Topological Analysis of Urban Street Networks

Transcription

1 NOT FOR CITATION, TO BE PUBLISHED IN ENVIRONMENT AND PLANNING B Topologcal Analyss of Urban Street Networks B. Jang () and C. Claramunt (2) () Dvson of Geomatcs, Dept. of Technology and Bult Envronment Unversty of Gävle, SE-8 76 Gävle, Sweden, Emal: bn.jang@hg.se (2) Naval Academy Research Insttute, BP 6, 2924, Brest Naval, France, Emal: claramunt@ecole-navale.fr Abstract Ths paper proposes a topologcal analyss of large urban street networks based on a computatonal and functonal graph representaton. Ths representaton gves a functonal vew n whch vertces represent named streets and edges represent street ntersectons. A range of graph measures ncludng street connectvty, average path length and clusterng coeffcent are computed for structural analyss. In order to characterze dfferent levels of clusterng degrees of nodes n a street network we generalze the clusterng coeffcent to a k-clusterng coeffcent that takes nto account k-neghbours. Based on valdatons appled to three ctes, we show that large urban street networks form small-world networks but do not exhbt scale-free property. Keywords: street network, small world, scale-free, power law, urban morphology Introducton Network analyss has long been basc functon of Geographc Informaton Systems (GIS) for a varety of applcatons such as n hydrology, facltes management, transportaton engneerng, busness and servce plannng. In network GIS, computatonal modellng of an urban network (e.g street network, underground) s based on a graph vew n whch the ntersectons of lnear features are regarded as nodes, whle connectons between par of nodes are represented as edges. Common network operatons nclude computatonal processes to fnd the shortest, least cost or most effcent path (pathfndng), to analyse network connectvty (tracng), and to assgn portons of a network to a locaton based on some gven crtera (allocaton) (Waters 999, Mller and Shaw 2). A network GIS can be also analysed wth respect to ts structural propertes, e.g., what are the mportant streets of a cty n terms of connectvty? What s the average level of ntegraton or segregaton of a street network? Overall how are those streets nterlnked? All these questons deal wth topologcal, logcal and structural propertes that are the scope of urban morphology. For nstance, space syntax (Hller and Hanson 984) adopts a graph-theoretc method to model how urban spaces are ntegrated or segregated usng so-called axal maps. Dervaton of axal maps to a great extent reles on the structural propertes of a gven street network and allocaton of buldngs wthn such an urban envronment.

2 NOT FOR CITATION, TO BE PUBLISHED IN ENVIRONMENT AND PLANNING B 2 In ths paper, we take a named-streets orented vew for topologcal analyss. Named streets represent a functonal modellng element of large urban street networks whose structure should be retaned by a structural analyss. Evaluatng to whch degree streets are nterconnected vs. segregated n a gven cty should mply, at a modellng level, desgnng a graph n whch the nodes model those named streets and edges connectons between those named streets (note that n such a vew a node models not a street segment but an entre named street). Wthout loss of generalty, a named street that s separated nto two or more parts (e.g. south queen street and north queen street) s semantcally aggregated. One of the objectves of ths paper s to explore such an alternatve graph model. Ths named-street centred network model s denoted as a topologcal network model, as t reflects at a hgher level of abstracton topologcal connectons n a gven street network (and not purely geometrcal connectons). The topologcal network model supports a street-orented computatonal analyss of the propertes of an urban street network. We develop a computaton based on a range of graphbased measures completed by a k-clusterng coeffcent that extends the current defnton of the clusterng coeffcent to the ntegraton of k-neghbours. The am of our model and study s to examne whether or not a named-street orented topology reveals small-world and scale free propertes for urban networks. A small-world network characterses a large network wth a short average dstance between any two randomly chosen nodes, and hghly clustered nodes when compared to a random network of equvalent sze (Watts and Strogatz 998). A scalefree property over a functon y = f (x) reveals the fact that whatever the range of x one looks at, the proporton of small to large y values s the same,.e., the slope of the curve on any secton of the log-log plot s the same. A scale-free network denotes a network where most nodes have a small number of lnks, and only a few have a large number of lnks (Barabas and Albert 999). Small-world and scale-free propertes have been nvestgated and llustrated n a varety of dscplnes and domans ncludng bology (Jeong et al. 2), ecology (Montoya and Sole 22) and lngustcs (Cancho and Sole 2) and computng n the Internet (Faloutsos et al. 999, Shode and Batty 2). Partcularly, t has been used n socal scence to analyse the structure of so-called socal networks (Scott 999). For a comprehensve overvew on smallworld and scale-free propertes readers can refer to for nstance Strogatz (2) and Barabas (22). The remander of ths paper s organzed as follows. Secton 2 brefly dscusses how smallworld and scale-free propertes apply to regular, random and real-work networks. Secton 3 consders an example of an urban dstrct for topologcal measures for ndvdual streets. Secton 4 ntroduces the concept of the k-clusterng coeffcent to characterse the clusterng degrees of nodes n large networks. Secton 5 reports on our expermental results and secton 6 draws some conclusons. 2. Regular, random, and real-world netwo rks: small-world and scale-free propertes A network can be represented as a graph, whch conssts of a fnte set of vertces (or nodes) V and a fnte set of edges (or lnks) E (note that we use vertces and nodes, and edges and lnks nterchangeably). A graph s often denoted as a par G(V,E) where V s the set of vertces, V = { v, v2,..., vn}, and E the set of edges, s a subset of the Cartesan product V V. For computatonal purposes we represent a connected, undrected and unweghted (.e. all lnks wth a unt dstance) graph by an adjacency matrx R(G):

