free property to thesystem. In accordance with Barabási and Albert (1999), if wetake all stations V of a network and identify those that hostone line, two lines and so on, and if the frequency plotdecays following a power law, as defined by the relationshipf ðℓÞpℓ� ε, then that network is referred to as scale-free,where the exponent ε is the scaling factor and ℓ is number oflines passing through a station. Such networks have plentyof monotonic stations (hosting only one line) and a fewdiatonic stations (hosting more than one line), as it usuallyhappens in ‘mature’ public transport systems. By performingan ordinary least-squares (OLS) regression, the frequencyfunction f takes the form lnðf ðℓÞÞ ¼ �ε lnðℓÞþ lnðαÞ, in whichthe factor ε is the slope and coefficient α is the intercept. Thescale-free property can then be identified by such statisticalmeasures as the magnitude of ε (large scale-free networkstypically have a factor in the range 2oεo3) and thegoodness-of-fit (R2).

The above measures provide useful insights about differentaspects of the conditions of serviceability and cost-effectiveness ofa proposed public mass transit design plan. Hence, they can beused to define planning policy priorities and support the evalua-tion of the development of such a system. In particular, anincreased average degree (ratio of total links to stations) and ratioof transfer stations to total stations are typically associated withhigher investment cost. However, they relate to higher levels ofservice and network reliability, since they reflect a larger numberof connections between stations and availability of paths betweenthem. Increased graph density and clustering coefficient (connec-tivity between stations) as well as reduced network diameter andaverage path length imply a decrease in the travel time of usersand improved reliability. The role of modularity on the perfor-mance of the system configuration is ambiguous, as it relies on theevenness (or the hierarchy) of nodes. In the case where the degree

T. Tsekeris, A.-Z. Souliotou / Transport Policy 32 (2014) 88–95 89

of nodes is more evenly distributed, increased modularity mayarguably improve efficiency, through suitably organizing (groupsof) neighboring nodes to reduce their dependency on other nodes(Langlois, 2002). In transit systems with strong hierarchicalstructure, i.e. increased dependence on a few large hub stations,the role of neighborhoods decreases. Namely, in such systems,modularity arguably constitutes a lower policy priority or designneed for enhancing its sustainable development.

A number of similarities in the above indices have been recog-nized for subway systems of large capital cities, which have alreadyreached a sufficient level of maturity, as they organize themselvesand converge to similar structures. It has been found that thesestructures evolve from tree-like to grid-like over their lifetime by

adding crossing linear axes to a mid-stage or eventual circle withbranches (Whitney, 2012). This type of structure has been consideredto be potentially similar to that of wide-area communication net-works, since passengers are treated like messages routed efficientlyfrom their origin to their destination. The most common character-istics generally refer to the scale-free and small-world topology.Specifically, in most capitals, subway systems possess characteristicsof scale-free networks rather than random ones, because they arehighly inhomogeneous and their design follows the phenomenon of‘preferential attachment’, by mostly relying their operation on largetransfer stations rather than small ones (Derrible and Kennedy,2010). They also possess small-world networking properties, sincethey gradually improve connectivity by introducing a few longer

Fig. 1. Current, scheduled and planned (metro line 4) extensions of the Athens fixed-route public transport system.Source: Athens Public Transport Organization

T. Tsekeris, A.-Z. Souliotou / Transport Policy 32 (2014) 88–9590

connections. Most ‘mature’ subway systems worldwide can bedefined as small-world networks, as their stations are connectedwith both long and short links, so that all of them are reachablethrough a reasonably short path, making the least number oftransfers possible.

It is noted that, in the current analysis, the span of edges is setconstant over the network and the trip length between any nodepair does not differentiate with the distance from the core and thenumber of connections (Barthélemy, 2011). Nevertheless, spatialinformation with use of node coordinates may well be inserted inthe proposed approach to employ geographical measures thatexamine further aspects of the fixed-route transport networkdevelopment. Such aspects may encompass the increased spacingbetween metro stations in the periphery, compared to the high-density core, the fractal dimension of network structure and itsself-organization properties, in response to land use changes.

