Analyzing the effect of the street network configuration on the efficiency of an urban transportation system

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  • Cities 31 (2013) 285297Contents lists available at SciVerse ScienceDirect


    journal homepage: www.elsevier .com/locate /c i t iesAnalyzing the effect of the street network configuration on the efficiencyof an urban transportation system

    Abbas Sheikh Mohammad Zadeh , Mohammad Ali RajabiDepartment of Surveying and Geomatics Engineering, University of Tehran, North Kargar St., Tehran, Iran

    a r t i c l e i n f oArticle history:Received 14 March 2012Received in revised form 4 July 2012Accepted 25 August 2012Available online 30 September 2012

    Keywords:Urban transportation networkSpatial configurationCentrality measureTraffic equilibrium0264-2751/$ - see front matter 2012 Elsevier Ltd. A

    Corresponding author. Address: Department oEngineering, University of Tehran, North Kargar St., PIran. Tel.: +98 21 88008841; fax: +98 21 88008837.

    E-mail addresses: (A. Sheikh (M.A. Rajabi).a b s t r a c t

    This paper proposes a new specialized centrality measure to quantify the importance and the contribu-tion amount of each street in an urban transportation network. Contrary to the previous general central-ity measures (e.g., degree, betweenness, and closeness), this measure considers three important vehiculartraffic characteristics: street capacity restrictions, the dominant pattern of travel demands, and thetraffic-flow equilibrium. Applying the developed centrality measure to simulated networks shows thattraffic flows more efficiently in urban networks with small-world configurations, where a certain num-ber of shortcut links and locally clustered streets are allowed. However, with regular grid networks, theapplication of the proposed measure shows little efficiency. This outcome suggests that the regular grid isnot suitable as a base structure in urban planning.

    2012 Elsevier Ltd. All rights reserved.Introduction

    In the last decade, it has been interesting to uncover the struc-tures of various real-world systems, including information, social,technology and biological systems, using topological analysis(Jiang, 2007). A majority of these analytical studies have modeledthe real-world systems as network graphs. In this context, a graphrefers to a collection of nodes (vertices) and links (edges) that con-nect pairs of related nodes in the real world. The graphical repre-sentation and related mathematical structure, i.e., graph theory,have been used widely to model pairwise relations between ob-jects in the systems. In this approach, measures are applied to esti-mate the importance or contribution amount of each component ina system. Among the various measures developed in graph theoryand network analysis, the concept of centrality is used to describethe structure of real-world systems. In this approach, the centralityof a node (link) determines the relative structural importance ofthat node (link) within the graph. In most networks, certain nodes(links) play special roles in the networks structure. In other words,there is an intuitive feeling that they are more central than others(Koschtzki et al., 2005). The centrality measures have been devel-oped to quantify this feeling. However, the term central is a gen-eral concept and can be interpreted in many ways. Thus, over time,ll rights reserved.

    f Surveying and Geomatics.O. Box 11365-4563, Tehran,

    hammad Zadeh), marajabi@dozens of centrality indices have been proposed based on differentinterpretations.

    Over the last five decades, a number of authors have developednumerous centrality measures, including degree centrality (Free-man, 1979), closeness and betweenness centralities (Freeman,1977), eigenvector centrality (Bonacich, 1972), information cen-trality (Stephenson & Zelen, 1989), and flow betweenness (Free-man, Borgatti, & White, 1991). Many of these concepts were firstdeveloped in social network analysis, and many related terms re-flect a sociological origin (Newman, 2010). However, the conceptshave also been widely applied in the topological analysis of variousreal-world systems, including transportation networks. In thesestudies, the streets are modeled as links and the intersections asnodes (or vice versa) in a graph. Then, the geometric or topologiccharacteristics of the graph are investigated using centrality mea-sures (Crucitti, Latora, & Porta, 2006; Gastner & Newman, 2006;Jiang, 2009; Jiang & Claramunt, 2004b).

    However, in introducing and applying numerous centralitymeasures to investigate urban street networks, researchers gener-ally neglect that the centrality measures typically make some im-plicit assumptions about the manner in which things flow in anetwork (Borgatti, 2005). Therefore, the measures can be appliedonly to flow processes that possess properties that are compatiblewith those assumptions. In other words, if the centrality measuresare applied to flow processes with incompatible properties, themeasures may result in wrong answers (Borgatti, 2005).

    Thus, applying a centrality measure to discover the configura-tional properties of a transportation network must observe themanner in which vehicles flow in a street network. Most of the im-plicit assumptions in certain applied centrality measures are rarely

  • 286 A. Sheikh Mohammad Zadeh, M.A. Rajabi / Cities 31 (2013) 285297compatible with the vehicular movements in urban areas. Forexample, Freemans betweenness, which has been applied as a cen-trality measure to analyze street networks (Crucitti et al., 2006;Jiang & Claramunt, 2004a; Tomko, Winter, & Claramunt, 2008), as-sumes that there is a flow between each possible pair of nodesthroughout the network that moves only along the shortest paths(geodesic trajectories). This assumption might be true for certainevents in social networks, which were the origin of Freemansbetweenness. However, Freemans betweenness is not valid for ur-ban traffic flow.