3 NOT FOR CITATION, TO BE PUBLISHED IN ENVIRONMENT AND PLANNING B 3 [ r j ] n n R( G) = where r j = f v v E otherwse j [] It should be noted that for an undrected graph G, ths adjacency matrx R(G) s symmetrc,.e. r r = r, and that all dagonal elements of R(G) are equal to zero. Then the lower or j j j upper trangular matrx of R(G) s suffcent for a complete descrpton of the graph G. In the context of ths research, we deal wth large networks wth hundreds or thousands nodes. A regular network s defned as a graph where each node has exactly the same number of lnks to ts neghbourng nodes. Examples of regular networks are a regular grd where all nodes have exactly four lnks, and a hexagonal lattce where each node s connected to three other nodes (Gross and Yellen 999). Such regularty s clearly absent n random networks, snce lnks are randomly placed among the nodes. Despte the fact that lnks don t show a regular pattern n large networks, t has been observed that the dstrbuton of connectvty n random graphs follows a bell curve where most nodes have the same number of lnks, and no hghly connected nodes exst (Barabas 22). Networks observed n realty are so-called real-world networks. They nclude for nstance Internet, scentfc collaboraton networks, cells and scentfc ctatons. Real-world networks are lkely to be small-world networks that demonstrate two basc propertes (Watts and Strogatz 998). The frst property of small-world networks s that the separaton between any two randomly chosen nodes s very short. The separaton s charactersed by the noton of path, whch s defned as the shortest dstance between nodes. The average path length for a socal network s lkely to reflect a short degree of separaton. Accordng to Mlgram (967), an emprcal study was carred out to testfy the concept of sx degrees of separaton n the Unted States. He frst randomly chose two target persons n Boston. Then he sent 6 letters to randomly chosen resdents of two areas respectvely n Kansas and Nebraska, whch are far away from the target persons. Surprsngly the frst letter arrved, passng through only two ntermedate lnks! Eventually he found that the medan number of ntermedate persons to get the letter to the target was 5.5, very small ndeed. Although the property of short separaton n socal network s generally accepted we can note that recent studes have mnored Mlgram s fndngs (Klenfeld 23, Watts 23). In partcular further experments queston the value of sx degree of separaton n a worldwde settng ( In the doman of the Internet, Albert et al. (999) found that the web forms a knd of small-world networks wth the separaton from page to page around 9 clcks. Mathematcally, such a separaton can be descrbed by average path length. Gven two vertces v v j V, let d mn (, j) be the shortest dstance between these two vertces. The average path length of a gven vertex v s gven by n L( v ) = dmn (, j) [2] n j= where n s the total number of vertces of the graph G. It should be noted that the average path length s a topologcal measure, whch s of nterest to structural analyss of large networks. The second property of small-world networks s ther hgh degree of clusterng. Ths can also be seen from our daly experence where for example our frends are lkely to be frends of each other as well, or n other words socal networks tend to be clustered. As to how hgh the

