5 th AGILE Conference on Geographc Informaton Scence, Palma (Mallorca, Span) Aprl 25 th -27 th 2002 A Structural Approach to Model Generalsaton of an Urban Street Network B. Jang () and C. Claramunt (2) () Dvson of Geomatcs, Insttutonen för Teknk Unversty of Gävle, SE-80 76 Gävle, Sweden Emal: bn.jang@hg.se (2) Naval Academy Research Insttute BP 600, 29240, Brest Naval, France Emal: claramunt@ecole-navale.fr Abstract. Ths paper proposes a novel generalsaton model for selectng characterstc streets n an urban street network. Ths model retans the central structure of a street network, t reles on a structural representaton of a street network usng graph prncples where vertces represent named streets and lnks represent street ntersectons. Based on ths representaton, so-called connectvty graph, local and global measures are ntroduced to qualfy the status of each ndvdual vertex wthn the graph. Two streets selecton algorthms based on these structural measures are ntroduced and mplemented. The proposed approach s valdated wth a case study appled to a mddle-szed Swedsh cty. Keywords : model generalsaton, structural analyss, space syntax, graph modellng Introducton Cartographc generalsaton s a constrant-based process used by cartographers to reduce the complexty n a map n a scale reducton process. It nvolves ntensve human knowledge obtaned through professonal cartographc expertse and practse. Automatc generalsaton has long been a research effort by both scentfc researchers and cartographc practtoners (Buttenfeld and McMaster 99, Muller et al. 995, AGENT 998). In partcular the dea of one sngle master database used to automatcally derve maps at dfferent scales has been a dlemma faced by many natonal mappng agences. Amongst many applcaton domans, cartographc generalsaton s for example used to reduce the complexty of an urban street network n a scale reducton process whle retanng ts general structure. Although generalsng an urban street network s often a cartographcal task, ths can be also consdered as an operaton where the objectve s rather to understand the structure and organsaton of the cty. Ths s an mportant am of many urban studes that focus on the understandng of urban structures and confguratons. In partcular, space syntax (Hller and Hanson 984) has developed as an mportant quanttatve way to analyse and understand the complexty of urban street networks usng a graph-theoretc method. These prncples support the belef that spatal layout or structure has great mpact on human socal actvtes. The applcaton of space syntax covers many urban studes such as modellng pedestran movement, vehcle flows, crme mappng, and human wayfndng process n complex bult envronments (Hller 996). Many emprcal studes have demonstrated the nterest of the space syntax for modellng and understandng of urban patterns and structures (Hller 997, Holanda 999, Jang et al. 999, Pepons et al. 200). Ths paper proposes a model generalsaton of an urban street network whose objectve s to retan the functonal structure of the cty. Frst our approach, whch s based on a computatonal applcaton of graph modellng prncples, uses vertces to represent named streets and edges to represent street ntersectons, so a form of derved graph nstead of a street network modelled as a graph. Integratng named streets (e.g. Kennedy avenue, 45th avenue) as a basc modellng unt gves a form of functonal representaton of the cty that complements the structural vew of the urban street network gven by the graph-based approach (let us remark that ths approach apples to ctes where streets are labelled usng ether names or dentfers). Ths functonal component comes from the observed fact that named streets often denote a logcal flow unt, or commercal envronment that s often perceved as a whole by people actng n the cty. Secondly, and from ths structural representaton, two flterng algorthms are ntroduced and mplemented. Our model consders both local and global measures to represent the structural property of each vertex n the graph, namely connectvty and average path length. Eventually, selecton of characterstcs streets s acheved through two flterng algorthms based on these structural measures. These algorthms are flexble n the sense that they support dfferent levels of generalsaton. The proposed approach s valdated wth a case study appled to a mddle-szed Swedsh cty. The remander of ths paper s organsed as follows. Secton 2 brefly ntroduces basc concepts of graph theory. Secton 3 ntroduces a structural representaton of a street network and the related structural measures. Secton 4
