Graph Analysis of Underground Transport Networks

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1 УДК 59.7:4.8 L. Sarycheva, K. Sergeeva Natonal Mnng Unversty Karl Marks ave., 9, 495, Dnepropetrovsk, Ukrane Graph Analyss of Underground Transport Networks Л.В. Сарычева, Е.Л. Сергеева ГВУЗ «Национальный горный университет» пр. Карла Маркса, 9, 495, Днепропетровск, Украина Анализ графов транспортных подземных сетей Л.В. Саричева, К.Л. Сергєєва ДВНЗ «Національний гірничий університет» пр. Карла Маркса, 9, 495, Дніпропетровськ, Україна Аналіз графів транспортних підземних мереж The methodc of cty subway networks analyss on the bass of graph characterstcs (centralty, connectvty and shape) s proposed. The subways characterstcs were calculated from subways ndexes (number of lnes, number of statons, length of lnes n klometers, rdershp per year) and from ndcators of ctes urbanzaton (area and populaton). The nterrelaton between graph (road) structures and weghts of ther edges, and between π -ndex descrbng the shape of the graph and the number of passengers s demonstrated. It s shown on a practcal example that the analyss of structure of proposed road network graphs can be useful n determnng the sequence of new roads constructon. Clusterng of underground transport networks based on characterstcs of network graph structure was performed for the frst tme. Keywords: graph analyss, transport network, underground, GIS Предложена методика анализа городских сетей метрополитенов на основе характеристик графов (центральность, связность и форма). Значения характеристик рассчитаны на основе индексов метрополитенов (количество линий, количество станций, протяженность линий в километрах, пассажиропоток в год) и показателей урбанизации городов (площадь и численности населения). Показана взаимосвязь между структурой графов транспортных сетей, весом их дуг и π -индексом для описания формы графа и количества пассажиров. На практическом примере показано, что анализ структуры представленных графов транспортных сетей может использоваться для определения последовательности этапов строительства новых линий. Впервые выполнена кластеризация транспортных подземных сетей на основе характеристик структуры графа сети. Ключевые слова: анализ графов, транспортные сети, метрополитен, ГИС Запропоновано методику аналізу міських мереж метрополітенів на основі характеристик графів (центральність, зв'язність і форма). Значення характеристик розраховані на основі індексів метрополітенів (кількість ліній, кількість станцій, протяжність ліній в кілометрах, пасажиропотік на рік) і показників урбанізації міст (площа та чисельність населення). Продемонстровано взаємозв'язок між структурою графів транспортних мереж, вагою їх дуг і π -індексом для опису форми графа і кількості пасажирів. На практичному прикладі показано, що аналіз структури представлених графів транспортних мереж може використовуватися для визначення послідовності етапів будівництва нових ліній. Вперше виконано кластеризацію транспортних підземних мереж на основі характеристик структури графа мережі. Ключові слова: аналіз графів, транспортні мережі, метрополітен, ГІС Introducton Researches on applyng graph theory to analyze transportaton networks have been carryng out snce the 96-s tll the present days. The most famous works were publshed by Davd Levnson, Mke Batty, Paul Longley etc. 58 ISSN «Искусственный интеллект» 4 4

