6.7 Network analysis Introduction. References - Network analysis. Topological analysis

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1 6.7 Network aalyss Le data that explctly store topologcal formato are called etwork data. Besdes spatal operatos, several methods of spatal aalyss are applcable to etwork data. Fgure: Network data Refereces - Network aalyss Itroducto 1. Haggett, P. ad Chorley, R. J. (1969): Network Aalyss Geography, Edward Arold. Some methods focus o the topologcal structure of the etwork, whle others cosder metrc propertes of the etwork. The former s called 'topologcal aalyss' whle the latter s 'metrc aalyss'. Topologcal aalyss I topology, a subfeld mathematcs, spatal objects are regarded as equvalet f they ca be trasformed wth each other by the rubber sheetg operato wthout chagg ther spatal structure. Equvalet objects are called somorphc objects. Fgure: Isomorphc etworks (metro etworks) 1

2 Topologcal aalyss focuses o the topologcal structure of a etwork; whether two odes are drectly coected, how may lks a ode s coected to, etc.. Cosequetly, aalyss of somorphc etworks yelds the same result. Topologcal aalyss s ofte called 'graph-theoretc', because t regards the etwork as a 'graph'. The term 'graph' refers to a represetato of the topologcal structure of a etwork, whch eglects the legth, shape, ad other attrbutes of lks. Metrc aalyss Termology Metrc aalyss, o the other had, cosders ot oly topologcal but also geometrc propertes of a etwork, say, the legth, drecto, ad curvature of lks. Graph A graph s a set of odes ad ther coectg lks. Subgraph: A subgraph s a part of a graph. It cossts of a subset of odes ad lks of the orgal graph. Coected graph A coected graph s a graph whose odes are coected drectly or drectly wth each other. Dscoected graph A dscoected graph s a graph whch some odes are ot coected ether drectly or drectly wth each other. A dscoected graph cossts of a set of coected graphs, whch are called coected elemets. Fgure: A graph ad ts subgraphs 2

3 Plaar graph A plaar graph s a graph whch lks tersect oly at odes. Three coected elemets No-plaar graph A o-plaar graph s a graph whch some lks tersect at pots betwee odes. Fgure: Coected ad dscoected graphs Note: A graph s a plaar graph f t ca be trasformed by a somorphc trasformato to a plaar graph. Fgure: Plaar ad o-plaar graphs Complete graph A complete graph s a graph whch every par of odes s coected drectly by oe lk. Fgure: Complete graphs

4 Crcut A crcut s a set of lks that starts from a ode, vsts several odes, ad returs to the startg ode. If a crcut vsts every ode oly oce, the crcut s called a smple crcut. Loop A loop s a lk whose eds are the same ode. Fgure: A graph ad ts cucuts Tree graph A tree graph s a graph that does ot cota a crcut. Fgure: Loops I graph theory, t s permtted that two odes are coected drectly by more tha oe lk. Loops ca also exst graph theory. Fgure: Tree graphs 4

5 6.7.2 Topologcal aalyss 1: coectvty measures I etwork aalyss, however, two odes ca be coected drectly by oly oe lk. Loops are ot permtted. Oe of the motvatos of etwork aalyss s to evaluate a etwork terms of the coectvty amog odes, whether lks are dese eough to provde a certa level of accessblty amog odes. Network aalyss s mportat trasportato plag, because the accessblty of resdets to urba facltes s evaluated o a road etwork. Dese etworks are more coveet tha sparse oes f they represet traffc etworks. Cosequetly, we evaluate the coectvty amog odes by measurg the desty of lks. Fgure: Dese ad sparse etworks 1) μ dex : Number of odes l: Number of lks c: Number of coected elemets A etwork s well-coected f l s relatvely larger tha. Coectvty measures thus evaluate l comparso wth. μ dex s defed by μ = l + c A dese etwork has a large μ, whch mples that odes are well coected. Amog coected graphs (c=1) a tree graph has the smallest μ. Ths dcates that, gve a set of odes, tree graphs are the most effcet graphs to coect all the odes. 5

6 2) α dex Gve the umber of odes, we ca calculate the maxmum umber of lks. It s gve by the complete graph, that s, (-1)/2. If we cosder oly plaar graphs, the maxmum umber of lks s -6, whch gves the maxmum μ, 2-5. The doma of μ s 0 μ 2 5 I etwork aalyss we ofte mplctly assume plaar graphs. α dex s a stadardzed verso of μ dex. Dvdg μ by ts maxmum 2-5, we obta μ l + c α = = The doma of α for plaar graphs s 0 α 1 ) β dex 4) γ dex β dex s defed by l μ = A dese etwork has a large β as well as μ. The doma of β for plaar graphs s 6 0 β γ dex s a stadardzed verso of β dex: β l γ = = 6 6 The doma of γ for plaar graphs s 0 γ 1 Comparso of four measures Propertes of coectvty measures The coectvty measures show how desely odes are coected by lks. They provde a smple ad effcet way of evaluatg accessblty amog odes. μ α β The measures cosder oly the umber of odes, lks, ad coected elemets. They drop detaled formato about etwork coecto, so they ofte caot dstgush dfferet graphs. γ