Finally, it is stressed that this paper deals with the graph-theoretic analysis of the fixed-route transport system, whichbasically involves the study of the topological properties relatedto the geometry of the network and not the flows. The use of datafor passenger flows between all the node pairs requires theavailability of detailed origin-destination trip demand matricesat each different stage of the system development (see Section 3).These forecasts and their application are impractical and beyondthe scope of this study. However, on the basis of relevantinformation availability, a future fixed-route transport networkanalysis could assign a particular weight to each link, according tothe prediction of the maximum number of passengers on that link,

to depict the effect of travel demand distribution on the variousmetrics. Additionally, the results of the current analysis could bepotentially employed to support a tool to forecast ridership orchanges in travel demand after the opening of new lines and tocontrol the distribution of flows, according to land use develop-ment and planning objectives.

3. Fixed-route transport system development in Athens

The basic structure of the fixed-route public transport systemof Athens is composed of three metro lines: 1, 2 and 3 (the lattertwo started their operation at the beginning of 2000). The metrosystem is supplemented by the tram and suburban railway net-work, which started their operation in mid-2004. Near-termextensions of metro lines 2 and 3 have been scheduled (within2013), as well as extensions of the tram system in the centers ofAthens and Piraeus. A new metro line 4 has been planned andalready designed, whose central part will cross/overlap withexisting lines and will create new transfer stations. Based on thecurrent and scheduled or hitherto designed line configurations ofthe Athens fixed-route public transport network, which are illu-strated in Fig. 1, the following five stages of the system develop-ment are set out:

� Metro (Existing Lines 1, 2 and 3).� Total Fixed-Route PT Network (Metro, Tram and Suburban Rail).� Total Fixed-Route PT Network with Extensions of Lines 2 and 3.

Fig. 2. Current and planned lines (indicated as L4, L5, L6, L7 and L8) of the Athens metro system.Source: Ministry of Infrastructure, Transport and Networks

T. Tsekeris, A.-Z. Souliotou / Transport Policy 32 (2014) 88–95 91

� Total Fixed-Route PT Network with Extensions of Lines 2 and3 and Tram.

� Total Fixed-Route PT Network with All Extensions and NewLine 4.

The above final configuration (including the new line 4) isexpected to constitute the backbone of the public transport systemof Athens in the next years. Each of the above stages of the systemdevelopment is coded and modeled with Gephi, which is an easy-to-implement open-source graph visualization and manipulation soft-ware. Nonetheless, several other software programs, such as the freeopen-source NodeXL, igraph, NetworkX, Cytoscape, Pajek and ORA,could also be well and relatively easily implemented to enable thecalculation of diverse graph-theoretic metrics and address differentvisualization needs. The system states resulting from each develop-ment stage are described and compared to each other, using theindices defined in Section 2.

In addition to the currently designed extension and develop-ment plan of the system, the recent master plan of the citydevelopment suggests four additional metro lines: three linearcrossing lines (numbered in sequence: 5, 6 and 7) and, afterwards,one semi-circumferential line (numbered: 8), which are illustratedin Fig. 2. The three linear (radial) axes crossing the core part of thesystem connect the various edge suburbs of the metropolitan area.The construction of these proposed metro lines aims at the long-term completion (with a time horizon extended beyond 2020) ofthe metropolitan public transport network, according to theprojected aggregate travel demand and urban growth trends.Based on the theoretical background mentioned before, this paperexamines an alternative plan for the system development, wherethe construction of the semi-circumferential line 8 precedes thatof the linear crossing lines (5, 6 and 7).

Provided that the specific approach serves for the quick-response evaluation of alternative network design options, allnodes are treated as both sources and sinks (attractors) of demandand density effects are considered in terms of such topologicalmeasures as the graph density, clustering coefficient, modularityand the ratio of transfer stations. The variations in urban popula-tion density essentially reflect the network structure, as describedby variations in the above measures, and the inner city (core) areais outlined by the semi-circumferential line. The following sectionevaluates the new proposed (alternative) system development,compared to the current development plan (with four metro lines)and the originally planned future development (where the semi-circumferential line 8 follows the construction of lines 5, 6 and 7).