4 NOT FOR CITATION, TO BE PUBLISHED IN ENVIRONMENT AND PLANNING B 4 possblty s, t depends on actual frendshp lnks among our frends. Let s assume that one has four frends. If they are all frends each other, then there wll be sx frendshp lnks. However actual lnks counted are less than 6 lnks - let s say, 4. In ths case, the lkelhood of any par beng frends of the four people s 4/6=.667. Watts and Strogatz (998) ntroduced a clusterng coeffcent to characterze the degree of clusterng. Mathematcally, The clusterng coeffcent s a measure of the extent to whch the neghbours of a vertex v are also lnked each other. The clusterng coeffcent of the vertex v s defned by 2 l C ( v ) = [3] m ( m ) where l s the number of edges among the mmedate neghbours of the vertex v, m the number of mmedate neghbours of the vertex v. We can note that C( v ) s drawn from the unt nterval [,]. The closer to one the clusterng coeffcent s, the more clustered the vertex v s. Fgure : Illustraton of a hghly clustered regular graph (the average clusterng coeffcent for all nodes s equal to.75) A hgh degree of clusterng s not a property of a random graph, but that of regular graphs, as long as the neghbourng nodes of a node are well lnked as shown n fgure. However, regular graphs are not lkely to reveal some small-world propertes, snce the shortest path length that connects two gven nodes s lkely to have a large degree of separaton for large graphs. Besdes small-world property, the scale-free property s another mportant feature of most real-world networks. Ths s related to the noton of connectvty denoted by m,.e. the number of lnks (or mmedate neghbours) of a node. The scale-free property reflects the fact that n a gven network most of the nodes have a small connectvty and only a few nodes have a very hgh connectvty. Overall, dstrbuton of connectvty n most real-world networks follows a power-law dstrbuton (Barabas 22). Power laws ntally ntroduced by Zpf (949) n the lngustcs have a typcal form of f(x) = cx α where c s a constant and α some parameter of the dstrbuton. Ths dfferentates a real-world network from a random network, snce the dstrbuton of connectvty of a random network essentally conforms to a bell-shaped exponental dstrbuton rather than power-law (Barabas and Albert 999). Many real-world networks such as the Internet (Faloutsos et al. 999), world wde web (Albert et al. 999), scentfc collaboraton networks (Newman 2), cells (Jeong et al. 2) and scentfc ctatons (Blke and Peterson 2) reveal a power law dstrbuton.

5 NOT FOR CITATION, TO BE PUBLISHED IN ENVIRONMENT AND PLANNING B 5 In a study of the formaton of scale-free networks, Barabas and Albert (999) fnd that most real-world networks are governed by two laws: growth and preferental attachment. Durng the growth of a real-world network, early nodes are lkely to have more lnks due to evoluton. If one smulates such a network growth, t does not lead to hghly connected nodes, but nstead, only a random network wth bell-shaped connectvty dstrbuton s created. Through careful observaton of web growth, Barabas and hs colleagues fnd that gven all possble news pages, we tend to lnk to those major news outlets such as cnn.com or bbc.co.uk. Ths mples that page lnks have potental preference n lnkng to other pages. In other words preferences are gven to the nodes, whch are already hghly connected. The two laws appear to be applcable to the evoluton of urban street networks as well. That s, early bult streets tend to be connected by more other streets on the one hand, and preferences are gven to the streets that are already hghly connected on the other. In the followng secton 5., we examne whether or not ths scale-free property exsts n urban street networks. 3. Topology of urban street networks a frst look Let us consder the example of a dstrct of the Swedsh cty Gävle as shown n fgure 2. To the left of the fgure s the street network of the dstrct Sätra; whle to the rght s the correspondng connectvty graph. The dervaton of the connectvty graph s based on the followng transformaton rule: named streets and ther ntersectons gve the nodes and lnks respectvely of the connectvty graph. One can remark that ths dstrct s a relatvely closed one: a bell-shaped street Sätrahöjden consttutes a form of boundary, and t s nternally connected by two streets (Norrbägen, Nyöstervägen) that form an nternal communcaton lnk. These three man streets form the man structure of ths dstrct to whch other short streets are connected. Fgure 2: Sätra dstrct network and ts connectvty graph (Note: every node s labelled by the correspondng street name) Table : Three measures for the streets of Sätra dstrct (part) (Note: m gves connectvty, L path length, C clusterng coeffcent)