    Similar implicit assumptions could be inferred from other cen-trality measures. For example, in certain centralities (e.g., flowbetweenness), it is assumed that each node is visited once,whereas in other measures (e.g., Bonacichs eigenvector), theassumption is that the movements can be circuitous and a travelercan revisit a node more than once (Borgatti, 2005). Sometimesthese assumptions are related to the nature of moving objects.For example, the betweenness centrality assumes that the movingobject is indivisible and cannot appear in more than one locationsimultaneously. The eigenvector assumption is the opposite; i.e.,the moving object can multiply and take multiple paths (Borgatti,2005). Therefore, according to the different implicit assumptionsthat are made by various centralities, it is essential to choose anappropriate measure whose assumptions are compatible with thesystem under study. For example, because a vehicle is an indivisi-ble object and cannot multiply, the betweenness centrality is morecompatible with vehicles than the eigenvector centrality, which ismore suitable for information or infection flows.

    In addition to the mentioned centralities, there are other typesof measure, such as mean first passage time and mean recurrencetime (Chung, 1997), which have been developed based on the ran-dom walker concept. A random walk is a type of movement inwhich the traveler moves from one node to a neighbor node ran-domly (Blanchard & Volchenkov, 2008). In other words, these mea-sures implicitly assume that the travelers are aimless and forgetful,which is contrary to the nature of aware and purposeful drivermovements. Therefore, because of their incompatible assumptions,these measures may not be adequate to depict the moving behav-ior of urban travelers.

    Even in the space syntax theories, which are a set of well-struc-tured theories and techniques for the analysis of spatial configura-tions that was originally introduced by Hillier and Hanson (1984),there are certain measures (e.g., depth distance) with implicitassumptions (e.g., the unlimited capacity of links) that are incom-patible with vehicular traffic flow in urban areas. As a result, mostof the studies performed on the configuration of transportationnetworks have applied centralities that are based on assumptionsthat are incompatible with the traffic flow in cities. As mentionedabove, most of these centralities originate in social network analy-sis (Crucitti et al., 2006). Therefore, the centralities are incompati-ble with the properties of all networks. Therefore, in selecting andapplying these centralities, one should consider the nature of thenetwork.

    As a result, a special centrality for studying transportation net-works seems essential. Therefore, this paper proposes a new mea-sure of centrality that encompasses three important characteristicsof urban vehicular traffic flows:

    The limited performance or capacity of urban streets (instead ofassuming that the streets can accommodate unlimited flow). The thrifty way-finding behavior of commuters, who eventuallyequilibrate the traffic distribution throughout the network(instead of assuming that the commuters are aimless orforgetful). The dominant pattern of travel demand in the network (insteadof assuming that travelers flow from anywhere to anywhere).In addition to the problem of centrality, in spatial-structurestudies of transportation networks, an issue that has long been atthe center of attention is the relationship between spatial configu-ration and traffic flows. A large body of research on this topic hasmainly attempted to study the structure of congestion flow fromthe perspective of the spatial configuration of streets. Accordingto these studies, spatial configuration is a factor that determineshuman activity and cognition in an urban area (Volchenkov,2008). Consequently, the behavior of commuters is more affectedby the topological and structural properties of a network than bythe networks metric characteristics (Hillier & Iida, 2005). Addi-tionally, it has been proposed that the goal-directed movementof individuals in a network does not significantly affect the generalflow distribution in that network (Jiang, Yin, & Zhao, 2009). Forexample, a recent study shows that the underlying street structureis the main determining factor in shaping the distribution of thetraffic flow (Jiang & Jia, 2011). The study concludes that in a givenstreet network, the movement patterns generated by purposefultravelers (e.g., human beings) and purposeless travelers (e.g., mon-keys) are the same.

    At first glance, it seems that the results of these studies contrastwith the idea that the pattern of land use in different parts of a net-work determines the movement density in that network (Dawson,2003). However, Hillier, Penn, Hanson, Grajewski, and Xu (1993)showed that the existence of land-use attractions in different partsof a network changes the linear relationship between centralityand movement density to a logarithmic relationship. In otherwords, these attractions serve as multipliers in the basic patternof movement density (Hillier et al., 1993). Concisely, all of thesestudies indicate that Human mobility in a street network is a sortof network-constrained movement (Jiang et al., 2009). In the citedstudies, the main purpose has been to determine the effect of con-figuration on movement density. No further studies have been per-formed regarding the quality of this effect. Therefore, consideringthe effect of configuration on flow distribution, the second partof this paper attempts to focus on the quality of this influenceusing the proposed measures.

    The organization of this paper is as follows. The characteristicsof vehicular traffic in urban areas are discussed, and the algorithmof the proposed targeted constrained betweenness (TCB) centralityis described. Additionally, the numerical results of applying theproposed TCB to simulated and real data are presented. Next, theTCB measure is used to evaluate the performance of different spa-tial configurations of networks. Finally, the last section presentsconclusions and discusses the results.Vehicular traffic flow characteristics

    Vehicular traffic is a complicated phenomenon that results frominteractions between trip makers, land-use patterns, socioeco-nomic factors and transportation infrastructure. Because of theunpredictable reactions of drivers, vehicular flow typically doesnot follow a specific rule. Therefore, to properly model traffic flows,it is important to consider the effects of the main factors involved.Transportation infrastructure properties, driver behavior and thetravel demand pattern are three determining factors in shapingthe traffic pattern. These important points are missing in manyof the previous structural studies on transportation networks.Therefore, this paper aims to evaluate the structural importanceof links in urban transportation networks subject to the three men-tioned constraints: the infrastructure properties, the behavior ofdrivers and the travel demand pattern.