4. Empirical analysis and discussion of results

The first set of results of the empirical analysis concerns thecurrent development plan of the Athens fixed-route transportsystem, which includes the extensions of tram and metro lines2 and 3 and the new line 4 (Section 3). The findings (Table 1) showthat the five stages of the current development will move thesystem towards small-world networking. This outcome reflectsthe increasing degree of connectivity, since the clustering coeffi-cient gradually increases by 12%, from C¼0.353 (for the existingmetro network with 54 stations and 55 links) to C¼0.394 (for thefinal configuration with 162 stations and 189 links), and theaverage shortest (minimum-cost) path is decreased by �18%, from14.9 (for the current total network with 107 stations and 118 links)to 12.2 (for the final configuration). Although the value of Ccoefficient lies within the lower range of values (about C¼0.4)for similar (but more ‘mature’) fixed-route transport networksworldwide, the current trend is consistent with the evolutionarystructural characteristics of those networks (Sienkiewicz and

Holyst, 2005; von Ferber et al., 2009; Derrible and Kennedy,2010). Such properties are associated with the fact that thenetwork becomes more clustered as it grows in size, whichreduces passenger travel times and improves the level of service.

These trends also reflect the changes in the network diameter,which substantially increases (from 32 to 48) when the tram andsuburban rail are included, but it decreases by the same magni-tude (from 48 to 32) after the completion of all scheduledextensions and the new line 4, as well as the increase of theaverage degree (from 2.2 to 2.3, for the total network) and theratio of transfer stations to total stations (from 7.4% to 16.7%, forthe total network). The reduction of the graph density from 0.04(for the current metro network) to 0.02 (for the current totalnetwork) and to 0.01 (for the final configuration) implies thegeographical expansion of the network (to increase coverage) andthat the planned topological arrangement of stations rather movesthe network away from being complete. Moreover, the networkdevelopment yields a higher modularity, from 0.72 (for the currentmetro network) to 0.78 (for the final configuration), which impliesthe increasing tendency of the system to be decomposed into sub-networks. This property depicts the emergence of local networkeffects, through the creation of transfer stations in the core cityand a few peripheral areas.

The incorporation of the tram and suburban railway lines, aswell as the extensions of existing lines, are found to reduce thesystem robustness, i.e., from r¼0.296 (for the current metronetwork) to r¼0.280 (for the current total network) andr¼0.065 (after all extensions). This outcome can be attributed tothe fact that those lines and their extensions do not generallycreate cycles, in terms of the number of loops due to existence ofalternative routes. Hence, the current network expansion plancannot enhance the system resilience, in terms of better absorbingdisturbances and avoiding failures in connecting one station withanother, dependent on size. However, the introduction of new line4 is found to increase the robustness to r¼0.111. After all thecurrently planned developments, the scaling factor ε remainsrelatively high, ε¼3.3 (R2¼63%), above the range of typical valuesascribing the scale-free property to similar types of networks. Thisoutcome shows the relatively even distribution of node connec-tions and reflects the lack of high reliance on the existing largetransfer stations. It may suggest the slow rate of ‘maturation’ or‘ageing’ of the network at these early stages of development,compared to other fixed-route transport networks which scale freeafter reaching a sufficient level of maturity.

Tables 2 and 3 present the results of the indices for the plannedsystem development (with the new lines in the order 5-6-7-8, i.e.the semi-circumferential line 8 last) and the alternative systemdevelopment (with the new lines in the order 8-5-6-7, i.e. thesemi-circumferential line 8 first), respectively. Despite both plansincluding the same number of stations (216) at their final networkconfiguration, the latter (alternative) development is found tosubstantially increase the number of links, compared to the former(planned) development. This outcome entails a considerablyincreased construction and operating cost for the alternativedevelopment, due to the need for building a higher numberof links between stations at the earlier stages to support theconnectivity of the system.