6 NOT FOR CITATION, TO BE PUBLISHED IN ENVIRONMENT AND PLANNING B 6 Street m L C Street m L C Sätrahöjden Klasbärs vägen Nyöstervägen Moränvägen Norrbågen Pnnmovägen Sadelvägen Scksackvägen Pngeltorpsvägen Skårängsvägen Skogvaktarvägen Smultronvägen Ulvsätersvägen Svalvägen Vnbärsvägen Tusslagovägen Fältspatvägen Bleckarvägen Kaveldunsvägen Blåbärsvägen 3.6. For llustraton purpose, table lsts three calculated measures for the frst 2 streets (n reverse order of connectvty) among 5 named streets of Sätra dstrct n fgure 2. Although, ths example of dstrct cannot be consdered as a large urban street network, some facts can be derved from table. One can remark that most of the streets have a small connectvty between and 4 and only 3 of them have very a hgh connectvty (column m). Ths s also reflected n fgure 2 where node szes show the degree of connectvty. We can notce that well connected streets have shorter path length (column L), whle less connected streets have longer path length. As far as clusterng coeffcent s concerned, one may notce that most of the streets n column C have a coeffcent of. Ths reflects that ether () street networks are not hghly clustered (ths wll be studed n secton 5) or (2) the coeffcent measure ntroduced by (Watts and Strogatz 998) does not dfferentate very well varous degrees of clusterng among the nodes of a network Ths leads us to propose a generalzaton of the clusterng coeffcent n the next secton. 4. K-clusterng coeffcent The clusterng coeffcent, hereafter denoted as -clusterng coeffcent, consders only mmedate neghbourng node. However a node wth a low -clusterng coeffcent can be relatvely hghly clustered among ts k-neghbours. In order to better characterse the stuaton we ntroduce a measure, k-clusterng coeffcent, that takes nto account the degree to whch the k-neghbours of a gven node are nterconnected each other or not. Let of edges among k-neghbours of vertex neghbourhood of vertex defned as follows. v, ( k ) v. K-clusterng coeffcent, denoted C ( v ) ( k ) ( k ) 2 l C ( v ) = [4] ( k ) ( k ) m ( m ) l (k ) be the number (k ) m be the number of nodes wthn k-, of a gven node v s One can remark that wth the above defnton, the -clusterng coeffcent s a specalzaton ( k ) of the k-clusterng coeffcent measure for mmedate neghbours,.e. C ( v ) = C ( v ) when k =. As for the -clusterng coeffcent, the k-clusterng coeffcent s also bounded by the unt nterval [, ]. From ts defnton, the 2-clusterng coeffcent descrbes to whch degree those streets wthn 2-neghbourhood are nterconnected each other. If all those streets are nterconnected each other, then the 2-clusterng coeffcent s equal to ; f all those streets are not nterconnected