    The first constraint concerns the capacity or physical restrictionof each street to pass traffic. Each street of a network has a limitedcapacity. Therefore, a street can only pass a certain number of

  • A. Sheikh Mohammad Zadeh, M.A. Rajabi / Cities 31 (2013) 285297 287vehicles fluently. As the number of moving vehicles increases, thetraffic fluency on that link decreases. In other words, the speed ofthe moving vehicles drops as the number of vehicles increases. Thisrelationship is expressed by a mathematical equation known as thelink-performance or cost-flow function (Ce = f(Ve)), in which thecost of link e is a monotonically increasing function of link es pass-ing flow Ve. Most centrality measures are based on seeking theshortest paths in networks. However, almost no centrality measureconsiders this restriction of links. Instead, the centrality measuresimplicitly assume an unlimited capacity for all links, which resultsin a constant cost for the links, whereas the cost of each link is var-iable and changes with the volume of the passing flow. Conse-quently, it could be inferred that a link in a given network mayhave two different centralities depending on the presence or theabsence of a capacity restriction.

    When addressing the flow distribution in a given network, it isnecessary to consider the behavior of moving objects in that net-work. The overall flow distribution in a network is typically theaggregation of all of the movements of each moving object. Accord-ing to its nature, a moving object behaves either consciously orunconsciously. For example, in certain networks, such as waterand oil pipeline networks, the moving objects are not conscious.Thus, they move through the network following physical rules(e.g., hydrodynamics). In contrast, in a traffic network, a travelerdetermines to move from an origin to a destination on an optimalpath. In this case, the moving object is conscious and intends toreach a certain destination on an optimal path. Thus, the pathmay change whenever the traveler feels that there is a betterway to reach the destination. It has been shown that in urban areasthe aggregation of such individual decisions finally forces trafficdistribution into a state of equilibrium, as when two identical tick-et windows sell tickets for a show. In this case, each new ticketbuyer typically chooses the ticket window with the shorter queue.Thus, over time, the queues reach a state of equilibrium in whichneither queue is significantly shorter than the other queue at anygiven time. According to Wardrops first principle, or Wardropsequilibrium, the urban traffic network comprises a similar set ofcircumstances: Each user non-cooperatively seeks to minimizehis cost of transportation. The traffic flows that satisfy this princi-ple are usually referred to as user equilibrium (UE) flows, sinceeach user chooses the route that is the best. Specifically, a user-optimized equilibrium is reached when no user may lower hisFig. 1. A comparison of the Betweenness centrality of links (atransportation cost through unilateral action (Wardrop, 1952).This particular behavior of urban commuters is another specialcharacteristic that must be considered in the structural study ofsuch networks. To illustrate this factors importance, in a samplenetwork, the betweenness centrality of each link and the normal-ized flow in the user-equilibrium state have been evaluated(Fig. 1a and b, respectively). As shown in Fig. 1, the comparison be-tween the equilibrium flows and the betweenness centralities re-veals that the way-finding behavior of travelers may lead thetravelers to use peripheral links that are not considered importantto the overall structure, which indicates that ignoring this con-straint in centrality evaluation can lead to unrealistic results.

    In general, an urban transportation system can be considered tobe a supplydemand system in which the infrastructure of thetransportation system can be further considered as the supply sideand the passenger movements as the demand side (Ortuzar & Wil-lumsen, 2004). The demand side of this system is generally shapedby human activity and influenced by socio-economic factors,demographic conditions, land use, and other factors. In the lastdecades, different theories have been presented for modeling tra-vel demand to achieve a balance between supply and demand inthis system. The first-generation travel demand model (the classicfour-step model) was developed in the 1950s. This model wasmainly designed to predict the effect of investment on the develop-ment of infrastructure, which was rapidly growing at that time(Jovicic, 2001). The classic four-step model satisfied this need atthat time. After two decades, because of considerable changes incongestion management, there was a need for models that couldpredict the reaction of individuals who were encountering traveldemand management policies such as single driver charges androad pricing. The classic four-step model was unable to predictthe reaction of individuals who were facing traffic managementpolicies. In the 1970s, second- (the disaggregated trip-based de-mand model) and third-generation (activity-based travel demandmodels) models were developed to satisfy the needs of planningpolicies rather than the needs of infrastructure expansion (Jovicic,2001). The purpose of all of these models is to estimate the traveldemands in different parts of the network. The travel demand pat-tern of commuters is another specification of a transportation sys-tem that should be considered when evaluating the differentcomponents of a network. For example, in the network sampleshown in Fig. 2, the betweenness centralities of each link are) with normalized flows in the user-equilibrium state (b).

  • (a) (b)Fig. 2. Different evaluated centralities of links in a network. (a) Depicts ordinary edge betweenness, in which it is implicitly assumed that the travel pattern is a matrix of oneswith zero diagonal entries. (b) Depicts constrained edge betweenness, in which the centrality is evaluated based on a given travel pattern.

    288 A. Sheikh Mohammad Zadeh, M.A. Rajabi / Cities 31 (2013) 285297evaluated using two ordinary and constrained methods. As shownin Fig. 2a and b, the structural participation of a given link mayvary with respect to different patterns of travel demand.

    Considering the described characteristics of urban traffic flow,this paper proposes a new centrality measure to evaluate the rela-tive contribution and importance of streets in urban transportationnetworks. This new centrality is denoted targeted constrainedbetweenness (TCB) and considers three main characteristics of ur-ban traffic flows: street capacity, user-equilibrated traffic flow, andthe travel demand pattern.