The difference in the total number of resulting links among thetwo final (currently planned and alternative) system configura-tions denotes the dynamic nature of the graph characteristics ofsuch planar networks as metro systems. Specifically, in thesenetworks, the topological properties are dependent not only onthe initial configuration but also on any intermediate connectivitychange over time, which signifies the complex nature of thesystem. Hence, the network structure changes at different ratesbetween the two metro expansion schemes. First, this is due to the

T. Tsekeris, A.-Z. Souliotou / Transport Policy 32 (2014) 88–9592

additional links which are necessary to be constructed at theinitial stage of the alternative network expansion (first column ofTable 3). The relatively high number of links required between thenew nodes of the semi-circumferential line 8 and the existingnodes, at the end of the current stage of development (last columnof Table 1), can be attributed to the fairly high sparsity of thepresent network. These additional links are not required for thenew nodes of the semi-circumferential line 8 at the plannednetwork expansion (last column of Table 2), due to the relativelyhigh graph density at this final stage of development.

Second, the additional transfer stations of the alternativeexpansion scheme (Table 3) correspond to nodes which aretopologically well located in the network, i.e. at the core part ofit, which denotes a higher proportion for any two links to cross

each other and create loops. This alternative configuration impliesthat the importance of a node as a transfer point to join pairs ofnodes is higher and the number of shortest paths grow faster withnetwork size, which finally reflects the difference in the resultingnumber of links and the other metrics, in relation to the plannedexpansion scheme (Table 2). The alternative expansion schemesuggests increased economies of scale, as the addition of a nodeprovides on average a higher benefit to all other nodes, in thesense that passengers could use more central nodes for transfer-ring in the network, compared to the planned expansion scheme.Due to these economies of scale, there is nonlinearity pertaining tothe dynamics of the system development, as the cost relationshipbetween any pair of stations is not static or directly proportional tothe increase of the network size.

Table 2Results of indices for the planned system development with new lines in the order 5-6-7-8.

Measure Athens total fixed-routePT network with all extensionsand lines 4 and 5

Athens total fixed-routePT network with all extensionsand lines 4, 5 and 6

Athens total fixed-routePT network with all extensionsand lines 4, 5, 6 and 7

Athens total fixed-routePT network with all extensionsand lines 4, 5, 6, 7 and 8

Links 209 237 256 276Stations 176 196 209 216Average degree 2.307 2.398 2.431 2.537Network diameter 32 32 32 29Graph density 0.013 0.012 0.011 0.012Modularity 0.777 0.773 0.781 0.758Average path length 11.810 11.459 11.004 10.195Clustering coefficient C 0.400 0.407 0.412 0.430Robustness index r 0.091 0.020 0.014 0.028Transfers/stations (%) 18.750 20.408 21.053 25.463Scaling factor ε (R²) 3.290 (0.695) 2.710 (0.540) 2.720 (0.568) 2.570 (0.508)

Table 3Results of indices for alternative system development with new lines in the order 8-5-6-7.

Measure Athens total fixed-routePT network with all extensionsand lines 4 and 8

Athens total fixed-routePT network with all extensionsand lines 4, 8 and 5

Athens total fixed-routePT network with all extensionsand lines 4, 8, 5 and 6

Athens total fixed-routePT Network with all extensionsand lines 4, 8, 5, 6 and 7

Links 296 316 344 363Stations 174 186 204 216Average degree 2.722 2.907 3.167 3.333Network diameter 27 24 22 21Graph density 0.012 0.013 0.014 0.014Modularity 0.726 0.703 0.688 0.661Average path length 8.803 8.186 7.313 6.891Clustering coefficient C 0.574 0.572 0.568 0.565Robustness index r 0.374 0.188 0.088 0.032Transfers/stations (%) 19.540 22.581 25.000 27.315Scaling factor ε (R²) 3.130 (0.610) 3.190 (0.701) 3.070 (0.611) 2.710 (0.604)

Table 1Results of indices for different (completed and scheduled) development stages of the Athens fixed-route transport system.