7 NOT FOR CITATION, TO BE PUBLISHED IN ENVIRONMENT AND PLANNING B 7 each other at all, then the 2-clusterng coeffcent s equal to (as for the -clusterng coeffcent). Table 2 shows that the 2-clusterng coeffcent makes a better dfference among the ndvdual nodes. For nstance, Scksackvägen s -clusterng coeffcent s equal to as ts mmedate neghbours have no lnk at all, whle ts 2-clusterng coeffcent s equal to.67 as there are two lnks among ts 3 2-neghbours. Table 2: - and 2-clusterng coeffcent for the streets of Sätra dstrct (part) Street C () C (2) Street C () C (2) Sätrahöjden..3 Klasbärsvägen..24 Nyöstervägen.7.5 Moränvägen..9 Norrbågen..5 Pnnmovägen..9 Sadelvägen..7 Scksackvägen..67 Pngeltorpsvägen..8 Skårängsvägen..8 Skogvaktarvägen..8 Smultronvägen..32 Ulvsätersvägen.33.4 Svalvägen..33 Vnbärsvägen..8 Tusslagovägen..8 Fältspatvägen..32 Bleckarvägen..67 Kaveldunsvägen..24 Blåbärsvägen Topology of urban street networks Experments In order to nvestgate the topology of urban street networks, further experments are appled to three ctes: Gävle (Sweden), Munch (Germany), and San Francsco (US). Note that only part of San Francsco network s used for the experments due to some constrants on datasets avalablty. The network data sets are composed of street central lnes topologcally nterconnected,.e. no solated streets exst. A computatonal scrpt determnes how each gven street ntersects to every other, and derves a matrx R to represent the connectvty graph (see for example Gävle case n fgure 3b). Informally the algorthm, wth reference to equaton [], can be read as follows: for each street (), check f t ntersects every other street (j), f yes, r =, otherwse r =. j j Usng the above algorthm, a street network can be mapped towards a matrx. The computatonal complexty of the algorthm s O(n 2 ), so a relatve costly process. The key to calculate the path length s to calculate the shortest dstance between any two vertces usng the Djkstra (959) algorthm. It should be noted that ths shortest dstance s not based on the geometry but rather on the topology. We frst calculate the shortest dstance from every vertex to every other and then summed all of them together to get the so-called total path length. The algorthm begns at the frst vertex and fnds ts neghbours, and then ther neghbours, and so on untl the algorthm has spanned throughout the graph and reached all vertces wthn a connectvty graph. Then ths process contnues wth the second and thrd vertex untl all vertces have been exhausted. In order to mplement ths algorthm, we adapted the Breadth- Frst Search (BFS) technque, whch s consdered to be qute an effcent method for fndng the dstance from one vertex to all the other vertces n a graph, and an extremely effectve method for sparse graphs (Buckley and Harary 99). The average path length s the total path length dvded by the total number of vertces n, and t s calculable n O(n 2 log n) tme. The computaton of connectvty and -clusterng coeffcent s performed accordng to the formula ntroduced n secton 2. They are calculable n O(nm) and O(nm 2 ) tme respectvely.

8 NOT FOR CITATION, TO BE PUBLISHED IN ENVIRONMENT AND PLANNING B 8 For k-clusterng coeffcent, t actually apples to the adjacency matrx of subgraph correspondng to each vertex neghborhood. So t s calculable n O(n 3 ) tme. Fgure 3: Gävle street network and ts connectvty graph 5. Dstrbuton of street connectvty The frst part of ths analyss s orented to the study of the scale-free property. Fgure 4a shows a lnear scale plot (Gävle case), where x and y axes represent street connectvty and cumulatve probablty, respectvely. One can observe that most streets have a small connectvty (e.g. about 75% streets of total 565 wth connectvty rangng from to 5), whle a few streets have a bg connectvty (e.g 25% streets wth connectvty rangng from 6 to 29, n partcular only one street wth connectvty of 29). Let y be the cumulatve probablty of occurrences per cumulated connectvty rank x. If the cumulatve probablty y and x a conforms to a power-law dstrbuton for connectvty, we have y = cx. Ths means that log( y) = log( c) a log( x), thus a power law wth exponent a s seen as a straght lne wth slope a on a log-log plot. The log-log scale plot of street connectvty versus cumulatve probablty for the case of Gävle s shown n fgure 4b. Ths llustrates the fact that the log-log plot does not reveal a strct lnear relatonshp, but nstead a nonlnear relatonshp.e. 2 y =.67x +.62x.2. In other words, the connectvty dstrbuton does not reflect a strct power law. The same concluson does apply for Munch network (fgure 5) and San Francsco (fgure 6). cumulatve probablty,2,8,6,4, connectvty Log(cumulatve probablty) y = -.67x x -.2 R 2 =.99 Log(connectvty)