    Targeted constrained betweenness (TCB)

    Beckman, McGuire, and Winsten (1956) proposed a rigorousframework to express Wardrops principles as a constrained mini-mization problem. In the proposed mathematical program, to ob-tain Wardrop equilibrium flows, an objective function isminimized subject to flow constraints. The problem can be writtenas (Beckman et al., 1956):

    Minimize ZfTijrg Xe

    Z Ve0

    Ceudu 1

    subject to:Xr

    Tijr Tij and Tijr P 0

    where Z{Tijr} is the objective function that corresponds to the sum ofthe areas under the cost-flow curves for all links, Tijr is the numberof trips between i and j via route r, Tij is the total number of tripsbetween i and j, Ve is the flow on link e and Ce(u) is the cost of linke when the flow of u is passing through link e. Incremental assign-ment, the method of successive average (MSA) and Frank-Wolf ap-proaches are methods commonly used to solve the constrainedminimization problem (Ortuzar & Willumsen, 2004).

    As mentioned above, to properly assess the importance and par-ticipation of a street in an urban transportation network, the ap-plied centrality measure must consider the balanced distributionof the traffic flow expressed in Eq. (1). Additionally, the appliedcentrality measure must consider the capacity of the streets andthe travel demand pattern. To fulfill these conditions, the proposedcentrality in this paper is developed based on the edge betweenness(EB) centrality. According to Freeman (1979), the EB centrality oflink e in a network corresponds to the fraction of all possible short-est paths between two nodes that pass through link e. Therefore,EB is the frequency of link occurrence on the shortest paths. TheEB of link e is given as follows (Freeman, 1979):

    EBe Xij

    nije=nij 2

    where nij is the total number of shortest paths from node i to node j,and nij(e) is the number of those paths that pass through link e. Asmentioned above, the assumption in Freemans betweenness offlow from each node to all others is incompatible with thesupplydemand paradigm of urban street networks. Therefore,Freemans betweenness must be modified to count only the flowsthat occur in the travel demand pattern instead of all possible flows.Furthermore, the proposed centrality must consider the capacity

  • A. Sheikh Mohammad Zadeh, M.A. Rajabi / Cities 31 (2013) 285297 289restriction of each link. When the flow on a given link exceeds athreshold, the cost of that link increases significantly, which com-pels the other drivers to choose a cheaper neighbor link instead ofpassing the congested link. This link-switching to find an optimalpath is continued by each commuter and finally forces the flow dis-tribution into Wardrops equilibrium.

    Taking these considerations into account, a new iterative cen-trality is proposed in the following algorithm. This algorithm yieldsthe new measure: TCB. This centrality measure evaluates the rela-tive participation and importance of each street in urban transpor-tation networks subject to the mentioned constraints.

    TCB centrality algorithmFipa1

    g. 3. The procettern derivedTCB0 = 0

    2 C0e CeTCB


    3 t = 0

    4 While (tol > e)

    5 t = t + 1

    6 For each commute in the travel demand pattern

    7 EBt Calculate EB based on Ct1e

    8 End For

    9 TCBt = (1 t1)TCBt1 + t1EBt10 Cte CeTCBt11 tol = TCBt TCBt1

    12 End While

    13 Output TCBtAs shown in the TCB algorithm, the proposed centrality is aniterative process that converges to the TCB measure. This processstarts by assigning an initial value (i.e., zero) to the TCB of each link(TCB0). Afterward, the initial cost of each link C0e is calculatedbased on the initial value of the TCB. The algorithm continuesthe iteration by calculating the EB of each link, counting only thespecified commutes in the travel demand pattern after updatingthe cost of each link based on the value of a links EB at the previ-ous iteration. Next, adopting the calculated EB, the corrected valueof the TCB is determined by applying a weighted average between(a)ss of applying travel demand patterns as an input in the TCB centrality. (from activity-based models (Feil, 2010).the current TCB and the calculated EB value of each link (line: 9).The weight of the EB in the weighted average is reciprocal to theiteration number. Therefore, the correction value scales down asthe iteration proceeds. Next, the cost of each link is recalculatedbased on the corrected TCB values ( line: 10). This cost updatingcauses the trip makers at the next iteration to choose an alternativepath to avoid the congested links. Finally, the TCB value is com-pared with the previous value to determine if the difference issmall enough to terminate the process (line: 11).Travel demand pattern as an input in the TCB algorithm

    TCB centrality is a measure that seeks to evaluate the structuraland topological importance of each link in an urban transportationnetwork considering three important features of an urban trans-portation system. One of these important features is the travel de-mand pattern. This pattern is the interface of the TCB centrality andthe travel demand models. Regardless of the model (trip-based oractivity-based) used to evaluate the travel demand, the results ofeach model in the form of an origindestination (OD) pattern canbe considered to be an input in the TCB centrality. The basis of thisinput in the TCB centrality is not a trip or an activity but the pre-dominant pattern in the origindestination of commutes that canbe the result of the classic four-step model or the aggregated formof the travel demands in an activity-based model. One can use theresults of a travel demand model in the TCB centrality in two ways:(1) If the model used is the four-step model or an aggregate activ-ity-based model, the results, which comprise an OD matrix, can bedirectly used in the TCB centrality (Fig. 3a). (2) If the model used isa disaggregate activity-based model, the results of this model canbe aggregated in an OD matrix (Feil, 2010) and then used in theTCB centrality (Fig. 3b). Because the TCB centrality is primarilyproposed to evaluate the planning and development scenarios ofinfrastructures, detailed information about the activity of individ-uals, which is necessary to evaluate the consequences of policies,is not required here. As a result, the aggregation of informationon the activities of individuals in the form of traveler flow (anOD matrix) and the use of this information as an input for theTCB centrality do not bias the result.(b)a) The travel demand pattern derived from trip-based models. (b) The travel demand