5 th AGILE Conference on Geographc Informaton Scence, Palma (Mallorca, Span) Aprl 25 th -27 th 2002 develops the prncples of the selecton algorthms and llustrates ther applcaton to a case study. Secton 5 dscusses some related work. Fnally secton 6 draws some conclusons. Graph theory prncples In order to develop a structural representaton of a street network, let s ntroduce some basc graph concepts. For a more complete ntroducton to graph theory, readers can refer for example to (Gross and Yellen 999). A graph G 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 G(V,E) where V s the set of vertces, V = v, v,..., v }, and E s the set of edges, E = v v }. For example, fgure shows a graph { 2 n { j { a, b, c, d, e, f, h, j, k G( V, E ) wth the set of vertces V = } and the set of edges E = { ab, ac, ad, cf, ch, de, df, dh, ej, ek, fh}. Let us remark that ths smplfed graph example s unweghted and undrected, and t s also connected as there s no solated vertex. Fgure : A smplfed example of graph We say that a graph H s a subgraph of a graph G f the vertces of H gve a subset of the vertces of G. Conversely, f H s a subgraph of G, we say that G s a supergraph of H. For a vertex subset U of a gven graph G, a subgraph whose vertces belong to U s sad to be nduced on the vertex subset U. Any two adjacent vertces v, v j of G (.e., v, v j E) are sad to be neghbours. The neghbourhood of a vertex v of a graph G, denoted N G ( v ), s the subgraph nduced by the set of vertces consstng of N ( v ) = { v v v E, j}. G j j v and all ts neghbours,.e., For computatonal purposes we represent a connected, undrected and unweghted (.e. all lnks wth a unt dstance) graph by an adjacency matrx R(G): [ r j ] n n R( G) = where rj = 0 f v v E otherwse It should be noted that for an undrected graph G, ths adjacency matrx R(G) s symmetrc,.e. r r = r j j j. Also all dagonal elements of R(G) are equal to zero so are not needed. Thus the lower or upper trangular matrx of R(G) s suffcent for a complete descrpton of the graph G. An approach to the structural representaton of street networks Gven an urban system, the underlyng street network can be consdered as a structurng element for many other cartographc objects (e.g. bult envronment, electrcty and gas networks) and soco-economcal actvtes n the cty. So reducng the complexty of an urban street network has many applcaton nterests. A street network has ts own ntrnsc logcal and spatal structure that must be represented and retaned n applyng a scale reducton process. We represent a street network usng some basc graph theoretc prncples; named streets (note that a named street s not a street segment but the entre named street consdered as a basc modellng unt) are represented as nodes and street ntersectons as lnks of a graph. One can remark that any graph derved usng such an approach s connected,.e. one can reach any vertex of the graph from any vertex. 2
5 th AGILE Conference on Geographc Informaton Scence, Palma (Mallorca, Span) Aprl 25 th -27 th 2002 To llustrate ths approach, let s consder the example shown n fgure 2. To the left of the fgure s the London underground map; whle to the rght s the correspondng connectvty graph. The dervaton of the connectvty graph s based on the followng rules: the underground lnes gve the nodes of the connectvty graph, two nodes of the derved graph (.e. two underground lnes) are connected f there s at least one drect lnk between these two nodes (.e. a connecton between these two underground lnes). It should be noted that multple ntersectons between pars of connected underground lnes are not represented n the connectvty graph. a b Fgure 2: London underground map (a) and ts connectvty graph (b) We ntroduce two measures for the descrpton of node status wthn a connectvty graph. Connectvty of a vertex v, denoted σ v ) (, s the number of vertces drectly lnked to ths vertex, so a local measure. For a gven graph G, the connectvty satsfes the followng condton: of edges, and n s the total number of vertces of the graph G. The average path