Measure Existing metrolines 1, 2 and 3

Existing network(metro lines 1, 2 and 3,tram and suburban rail)

Existing networkwith extensions oflines 2 and 3

Existing network withextensions of lines 2 and3 and Tram

Existing networkwith all extensionsand new line 4

Links 55 118 141 159 189Stations 54 107 127 139 162Average degree 2.037 2.206 2.205 2.259 2.309Network diameter 32 48 36 32 32Graph density 0.038 0.020 0.017 0.016 0.014Modularity 0.716 0.757 0.779 0.781 0.784Average path length 10.449 14.851 13.535 12.863 12.172Clustering coefficient C 0.353 0.375 0.376 0.387 0.394Robustness index r 0.296 0.280 0.205 0.065 0.111Transfers/stations (%) 7.407 12.150 11.811 13.669 16.667Scaling factor ε (R²) 3.490 (1.000) 3.840 (0.495) 4.160 (0.620) 3.680 (0.736) 3.300 (0.634)

Note: All measurements result from Gephi 0.7a© (http://gephi.org/).

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The resulting indices clearly demonstrate the higher level ofmaturity and the increased level of service and reliability asso-ciated with the alternative system development. More specifically,the final network configuration obtained from the alternativedevelopment (last column of Table 3) is more dense (higher graphdensity equal to 0.014) and it involves a higher ratio of transferstations to total stations (27.3%), compared to the planned devel-opment (last column of Table 2), with corresponding values equalto 0.012 and 25.5%. Furthermore, it has a significantly higheraverage degree (3.33 versus 2.54) and significantly lower networkdiameter (21 compared to 29), which imply lower time costsincurred to passengers to travel between any two points (stations)in the system.

The findings show that both the planned (Table 2) and,particularly, the alternative (Table 3) system developments finallyyield lower modularity scores than the current one (Table 1) (0.76and 0.66, respectively, compared to 0.78). Namely, increasedmodularity is found to be more appropriate for the system at itsearly stages of development than the later ones, when itapproaches a complete (and more hierarchical) graph structure.Also, a future system development originally based on radial axesfits better to planning policies whose priority is a networkstructure with increased modularity, compared to a future devel-opment originally based on a circumferential line. The lowermodularity associated with the future development plans possiblydenotes that, as the system gradually tends to obtain scale-freeand, particularly, small-world network properties (see below), itssmooth operation becomes more dependent on a few largetransfer stations rather than on neighboring stations of similardegree.

In the final network configuration of both future developmentplans, the system robustness is found to decrease, namely,r¼0.028 for the original plan (Table 2) and r¼0.032 for thealternative plan (Table 3), compared to the current developmentplan where r¼0.111 (Table 1). Therefore, the future developmentoriginally based on the construction of the semi-circumferentialline would provide slightly higher robustness, in comparison tothe future development originally based on the construction ofradial axes. This overall reduction of robustness denotes that thefuture expansion of the network through constructing new metrolines is generally anticipated to deteriorate the avoidance of failurein connecting one station with another. It can be attributed to theneed for increasing coverage and the interconnectivity and inter-modality of the urban transport system, and enhancing theaccessibility of the residents living in the growing city suburbs,especially at the north and west parts of the metropolitan area, tothe central city.

The change of network robustness with respect to the ratio oftransfer to total stations should be explored in accordance withthe underlying dynamics pertaining to the topological features ofthe system. Specifically, the relationship between connectivity, onthe one hand, and robustness (and resilience), on the other hand,can be considered as nonlinear (Wallace and Wallace, 2008;Reggiani, 2012). This is because, during the system0s transitiontowards a scale-free network, the robustness decreases (or thevulnerability increases) as the connectivity grows, due to thefragility of the small and isolated communities (sub-networks)formed in the periphery of the system. At these stages of the metrodevelopment, the network is regarded as being still immature andthe average degree of transfer stations in the core increases with arate higher than that of the average degree of stations in theperiphery. This decreasing relationship between the robustnessand the ratio of transfer to total stations will presumably continue,until such a critical level of connectivity is reached that will signifythe realization of sufficiently large hubs (transfer stations) ofnetwork-wide extent. Above this level (of maturity), the increase

of network connectivity and creation of more transfer stationswould retain the scale-free property and enhance the systemrobustness and its resilience, because shocks and contagiousprocesses could then be prevented by considerably increasingthe linkages among nodes and the graph density.