  • 290 A. Sheikh Mohammad Zadeh, M.A. Rajabi / Cities 31 (2013) 285297Numerical results

    To examine the TCB centrality measure, three different struc-tured networks are simulated, and the importance of their linksin the entire structure is estimated by applying the proposed algo-rithm. Then, the estimated importance values are compared withthe link flows in the user-equilibrium state. As shown in the left-hand column of Fig. 4, each simulated network has 16 nodes and24 undirected links structured in three different regularity levels,i.e., a regular grid (a), a semi-regular structure (b) and a randomnetwork (c). Network (a) is a 4 4 square grid, whereas network(b) is a rewired version of the regular grid with two shortcuts. Net-work (c) is a network with a random structure that has been gen-erated by the ErdosRenyi model (Newman, 2010). These differentregularity levels are chosen to examine the TCBs sensitivity to dif-ferent structures. The dual graph representations of the simulatednetworks are depicted in the right-hand column of Fig. 4. In thesedual graphs, each node represents a street. Therefore, if two streetsFig. 4. Simulated graphs (left-hand column) and the related dual graphs (right-hand ccentrality of the links.intersect, the streets nodes in the dual graph are connected via alink. Additionally, the size of the nodes in the dual graphs is pro-portional to the betweenness centrality of each street in the simu-lated networks.

    The travel demand pattern of each study network is modeled asa 16 16 OD matrix in which the 30 trips are simulated in a man-ner that randomly distributes the origins and destinations of thetrips throughout the network.

    To model the relationship between cost and flow, many differ-ent functions, such as polynomial, exponential, and logarithmicfunctions, have been presented. Each of these functions expressesa different cost growth rate as a function of the flow. Because inthe suggested centrality the moment of the congestion in a linkwhen the flow switches to the neighboring link is important, therate of the transition from a free-flow state to a congested statein the network links may affect the overall flow distribution andconsequently influence the evaluated structural importance ofeach link. To examine the effect of this factor, three functionsolumn). The size of nodes in the dual graphs is proportional to the betweenness

  • 0.5 1 1.5 2 2.5 30.5





    3Guide lineC = F + 1C = F 2 + 1

    C = eF - 1

    0 0.5 1 1.5 2 2.5 30






    3Guide lineC = F + 1C = F2 + 1

    C = eF - 1

    0 0.5 1 1.5 2 2.5 3 3.50







    3.5Guide lineC = F + 1C = F2 + 1

    C = eF - 1

    Fig. 5. The correlation between the TCB and the Wardrop equilibrium in threesimulated networks.

    0 5 10 15 20 25 30 35 40 45 500







    1.4C = F + 1C = F 2 + 1

    C = eF - 1

    0 5 10 15 20 25 30 35 40 45 500







    0.7C = F + 1C = F 2 + 1

    C = eF - 1

    0 5 10 15 20 25 30 35 40 45 500







    1.4C = F + 1C = F 2 + 1

    C = eF - 1

    Fig. 6. The root mean square error (RMSE) of the calculated TCB for the first 50iterations.

    A. Sheikh Mohammad Zadeh, M.A. Rajabi / Cities 31 (2013) 285297 291(linear, quadratic, and exponential) with different rates of transi-tion from the free-flow to the congested state have been chosenfor each network. These functions are shown in Fig. 5. In real-worldstudies, the parameters of these functions have been calibratedconsidering the physical characteristics of each link and then usedin modeling and predictions.

    Using a simulated travel demand pattern and the cost-flowfunctions, the importance of the links and the flow on the links

  • 292 A. Sheikh Mohammad Zadeh, M.A. Rajabi / Cities 31 (2013) 285297in the Wardrop equilibrium state are calculated in three simulatednetworks. The importance of each link is evaluated using the TCBcentrality measure, and the balanced flows in the Wardrop equilib-rium are calculated by adopting the disaggregated simplicialdecomposition algorithm to solve the constrained minimizationproblem expressed in Eq. (1). The calculated values of the TCBand the Wardrop equilibrium flows for each simulation networklink are shown in Fig. 5.

    As shown in Fig. 5, in each configuration and for different cost-flow functions, there is a strict correlation between the TCB cen-trality and the Wardrop equilibrium flows, which indicates thateach link with a higher TCB centrality passes more flow in the equi-librium state. In other words, in the overall structure of each net-work, there are links that play a more important role than theothers. The importance of the links are quantified using the TCBcentrality subject to the mentioned constraints.

    To illustrate the correlation between the TCB centrality and theWardrop equilibrium flows, the root mean square error (RMSE) ofthe TCB relative to the normalized Wardrop equilibrium flows isshown in Fig. 6 for the first 50 iterations.

    As shown in Figs. 5 and 6, the TCB centrality strictly correlateswith the flow on the links in the Wardrop equilibrium state. There-fore, the TCB centrality can be regarded as a specialized indicator tomeasure the importance of streets in an urban transportation net-work where the streets are capacity restricted, the starting and endpoints of travels are determined by the travel demand pattern, andthe traffic flow distributes to reach equilibrium.Case study: computing the TCB centrality in the Isfahan streetnetwork

    To illustrate how the TCB centrality can be used to analyze ur-ban street networks, a case study is performed using the transpor-tation network of Isfahan, Iran. Isfahan is the capital of Isfahanprovince and located in the center of Iran. Isfahan has a populationof 1,600,000 and is Irans third largest city. The Isfahan networkstudied in this paper consists of highways and major and minorarterials. The network is composed of 878 links and 705 nodes.The streets central lines and the related dual graph that depictsthe spatial connectivity between the streets are illustrated inFig. 7a and b, respectively.