length of a gven vertex v, denoted L v ) ( n = σ ( v ) = 2m, where m s the total number, consders not only those drectly connected vertces, but also those wthn a few steps, so a form of global measure when the number of steps consdered s hgh. Gven two vertces average path length of a gven vertex v v j V, let d (, j) mn be the shortest dstance between these two vertces. The v s gven by (n beng the total number of vertces of the graph G): L v ) = n n (, where n s the total number of vertces of the graph G. j= dmn (, j) Table : Two measures for the nodes of graph G Node ID Connectvty Average path length A 3,6667 B 2,4444 C 3 2,0000 D 4,3333 E 3,6667 F 3,7778 H 3,7778 J 2,4444 K 2,4444 The above two measures (connectvty and average path length) present respectvely some local and global propertes of each consdered vertex wthn ts connectvty graph. For llustraton purpose, table lsts the two calculated measures for the graph G shown n fgure. We can remark that less connected nodes are less mportant from a structural pont of than those well connected at the local level. From a global perspectve, the average path length measures how each node connects to every other n the connectvty graph. Ths gves a sense to what extent any vertex s ntegrated or segregated to every other wthn a connectvty graph. The lower 3
5 th AGILE Conference on Geographc Informaton Scence, Palma (Mallorca, Span) Aprl 25 th -27 th 2002 the value of that measure s, the more ntegrated the node s. Ths property can be llustrated n fgure 3, where all nodes are arranged n terms of how far (shortest dstance) every other node s from the two nodes: a and d respectvely. We can observe that node d s better ntegrated to every other than node a s. Conversely we can remark that node a s relatvely far from every other, whle node d s close to every other. The connectvty structure of the graph s mportant n dervng a seres of subgraphs whch retan the man structure of ntal graph. For nstance, the node d should have a hgher probablty than a to be kept durng the processng of the reducton scale algorthm, as t s better ntegrated to every other node at the global level, and also better connected to other nodes at the local level. Fgure 3: Respectve ntegraton/connecton of nodes a and d The above example llustrates how connectvty gves a sense on nodes ntegraton wth mmedate neghbors (local level), whle the average path length reflects the way each node s ntegrated to ts k-neghbors (global level). Overall a relevant structural approach to model generalzaton of urban street network should keep wellntegrated nodes (or n other words to elmnate less ntegrated nodes). Logcally we can remark that wellconnected and -ntegrated streets tend to be more mportant from a structural pont of vew than those less connected and ntegrated. Structural generalsaton of an urban street network We propose a generalzaton process based on the dervaton of a seres of subgraphs from the ntal connectvty graph. In a related work we have llustrated the fact that street connectvty conforms to a power law dstrbuton, that s, most named streets have low connectvty values whle a few have hgh connectvty values (Jang et al. 200). Ths provdes an nterestng clue for the defnton of a reducton scale algorthm based on those connectvty values at the local level (such an algorthm wll be very much selectve for average connectvty values). By contrast applyng a flterng algorthm on the average path length reflects some global propertes of the network. The followng sub-sectons ntroduce these approaches: frstly wth a connectvty-based selectve algorthm, secondly wth a selectve algorthm proposed for average path length selecton, fnally wth the applcaton of a recursve algorthm that consders the herarchcal nature of a street network. The respectve propertes and advantages of these algorthms are dscussed. Connectvty-based generalsaton Let s use a Gävle cty network for example to llustrate the dfferent prncples used for the selecton of streets n a reducton scale process. Ths network nvolves 565 named streets, so 565 nodes n the connectvty graph (fgure 4 a and b). It s composed of street central lnes topologcally nterconnected,.e. no solated streets. A scrpt determnes how each gven street ntersects to every other, and then creates a connectvty graph. Informally the algorthm can be read as follows: for each street, check f t ntersects another street, f yes, r =, otherwse r = 0 j j. Ths algorthm s as follows: Algorthm create-connectvty-graph // V s a set of streets // assume that street ID range from 0 to n- // R s the output matrx of the connectvty graph Begn 4