9 NOT FOR CITATION, TO BE PUBLISHED IN ENVIRONMENT AND PLANNING B 9 Fgure 4: Gävle Lnear scale plot connectvty versus cumulatve probablty and log-log scale plot connectvty versus cumulatve probablty Cumulatve probablty Connectvty Log(cumulatve probablty) Log(connectvty) y = -2.35x 2 +.5x -.9 R 2 =.97 Fgure 5: Munch Lnear scale plot connectvty versus cumulatve probablty and loglog scale plot connectvty versus cumulatve probablty Cumulatve probablt,2,8,6,4, Connectvty y = -5.27x x x x R 2 =.99 Fgure 6: San Francsco Lnear scale plot connectvty versus cumulatve probablty and log-log scale plot connectvty versus cumulatve probablty Log(cumulatve probablty) Log(connectvty) 5.2 Small-world propertes The second part of our analyss concerns the evaluaton of small-world behavours n urban street networks. We calculated the average path length and clusterng coeffcent of three cty () networks as shown n the columns of L actual and C áctual of table 3. For comparson purposes, we calculated the two measures for a random graph wth the same number of nodes and () connectvty per vertex (m ) n the column of L and C. They are approxmately gven by ln( n) L random = and ln( m) () C random = m n random random respectvely, where n s total number of vertces of the random graph, m s average number of edges per vertex (c.f. Watts and Strogatz 998). The calculated results show that all three networks have small degrees of separaton,.e. the average separaton between any two randomly chosen streets s less than 7 n all cases. Ths means that on average any two streets are just several streets away. In addton, clusterng coeffcents for the three ctes meet condton of C >> Crandom, wth 27-, 36-, and 2-fold respectvely for Gävle, Munch and San Francsco networks. These two results reveal the fact that these street networks are small-world networks.

10 NOT FOR CITATION, TO BE PUBLISHED IN ENVIRONMENT AND PLANNING B Table 3: Calculaton results streets m actual () L L random C actual () C random Gävle Munch San Francsco Dstrbuton of 2-clusterng coeffcent The thrd part of our analyss concerns the dstrbuton of the 2-clusterng coeffcent among the streets of the three ctes of our case study. Fgures 7, 8 and 9 show the dstrbuton of - and 2-clusterng coeffcents. These fgures show that the 2-clusterng coeffcent s more smoothly dstrbuted than the -clusterng coeffcent n those three networks. It should be noted that n these fgures x axes represents the sorted sequence of streets per reverse order of 2-clusterng coeffcent values, and as the orderng of values are dfferent, no correlaton exst between the two measures. Furthermore the smooth dstrbuton reflects the fact that 2- clusterng coeffcent can better dfferentate clusterng degrees among streets.,2,8,6,4, street ID 2-clusterng coeffcent,2,8,6,4, Street ID Fgure 7: Gävle Dstrbuton of -clusterng coeffcent and 2-clusterng coeffcent of streets,2,2,,8,6,4,2 2-clusterng coeffcent,8,6,4,2, Street ID Street ID Fgure 8: Munch Dstrbuton of -clusterng coeffcent and 2-clusterng coeffcent of streets

11 NOT FOR CITATION, TO BE PUBLISHED IN ENVIRONMENT AND PLANNING B,2,2,,8,6,4,2 2-clusterng coeffcent,8,6,4,2, Street ID Street ID Fgure 9: San Francsco Dstrbuton of -clusterng coeffcent and 2-clusterng coeffcent of streets Log-log scale plots for 2-clusterng coeffcent versus cumulatve probablty show that there do exst a strct power law dstrbuton for San Francsco cty but not for the two other ctes (fgures,, and 2). Ths reflects the property that for a majorty of streets n San Francsco, ther neghbourng streets wthn 2-neghbourhood are not well nterconnected, and only for a few streets, ther neghbourng streets wthn 2-neghbourhood are extremely well nterconnected. Ths property may dfferentate grd-lke ctes from other rregular ctes Log(cumulatve probablty) y = -.3x x R 2 = Log(2-clusterng coeffcent) Fgure : Gävle Log-log scale plot 2-clusterng coeffcent versus cumulatve probablty