    To calculate the TCB values of the streets in the case-studynetwork, two additional data sets are required, including the ODpattern of the commutes and the cost-flow (link performance)function of each street.Fig. 7. Isfahans street central lines (aBased on sample data acquired using face-to-face interviews,the Department of Transportation in Isfahan Municipality has com-piled the Isfahan travel demand model. During the interviews, therespondents were asked about activities and related trips. Finally,the results are aggregated into 146 traffic analysis zones, and theOD matrix was derived from the gathered data in which each ele-ment of the matrix indicates the number of commutes betweenzones.

    The capacity and the performance of streets are additional datathat are required to calculate the TCB centrality of each street. Thelink performance function is typically represented by a mathemat-ical equation between traffic volume and travel time for any givenlink in the network. In the study network, the widely used link per-formance function developed by the Bureau of Public Roads (BPR)is applied to represent the performance of the links. The BPR linkperformance function is stated as follows (BPR, 1964):Tava ta1 ava=Cab 3where va is the flow attempting to use link a, Ta(va) is the travel timeof link a at traffic flow va, ta is the free-flow travel time on link a, Cais the capacity of link a and a, b are calibration parameters.

    The free-flow travel time and the capacity of the links in thestudy network are assessed based on the estimated saturation flowin any given link. Additionally, the calibration parameters (a, b) areestimated based on the link functional characteristics (highway,major arterial or minor arterial).

    To validate the calculated structural importance (TCB central-ity) of each link, actual traffic flow data of 20 links in the study net-work have been observed at peak hours, and the mean weekly flowvolume of the links is calculated. Using the observed traffic data,the TCB centrality of each of the 20 links is correlated to the relatedmean weekly traffic flow. The TCB-flow comparison reveals a highdegree of correlation (R2 = 0.76) between the TCB centrality of thelinks and the related mean weekly traffic flow.

    Furthermore, to compare the TCB, which is a specialized mea-sure, with the general centralities, the value of three widely usedcentralities (betweenness, closeness and the clustering coefficient)are calculated for each link in the study network. Next, the generalcentralities are correlated with the observed traffic flows in the 20links. As shown in Table 1, correlating the centralities with actualflows reveals that the constrained TCB centrality quantifies thestructural participation of links more realistically than the generalcentralities.) and the related dual graph (b).

  • Table 1A comparison between the TCB and some general centralities in the case studynetwork.

    Betweenness Closeness Clustering coefficient TCB

    R2 0.57 0.42 0.36 0.76

    A. Sheikh Mohammad Zadeh, M.A. Rajabi / Cities 31 (2013) 285297 293The next section investigates the functional significance of net-work spatial configurations by applying the proposed TCB central-ity measure.

    Different structures prove different efficiencies

    Worldwide, in each city, streets have been established in differ-ent spatial configurations. In other words, each city has a specificarrangement of streets. Certain cities are planned and possess astrict regular configuration, such as a square grid. Other cities orga-nize themselves during their development and do not display reg-ularity in their spatial configurations (Fig. 8). Investigating thesedifferent spatial configurations raises an important question aboutthe best configuration of networks: How is it possible to arrangestreets so that the final configuration provides the best efficiency?Fig. 8. Different spatial configurations of urban streets: square gridFinding a reasonable answer to this question requires definingand establishing a measure to evaluate network efficiency. To de-fine network efficiency, this paper appliesWardrops second princi-ple, which is essentially a design principle. In contrast to the firstprinciple, the Wardrop equilibrium, which seeks to model driverbehavior on the assumption that drivers attempt tominimize travelcost, the secondprinciple is planning- and engineering-oriented andtries to manage traffic to minimize travel costs (Ortuzar & Willum-sen, 2004).

    Wardrops second principle proposes the optimal distribution ofthe traffic flow throughout the network such that the overallnetwork cost is minimal. This condition, which is known assystem-optimal (Ortuzar & Willumsen, 2004) can be consideredto be the best performance of a network. Therefore, this paperadopts Wardrops second principle to define the efficiency of anetwork.

    According to Wardrops second principle, the system-optimalflow distribution of a given network can be formulated asfollows:


    CaVa Va 4(a), radial (b), triangular (c), and irregular (d) (Google maps).

  • 294 A. Sheikh Mohammad Zadeh, M.A. Rajabi / Cities 31 (2013) 285297subject to:Xr

    Tijr Tij and Tijr P 0

    where Tijr is the number of trips between i and j via route r, Tij isthe total number of trips between i and j, Va is the flow on link a,and Ca(Va) is the cost-flow relationship for link a.

    The target function of minimization problem (3), i.e.,P

    aCa(Va) - Va, which is known as overall cost (OC), is adopted in this paperto quantify the efficiency of a network. As mentioned above, thebest flow distribution in a given network occurs at the minimumpoint of the OC function. Therefore, this optimum is the goal offlow distribution in a network. Therefore, the OC value of each pos-sible flow distribution can be considered to be a measure withwhich to compare a possible flow distribution with the optimaldistribution. In this paper, the natural flow distribution of a net-work (i.e., the user equilibrium) is considered to be the networksperformance and is compared with the optimal distribution.1.5 2 2.5 3 3.5 4 4.5 50











    1.5 2 2.5 30













    Fig. 9. The efficiency of different spatial configurations for three different cost-flow frepresents a network, and the color (size) of a point indicates the OC value of the relatedAdditionally, the size of each point is proportional to the OC value of the related network(For interpretation of the references to color in this figure legend, the reader is referredBecause the user-equilibrium state of flows in a network is notnecessarily the optimal distribution, the OC in the user-equilib-rium state is always larger than (or equal to) the OC in thesystem-optimal condition.