5 th AGILE Conference on Geographc Informaton Scence, Palma (Mallorca, Span) Aprl 25 th -27 th 2002 [ 0... n ][ 0... n ] an empty matrx wth all elements as zero R // V ' V // t arget set of vertces v V do w V do INTERSECTI ON ( v, w then for every for every f ) R[ v ][ w] else R[ v ][ w] 0 end f end for end for end create-connectvty-graph a Fgure 4: Gävle street network (a), and ts connectvty graph (b) Based on ths connectvty graph, or more specfcally on the connectvty measure, we can start to select or elmnate streets for generalsaton purposes. As mentoned n the prevous secton, well-connected streets tend to more mportant then less connected. Therefore the frst rule for the selecton s defned as follows: If a street connectvty s greater than a gven threshold, then keep t; otherwse elmnate t. For llustraton purpose, fgure 5 shows a seres of generalsed maps wth threshold values respectvely equal to, 2, 3 and 4. b (a) (b) 5
5 th AGILE Conference on Geographc Informaton Scence, Palma (Mallorca, Span) Aprl 25 th -27 th 2002 (c) (d) Fgure 5: Streets generalsaton wth connectvty values (a), 2 (b), 3 (c) and 4 (d) Average path length-based generalsaton Connectvty consders local streets drectly connected, that s streets wthn a range of one step. On the other hand, average path length consders streets wthn k steps, and ths reflects how a gven street s ntegrated to every other wthn an urban street network. So an average path length-based algorthm selects those wellntegrated streets. The rule for the selecton of these streets s defned as follows: If the average path length of a street s less than a gven threshold, then keep t; otherwse elmnate t. Fgure 6 llustrates a seres of generalsed maps wth the threshold values of average path length equal to 6.5, 6.25, 5.75, and 5.5 (respectvely fgures (a) (b) (c) and (d)). It should be noted that n both fgure 5 and 6, thresholds are defned for llustraton purpose. End users can choose approprate thresholds accordng to ther partcular objectves n applyng such a reducton algorthm (ths mght be an exploratory and nteractve process). (a) (b) 6
5 th AGILE Conference on Geographc Informaton Scence, Palma (Mallorca, Span) Aprl 25 th -27 th 2002 (c) (d) Fgure 6: Streets generalsaton wth threshold values of average path length equal to 6.5 (a), 6.25 (b), 5.75 (c) and 5.5 (d) Herarchy-based generalsaton Streets are herarchcally organsed n terms of connectvty and average path length measures. To llustrate ths, let s consder the connectvty graph whose nodes are represented by dfferent sze n terms of the magntude of connectvty of each node (.e. ndvdual street mapped from the street network) (fgure 7a). These fgures dsplay well-connected streets usng larger node szes, and less connected streets usng smaller node szes. These patterns llustrate the fact that these nodes are arranged at dfferent levels of a herarchy. So another way of generalsng a street network can be based on such a herarchcal property. We ntroduce a recursve algorthm to reduce the number of nodes based on that herarchcal structure. The proposed algorthm follows a recursve process. For example, f one set a connectvty threshold of 2, then n the generalsed graph the mnmum connectvty of nodes s equal to two as shown n fgure 7(b) (so no nodes wth a connectvty value of are left). Smlarly, one can then generalse ths resultng graph wth a connectvty threshold beng equal to 3 (7 nodes left as n fgure 7c), and then 4 (65 nodes left as n fgure 7d), and then 5 (9 nodes left as n fgure 7e) and then 6 (8 nodes left as n fgure 7f). It should be noted that from ths stage, an addtonal teraton of the algorthm gves no nodes left at all, because of the recursve nature of ths approach. We can also remark that ths algorthm retans the central structure of the cty whle the mportance of outlyng streets s dmnshed. (a) 565 nodes (b) 408 nodes (c) 7 nodes (d) 65 nodes (e) 9 nodes (f) 8 nodes Fgure 7: A herarchy-based connectvty streets generalsaton Let us map the schematc graph n fgure 7f nto the street network. Fgure 8 shows the most generalsed network (represented as thcker lnes) derved from the 8 resultng nodes of the scale reducng process. A cross-check of the roles of those resultng streets n the cty of Gävle shows that these streets consttute the central structurng part of the cty, and are most accessble n terms of transportaton and commercal actvtes allocaton. For 7