12 NOT FOR CITATION, TO BE PUBLISHED IN ENVIRONMENT AND PLANNING B 2.5 Log(cumulatve probablty) y = -.77x 2-4.4x R 2 =.98-3 Log(2-clusterng coeffcent) Fgure : Munch Log-log scale plot 2-clusterng coeffcent versus cumulatve probablty Log(cumulatve probablty) y = -.44x R 2 = Log(2-clusterng coeffcent) -2.5 Fgure 2: San Francsco Log-log scale plot 2-clusterng coeffcent versus cumulatve probablty 6. Concluson The research presented n ths paper combnes expermental and computatonal fndngs. At the expermental level, we show the nterest of a topologcal-based analyss of an urban street network. Two conclusons can be drawn at ths pont. Frstly, street connectvty does not conform to a strct power-law dstrbuton, and no scale-free property exhbts from the pont of vew of street connectvty. Secondly, the topology of street networks reveals a small-world property wth a short separaton and hghly clusterng coeffcent. Ths presents a nce analogy wth small-world networks n the sense that the number of steps requred to connect from any street to any other street of a cty s generally very short.

13 NOT FOR CITATION, TO BE PUBLISHED IN ENVIRONMENT AND PLANNING B 3 From a computatonal perspectve, we ntroduce a new k-clusterng coeffcent that generalses the clusterng coeffcent by ncludng k-neghbours rather than mmedate neghbours only. In partcular our experment shows that the 2-clusterng coeffcent can better dfferentate the clusterng degrees of named-streets n a network, and that ths coeffcent shows power law dstrbuton n grd-lke networks. The prelmnary computatonal experments show that hgher values of k for the k-clusterng coeffcent (k>2) seems to confrm the propertes observed for k=2. There s also a need to generalse the computaton of 2-clusterng coeffcent to a whole network. Those ssues wll be addressed n further work. The topologcal propertes llustrated may provde some nce evdence to urban studes. For example, ths computatonal approach provdes a dfferent perspectve to urban morphology studes, by offerng another level of abstracton from those often used n network GIS, snce our computatonal model uses a street-centred modellng vew. It should be emphaszed that the defnton of named street s rather cultural dependent. Our future work should focus on the mpact of such cultural mpact on overall topologcal propertes. For example, whether or not US and European ctes demonstrate dfferent topologcal propertes. Acknowledgements The authors thank the referees for ther constructve comments and suggestons that sgnfcantly mproved the qualty of ths paper. We also would lke to thank Vt Vasek for an avenue scrpt, Mng Zhao for hs dscussons on power law and Andrej Mrvar for hs tmely advce and support n usng the Pajek software. The Munch dataset was provded by NavTech from the year 2, San Francsco data set from ESRI sample data, and Gävle dataset by Gävle cty. References Albert R., Jeong H. and Barabas A. L. (999), Dameter of the World-Wde-Web, Nature, 4, pp Barabas A. L. and Albert R. (999), Emergence of Scalng n Random Networks, Scence, 286, pp Barabas A. L. (22), Lnked: The New Scence of Networks, Perseus Publshng: Cambrdge, Massachusetts. Blke S. and Peterson C. (2), Topologcal Propertes of Ctaton and Metabolc Networks, Physcal Revew, E64, pp Buckley F. and Harary F. (99), Dstance n Graphs, Addson-Wesley Publshng Company. Cancho R. F. and Sole R. V. (2), The Small-World of Human Language, The Small World of Human Language, Proc. Roy. Soc. London B 268 (2), pp Djkstra E. W. (959), A Note on Two Problems n Connecton wth Graphs, Numersche Mathematk,, pp

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