    To evaluate the OC value of a given network, one must calculatethe link flows in the equilibrium condition by solving the minimi-zation problem expressed in Eq. (1). However, as discussed above,the proposed TCB centrality strictly correlates with the link flowsin the Wardrop equilibrium. Therefore, the equilibrium flow inthe OC function could be replaced by the TCB measure, whichyields a new efficiency evaluation function as follows:

    OC Xa

    CaTCBa TCBa 5

    Different cities not only possess different street configurationsbut most cities also have different configurations at differentlocations. To quantify these different configurations, one canclassify the different regularity levels of street configurations in a1.5 2 2.5 3 3.5 4 4.5 50











    3.5 4 4.5 5



    unctions: linear (a), quadratic (b) and exponential (c). Each point on the graphsnetwork. Dark red indicates the highest OC, and dark blue indicates the lowest OC.

    . The small-world structure has the highest and the regular grid the lowest the web version of this article.)

  • 0 100 200 300 400 500 6000










    0 5 10 15 20 25 30 35 40 45 500










    (a) (b)Fig. A1. The computation time of the implemented algorithm: (a) 15 origindestinations on different network sizes and (b) different numbers of origindestinations on anetwork with 168 links.

    A. Sheikh Mohammad Zadeh, M.A. Rajabi / Cities 31 (2013) 285297 295continuum from strict regular grid networks to completely ran-domly connected streets. In this section, this regularity continuumis adopted to compare the efficiency of different configurations.

    To achieve this goal, a 10 10 regular grid network that con-tains 100 nodes and 180 undirected links is chosen as a regular ex-treme point on the mentioned continuum. Then, this grid isrewired by deleting certain links and randomly forming new linksbetween two distinct nodes. This rewiring process not only wors-ens the network regularity; the rewiring also changes certainstructural properties of the network. During the rewiring, to ob-serve structural changes, two structural properties of rewiring net-works are tracked. These properties include the geodesic distanceaverage, that is, the average of all shortest paths between allstreets, and the clustering coefficient average in the dual graph, thatis, the average probability that adjacent nodes of a single node areconnected. Through the rewiring process, twenty new networkswith different configurations are generated. Next, the TCB-basedOC value is calculated for each network. The resulting OC valuesfor the twenty rewired networks with three different cost-flowfunctions are provided in Fig. 9. In these graphs, the measuredstructural properties, i.e., the geodesic distance average and the clus-tering coefficient average, are shown on the x and y axes, respec-tively. Each colored point on the graphs corresponds to a rewirednetwork. Dark red indicates the highest OC value, and dark blueindicates the lowest OC value. Moreover, the size of each pointon the graphs is proportional to the OC value of that network.

    As shown in Fig. 9, the OC result for rewired networks revealsthat the networks with smaller geodesic distance averages andgreater clustering coefficient averages have lower OC values. In con-trast, the initial configuration, which is a 10 10 regular grid net-work, has the highest OC value. According to graph theory, thestructure that has the lowest OC value is a specific configurationthat is known as small world. This configuration is an intermediatestatus between an order and a disorder. In other words, a small-world network is a large network in which the average distance be-tween all possible pairs of nodes is small and the neighbors of eachnode are most likely connected (Jiang & Claramunt, 2004b).

    The rewiring process showed that a small-world configuration, incomparisonwith the small-world configurations regular grid coun-terparts, is a structure that is more efficient in transporting vehiclessubject to urban transportation constraints. The high efficiency ofthese structures results from the structures distinctive properties,i.e., the small separation throughout the network, and the highdegree of clustering between local nodes (Jiang, 2008).

    Conclusion and discussion

    The structure and configuration of transportation networkshave been the focus of many studies. This paper has attemptedto address two important but neglected points in these studies.The first point is that in most of these studies general centralitiesdeveloped in other fields (for example, the study of social net-works), have been employed to quantify the structural importanceor participation of the links in transportation networks. However,these centralities might not be suitable for evaluating the structureof every network, including transportation networks, because ofthe implicit assumptions that these centralities make. Further-more, the transportation network, observed independently, pos-sesses certain characteristics that the applied centrality measuresshould consider. The second point is that previous studies haveextensively studied the correlation between the traffic distributionand the configuration of the network. However, none of these stud-ies have considered the quality of this influence in transportationnetworks. Therefore, this article sought to evaluate the quality ofthis influence on traffic distribution.

    In urban transportation systems, the centrality or importance(or participation) of a street is a function of both demand-sideand infrastructure properties. The demand side encompasses allof the parameters that are related to the drivers, the drivers originor destination, and the drivers way-finding behavior. Theseparameters are mainly determined by socio-economic factors,household attributes and the land-use pattern of the city. Theawareness of drivers is another factor on the demand side that fur-ther complicates the traffic flow distribution: each driver tends toindividually find the optimal path to a destination. Eventually, thisbehavior forces the overall traffic flow into a state of equilibrium.