5 th AGILE Conference on Geographc Informaton Scence, Palma (Mallorca, Span) Aprl 25 th -27 th 2002 example, the fours streets quoted n fgure 8 namely Nygatan, Drottnngatan, Kungsgatan and Rådmansgatan are most mportant commercal streets. Important landmarks such as central staton, theatre, cty hall, and central shoppng mall are also located wthn ths generalsed street network. Nygatan, Drottnngatan Dscusson Kungsgatan, Rådmansgatan Fgure 8: A fnal generalsed map wth 8 major streets n the centre of Gâvle Let us brefly dscuss related work n the domans of cartographcal generalsaton and space syntax studes. In the cartography and GIS research communty, one can make a dstncton between two types of generalsaton, namely model and graphc generalsaton (Muller et al. 995). Model generalsaton s manly orented to data flterng n a scale reducton process, whle graphc generalsaton s more concerned wth graphc representaton or vsualsaton at the vsual output level (Webel 995). These two generalsaton approaches are closely related, often model generalsaton beng a pre-process of graphc generalsaton. Combnng the two permts not only to consder geometrc smplfcaton but also to ntegrate spatal structure factors n the generalsaton process. Street network generalsaton consttutes an mportant research challenge n cartographc generalsaton, as t has also an nfluence on other cartographc object generalsaton. There have been many research efforts snce the appearance of the semnal Douglas-Peucker algorthm for lne smplfcaton (Douglas and Peucker 973). Recently, graph-based approaches have been nvestgated for lnear object generalsaton lke street and hydrologcal networks. Mackaness and Beard (993) dscussed the potental of graph theory prncples for dervaton of nformaton at the topologcal level to support generalsaton of lnear objects. They appled weghted graph, drected graph, and mnmum spannng trees n the descrpton of street and dranage networks, and derved some prelmnary rules for generalsaton process. Thomson and Rchardson (995) used the concept of mnmum spannng tree n road network generalsaton. More recently Kreveld and Pescher (998) proposed a three-step approach to road network generalsaton by consderng basc geometrc, topologcal and semantc requrements n lne smplfcaton. Mackaness (995) appled space syntax prncples and demonstrated how they can be used to derve herarches of urban road networks. Hs study shows that street segment nter-connectons (note heren street segments rather than named streets) and space syntax parameters can be used to llustrate the structure of an urban street network. Although no mplementatons of these prncples have been acheved so far, ths proposal llustrates the potental of space syntax for the structural analyss of an urban street network. Space syntax has also developed many computatonal solutons to the analyss of an urban street network. Space syntax studes often derve some local or global propertes of a gven urban network. For example, a long straght street tends to have many streets nterconnected, thus t has a hgh connectvty; smlarly, the same street tends to be well ntegrated to every other street wth a shorter average path length. However, and to the best of our knowledge, space syntax prncples have not been appled so far to generalse an urban street network whle retanng ts man structure. We can remark that our approach to the structural representaton bears some smlarty to Rchardson s (2000) approach based on human s spontaneously perceptual organsaton (or groupng) on lnear objects. She used a term stroke to defne the elementary unts of a network based on movement contnuty. Each stroke s actually a basc modellng unt, a smlar concept to the one we adopt by consderng named street as vertces of the 8