In addition, the relatively high clustering coefficient, in conjunc-tionwith the reduced average path length, makes the system a small-world network, for both future development plans. The small-worldnetworking property is notably evident for the alternative future plan(Table 3), compared to the original future plan (Table 2). In the formercase, the average path length rapidly decreases from 8.8 (after theconstruction of the semi-circumferential line) to 6.9 (at the finalconfiguration), whereas the clustering coefficient remains almost thesame (approximately equal to C¼0.57) until the completion of alllines. In the original plan (Table 2), the average path length is quitefar from those values (it decreases only from 11.8 to 10.2), whereasthe clustering coefficient is considerably lower and slightly increasesfrom 0.40 to 0.43.

Last, in the final configuration of both future developmentplans, the value of scaling factor ε is found to lie within the typicalrange (2oεo3), i.e., ε¼2.57 for the planned network expansion(Table 2) and ε¼2.71 for the alternative network expansion(Table 3). The specific range signifies the transition of both systemdevelopment plans towards a scale-free network setting. Thisoutcome demonstrates the critical dependence of the resultingsystem on a few major transfer stations (hubs) and it suggests theso-called ‘preferential attachment’ behavior. According to it, as thenetwork grows, the probability that a new node will be connectedto an existing node relies on the connectivity (node degree) of thatnode. Consequently, prioritization of infrastructure and serviceinvestments (for protection, maintenance and upgrading) in thosecritical nodes would enhance the resilience and scalability ofpublic transport operations and the efficient allocation of networkresources.

Additionally, the scaling factor values can offer a reference basefor comparing the results among alternative plans, in order tounderstand the role of topological characteristics and their evolu-tion, and verifying their consistency with other metro networksworldwide. In particular, it is confirmed that the node degreefrequency is distributed according to a power law with theexponent to vary with the network size, structure and rate ofgrowth. It is further shown that, for a growing metro network, asthe one corresponding to the alternative development plan(Table 3), the scale-free property can be attained even if a decayrelationship between the node connectivity (or clustering) andaverage path length is not always observed.

5. Conclusions

Public investment in metro lines and other fixed-route publictransport infrastructure is considered to be of high priority forpromoting the efficient, resilient, fair and environmentally sus-tainable mobility in large urbanized areas. By applying a graph-theoretic analysis to the various (past, current and future) stages ofthe fixed-route transport system development in the Athensmetropolitan area, a number of important implications come upabout the formulation, programming and evaluation of suchinvestment plans. The use of graph-theoretic tools and metricsmay plausibly capture network economies of scale and density aswell as interaction effects that arise in the topological structureand operational conditions of the system. The adoption of suchmeasures as connectivity, modularity and robustness increase theflexibility of the decision-making process according to a varietyof policy objectives, in contrast with the strict criterion of traveltime minimization. Hence, they can provide useful inputs into the

T. Tsekeris, A.-Z. Souliotou / Transport Policy 32 (2014) 88–9594

optimal network design and investment project evaluation mod-els, allowing the examination of different urban planning anddevelopment scenarios.

The present findings can offer valuable insights into how newmetro lines, as proposed in the master plan of the city development,should be configured with respect to each other and evolve overtime. The current system can be regarded as relatively ‘immature’ orbeing at the early stages of its development, compared to othersimilar large-scale public transport systems worldwide. Suitabletransit network design options for future development involve theincrease of the ratio of transfer stations to the total stations(through more crossings of the existing lines by the extended andnew lines) and a new (semi-)circumferential metro line, in order toincrease the network density, connectivity and level of service topassengers.

Prospective planning directions can be reformulated and prior-itized in terms of trading investment cost for enhancing systemefficiency and resilience. For the metropolitan area of Athens, theconstruction of a semi-circumferential line in the middle of the overallsystem development (including 8 metro lines) constitutes a mega-infrastructure development option which trades cost for increasedlevels of service and robustness, compared to the scenario of con-structing the semi-circumferential line at the final stage. The increasedlevels of service relate to higher connectivity and reduced travel times,namely, greater small-world networking with scale-free characteris-tics. The latter features denote the growth of network size withoutcompromising the ability to manage or control operations withincreasing scale. Finally, new tram line extensions could additionallyoffer cost-effective design solutions for increasing network density andconnectivity between metro lines, by raising the number of transferstations both at the core and the periphery of the city.

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