    In contrast, on the supply side, there are specific, limited, phys-ical infrastructures that commuters pass through on the way to-ward their destinations. These facilities are spatial constraintsthat affect the way-finding behavior of the commuters. In thiscontext, the movement does not occur in an unconstrained space.Instead, it occurs in a spatially constrained environment in whichthe travelers choose a path from a limited number of options.Therefore, the spatial configuration of the options (streets) could

  • 296 A. Sheikh Mohammad Zadeh, M.A. Rajabi / Cities 31 (2013) 285297independently determine the manner in which a commuter movesin the network.

    As an immediate result, both the demand-side and supply-sideproperties of an urban transportation system are important to theflow composition. In many studies that investigate urban transpor-tation structurally, one of these sides has been missing. Therefore,this paper proposed a new centrality measure (i.e., the TCB) thatconsiders both the demand and supply properties. Contrary tothe general centralities (e.g., degree, betweenness, and closeness),the TCB is a specialized centrality that considers network and traf-fic-related constraints to quantify the importance of each street inan urban network structure.

    Applying the proposed OC measure to examine the quality ofthe configuration impact on different networks revealed that anurban network with a small-world configuration has high efficiencyin transporting vehicles, whereas the regular grid network has thelowest efficiency among the investigated networks. Therefore, theregular grid is not suitable as a base structure in urban planning.

    From the urban traffic flow perspective, in small-world net-works, globally, the commuters can access most locations in thenetwork quickly. Additionally, the commuters can efficiently avoidcongested streets via the loops of local streets that result from thehigh clustering degree in small-world networks. Typically, the com-muters attempt to avoid congested streets. Therefore, in congestedsituations, the commuters prefer to switch to a path that has lesstraffic. Therefore, strict locally connected streets in small-worldconfigurations enable commuters to efficiently bypass traffic bot-tlenecks. Furthermore, in small-world networks, the small separa-tion between streets, which results from shortcut links, preventsthe commuters from passing a large number of streets on theway to their destinations. These unique characteristics causesmall-world networks to be more efficient than regular grids, inwhich the lack of shortcut links and locally clustered streets resultsin a high overall cost in the network.

    The results of this study are important for designers and urbanplanners in two ways: in designing or developing urban transpor-tation infrastructure and in planning urban land-use patterns.According to the results of this study and considering the natureof urban passengers, the small-world structure for the urban trans-portation network is more cost effective for urban passengers thanother network structures. Therefore, in designing transportationnetworks, designers should design networks that reflect thesmall-world structure. In such a structure, shortcuts and locallyclustered streets can reduce the traffic to a considerable extent be-cause urban passengers attempt to find the optimal path in a net-work. Additionally, the infrastructure in a network should bedeveloped to promote the small-world structure. Therefore, if, forexample, new highways, tunnels or bridges are to be constructedin an urban network, they should be constructed with consider-ation for the small-world structure. To evaluate the extent towhich the newly developed network reflects the small-worldstructure, the proposed measure to evaluate the overall cost (OC)should be applied. Because a small-world structure displays smallgeodesic distances and a large clustering coefficient, a small-worldstructure has a small OC value. In addition, by employing the rec-ommended TCB and OC, the efficiency of certain operations, suchas widening the streets, can be further evaluated. Such operationschange the performance of the links in the network, and the effectof these changes can be evaluated using the TCB and OC.

    Additionally, the results of this study can be used in planningand designing the pattern of urban land use. The TCB centralityproposed in this article can also evaluate the effect of different pat-terns of land use on traffic congestion in different parts of a net-work. Different land-use patterns will change the OD pattern ofcommuting. This change could be evaluated by applying the sug-gested centrality.Appendix. The computational complexity of the proposedalgorithm

    To compute the proposed centrality, it is necessary to deter-mine all possible shortest paths between all traveler origindesti-nations in each iteration. These shortest paths are used to evaluatethe related EB of links. To find all possible shortest paths, the algo-rithm introduced by Yen (1971) to compute the k-shortest pathwas applied in the implementation of the proposed algorithm.The Yen algorithm yields all possible shortest paths between twoassumed nodes by considering k = 1. The main reason to use thisalgorithm is that the other shortest path algorithms, such as Dijk-stra (1959), are designed in such a way that they are able to detectonly one possible shortest path between two nodes and becausethe problem becomes more computationally complex if the algo-rithms must compute more than one shortest path. Based onHershberger, Maxel, and Suri (2007), the computational complex-ity of the Yen algorithm under the worst circumstances will be ofthe order of O(kn(m + n logn)). In this formula, n is the number ofnodes, m is the number of links, and k is the number of k-shortestpaths. Given that in the implementation of the proposed algorithmk is equal to one, the algorithms computational complexity in eachiteration for each OD will be of the order of O(n(m + n logn)). Todemonstrate the computational complexity of the suggested algo-rithm, the computation time for two scenarios has been calculated(Fig. A1). In the first scenario, considering 15 fixed origindestina-tions, the computation time for networks with different numbersof links has been evaluated (Fig. A1a). In the second scenario(Fig. A1b), for a network with 168 links, the computation timehas been evaluated and shown for different numbers of origindestinations.References

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    Analyzing the effect of the street network configuration on the efficiency of an urban transportation systemIntroductionVehicular traffic flow characteristicsTargeted constrained betweenness (TCB)Travel demand pattern as an input in the TCB algorithmNumerical resultsCase study: computing the TCB centrality in the Isfahan street networkDifferent structures prove different efficienciesConclusion and discussionAppendix. The computational complexity of the proposed algorithmReferences


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