5 th AGILE Conference on Geographc Informaton Scence, Palma (Mallorca, Span) Aprl 25 th -27 th 2002 connectvty graph. However, ths approach that consders a cogntve-based graphc representaton s dfferent n essence from the structural and model-based generalsaton we propose n ths paper, and t s also not drectly computable. For nstance, and at the excepton of regular and orthogonal networks, street networks may not appear to have such mmedate and structurng vsual propertes. Concluson An urban street network s a structurng component of the cty so defnng and mplementng flterng algorthms that keep the man and central structure of an urban street network s of much nterest for many urban applcatons and studes. Ths paper proposes a model generalsaton approach for the selecton of characterstc streets n an urban street network. It s based on the applcaton of selectve and flexble algorthms that consders both local and global structurng propertes of named streets that correspond to basc functonal elements n the cty. The proposed algorthms are flexble as thresholds are user-defned and controlled gvng thus an nteractve soluton to the applcaton of such algorthms to a street network. The case study presented n the paper shows how the structure of a street network s retaned wth subsequent flterng of streets. Our model and approach extend Mackaness (995) proposal by an mplementaton of two model generalsaton algorthms. Ths model generalsaton can also be treated as a prerequste for further generalsaton processes such as lne smplfcaton or buldng blocks generalsaton. Further work concerns the ntegraton of weghted graph wth the connectvty graph by consderng a semantcs classfcaton of streets and the applcaton of the method to other urban contexts. References AGENT (998), Constrant Analyss, ESPRIT report, Department of Geography, Unversty of Zurch. Buttenfeld B. P. and McMaster R. B. (99), Map Generalsaton: Makng Rules for Knowledge Representaton, Longman Scentfc & Techncal. Douglas D. and Peucker T. (973), Algorthms for the Reducton of the Number of Ponts Requred to Represent a Dgtal Lne or ts Carcature, The Canadan Cartographer, Vol. 0, pp. 2 22. Gross J. and Yellen J. (999), Graph Theory and Its Applcaton, CRC Press: London. Hller B. (996), Space s the Machne: A Confguratonal Theory of Archtecture, Cambrdge Unversty Press, Cambrdge, UK. Hller B. and Hanson J. (984), The Socal Logc of Space, Cambrdge Unversty Press: Cambrdge. Hller, B. (edtor, 997), Proceedngs, Frst Internatonal Symposum on Space Syntax, Unversty College London, London, 6-8 Aprl, 997. Holanda F. (edtor, 999), Proceedngs, Second Internatonal Symposum on Space Syntax, Unversdade de Brasla, Brasla, 29 March-2 Aprl 999. Jang, B., Claramunt, C. and Batty, M. (999), Geometrc Accessblty and Geographc Informaton: extendng desktop GIS to space syntax, Computer Envronment and Urban Systems, Elsever Scence (Pub.), 23 (2), pp. 27-46. Jang B. Claramunt C. and Vasek V. (200), Small World Patterns n Urban Street Networks, Workng paper at the Unversty of Gävle. Kreveld M. V. and Pescher J. (998), On the Automated Generalzaton of Road Network Maps, GeoComputaton 98, avalable at http://dvcom.otago.ac.nz/sirc/webpages/conferences/geocomp/geocomp98/geocomp98.htm Mackaness W. A. and Beard M. K. (993), Use of Graph Theory to Support Map Generalsaton, Cartography and Geographc Informaton Systems, Vol. 20, pp. 20 22. Mackaness W. A. (995), Analyss of Urban Road Networks to Support Cartographc Generalzaton, Cartography and Geographc Informaton Systems, Vol. 22, pp. 306 36. Muller J. C., Lagrange J. P. and Webel R. (995, eds.), GIS and Generalzaton: Methodology and Practce, Taylor and Francs: London. Pepons J., Wneman J. and Bafna S. (edtors, 200), Proceedngs, Thrd Internatonal Symposum on Space Syntax, Georga Insttute of Technology Atlanta, May 7-, 200. Rchardson D. (2000), Generalzaton of Road Networks, avalable at http://www.ccrs.nrcan.gc.ca/ccrs/tekrd/rd/apps/map/current/genrne.html Thomson R. C. and Rchardson D. E. (995), A Graph Theory Approach to Road Network Generalsaton, n: Proceedng of the 7 th Internatonal Cartographc Conference, pp. 87 880. 9
5 th AGILE Conference on Geographc Informaton Scence, Palma (Mallorca, Span) Aprl 25 th -27 th 2002 Webel R. (995), Three Essental Buldng Blocks for Automated Generalsaton, n: Muller J. C., Lagrange J. P. and Webel R. (eds.), GIS and Generalzaton: Methodology and Practce, Taylor and Francs: London, pp. 56-69. 0