7 6.7. Topologcal aalyss 2: accessblty measures Coectvty measures descrbe the total (average) coectvty amog odes. I ths sese they are global measures. Accessblty measures, o the other had, are local measures because they are defed for every ode. Accessblty measures evaluate the accessblty from a ode to the other odes. Fgure: Graphs dstgushable by coectvty measures Termology Dstace betwee two odes I graph theory, the dstace betwee two odes s defed as the mmum dstace o the etwork. Topologcal dstace betwee two odes Topologcal dstace betwee two odes s defed as the mmum dstace o the etwork, where the legth of all the lks s set to oe. Cosequetly, topologcal dstace betwee two odes s the mmum umber of lks that coect the odes Fgure: Topologcal dstace from a ode 1) Kög umber 4 Kög umber of a ode s the topologcal dstace to ts farthest ode. Ay ode s located wth the topologcal dstace gve by the Kög umber Fgure: Kög umbers 7

8 Small Kög umber dcates that the ode has hgh accessblty the etwork, that s, t s located at the 'ceter' of the etwork. I trasportato etwork, odes of small Kög umbers are coveet locatos terms of accessblty to other odes. Usg the Kög umber we defe the topologcal dameter of a etwork. The dameter of a etwork s the dstace betwee the farthest par of odes, that s, the maxmum Kög umber. If a graph has a roud shape, ts dameter s relatvely small. A graph of a elogated shape has a large dameter. 2) Smbel umber Smbel umber of a ode s the sum of the topologcal dstaces to the other odes. 2 7 As well as the Kög umber, the Smbel umber descrbes the accessblty of a ode to other odes, ad cosequetly, evaluates the locatoal coveece of the ode. Fgure: Topologcal dameter of etworks Fgure: Smbel umbers Metrc aalyss 1: coectvty measures I topologcal aalyss of a etwork, we cosder oly the topologcal structure of etworks. Coectvty measures are defed by oly the umber of odes, lks, ad coected elemets. Accessblty measures are calculated from topologcal dstace betwee odes. I metrc aalyss, o the other had, we cosder ot oly topologcal but also metrc propertes of a etwork, say, the legth, curvature, ad shape of lks ad the flow o the etwork. 8

9 Termology Notato Legth of a lk I topologcal aalyss, the legth of a lk s always set to oe because t s the topologcal dstace betwee adjacet two odes. I metrc aalyss, o the other had, the legth of a lk s defed by a metrc measure such as the Eucldea dstace, etwork dstace, or tme dstace. : The umber of odes l: The umber of lks d : The legth of lk D j : The dstace betwee odes ad j D: The dameter of a etwork ( D= max D j ) f : The flow o lk, j 1) π dex π dex s the rato of the total legth of lks to the dameter of the etwork. Mathematcally t s defed as π = D d A dese etwork has a large π, whch dcates that odes are well-coected ad that the etwork s coveetly structured. The doma of π s 1 π If a graph s dscoected, we calculate π for each coected elemet separately ad average the dces. 2) θ dex There are two types of θ dex: θ 1 ad θ 2. The former s a coectvty measure whle the latter a accessblty measure. θ 1 dex s the average flow per ode: θ = 1 f I the real world, a large flow o a etwork mples that the odes are closely related wth each other. A large θ 1, cosequetly, whch reflects a large flow o the etwork, suggests that the odes are well coected. 9

10 6.7.5 Metrc aalyss 2: accessblty measures 1) η dex As well as topologcal aalyss, metrc aalyss dscusses the accessblty of odes o a etwork. η dex s the average legth of lks: η = l d If odes are coected by short lks, η shows a small value, whch mples hgh accessblty amog odes. 2) θ dex ) Degree of crcuty θ 2 dex s the average legth of lks per ode: θ = 2 l A small θ 2 dcates that odes are coected by short lks, ad cosequetly, hgh accessblty amog odes. The degree of crcuty s defed by ( l e ) 2 where e s the Eucldea dstace betwee ed odes of lk. If lks are close to straght les, ths dex shows a small value, whch dcates that hgh accessblty amog odes Applcato Coectvty ad accessblty measures are used 1. to evaluate exstg trasportato etworks, 2. to aalyze trasportato etworks relato to laduse patters, ad. to aalyze urba developmet process. Fgure: Urba developmet process 10

11 Homework Q.6.5 Take a traffc etwork a cty such as subways ad expressways, ad evaluate ts coectvty by usg quattatve measures. 11