2 Graph Analyss of Underground Transport Networks 3S Analyss of transport networks graphs nvolves solvng problems: forecastng and evaluatng transport network growth [, ]; studyng the nfluence of transport network structure and topology on quanttatve ndcators of traffc flows [3]; nvestgaton of dependence of network structure from the sze of ctes ground transportaton and urban structure [4]; constructon of dynamc models of urban systems usng GIS technologes based on cellular automata, agent-based modelng and fractal analyss [5, 6, 7]; nvestgaton relatonshps between quanttatve ndcators of transport network structure and ts performance, densty and urban spatal pattern, and the trps dstance for early soluton of transport problems etc. The up-to-date researches don t pay enough attenton to network bandwdth analyss dependng on the parameters of structure of the network graph. Ths aspect s nvestgated n the paper. Basc Defntons Graph ( V, E ) of unordered pars of dstnct elements of V set [8]: ( ) G s a par of two sets non-empty set of nodes V and an assemblage E G V, E = V ; E, v, E V V. Pars of the E assemblage are called as rbs. The number of nodes of a graph G s denoted n = n G, m = m G : n G = V, m G = E, as n and the number of edges m ( ( ) ( )) ( ) ( ) where V, E, v, nodes, ( v, ) V E cardnal number. Assume that v edge between them. Maxmum dstance for a gven graph G s called as dameter: D(G) = max d(u, v). u,v G e = v The set of nodes at the same dstance g from the node v s called as ter: { u V d ( v, u ) g} D ( v, g ) = =. The dstance matrx ( d, j ),, j,,..., n s defned as: d, j = d( v, v j ). Characterstcs of Graph Structure = of the G graph The prncple characterstcs of graph structure are ts centralty, connectvty and shape. Centralty characterzes the postonalty of graph nodes. Absolute ndex S of the v node accessblty s the sum of dstances from ths node to other nodes [9, ]: S = d j. j The v * node s called a central f t possesses the smallest absolute value of reachablty ndex S * = mns. The Köng's number K = max d. j n j n K j s also the absolute ndex of the node The central v * node possesses the least Köng's number: K * j v reachablty: = mn K. n «Штучний інтелект»

3 L. Sarycheva, K. Sergeeva S The degree of devaton of the -th node from the central one: η =. S * The ndex of herarchy y = d( v, v * ) shows the topologcal dstance from the central node. Nodes wth the same values of ndex of herarchy form a ter. Centralty s useful for analyss of centers locaton of local or regonal enttes, transportaton nodes. A measure of centralty on the set of centraltes { S } of graph nodes s ntegraton of: S = d j = S., j The S * ndex of centralty of the center node defnes the graph unpolarty: S * = mn S. n The varance on a set of central nodes descrbes the graph centralzaton: H = (S S ) = S ns. * The terrtory of the cty conssts of fnte sets of objects: the set of enterprses, buldngs, roads, etc. The essence of the terrtoral communty conssts n exstng mathematcal relatonshps between these objects. The confguraton of lne-node structure of terrtory objects placement and relatonshp s modeled usng graphs []. For example, structures smulated by graphs n Fg. descrbe three stages of terrtory development: ntal (a), medum (b), rpe (c) on a qualtatve level. * а) b) c) Fgure Graphs model for three stages of the terrtory development: ntal (a), medum (b) and rpe (c) Connected graph s modelng the structure of urban subway and road network. The graph s connected f there s a path from any node to any other. The graph connectvty parameters characterze t ntenseness wth rbs and degree of trangulaton. The best known of them are three connectvty parameters α,β,ϕ -ndexes: m n + k α = ; n 5 < α < ; () m β = ; n β 3; () m ϕ = ; 3(n ) ϕ, (3) where m, n, k are the number of edges, nodes and graph connected components respectvely. To calculate the graph shape parameters the matrx M = ( µ j), =,,...,n, j =,,...,n of ordnal vcnty can be used: µ = j { } the number of vcnal vertces of the j-th order for the node v, 6 «Искусственный интеллект» 4 4

4 Graph Analyss of Underground Transport Networks 3S where vcnal nodes of the j -th order for node D v, j ter (neghbors of the -st order adjacent nodes). For example, the sequence neghborhood matrxes for graphs from Fg. (a), Fg. (b), Fg. (c) are represented n Fg a) b) c) v are nodes of the ( ) Fgure Sequence neghborhood matrx for graphs from Fg. If two graphs possess same number of nodes the more compact of them s the one havng more zero columns n the M matrx. The graph on Fg.(c) s more compact than the graphs n Fg.(a) and Fg.(b) ( tmes and 4/3 tmes, respectvely). The rato of the graph edges total length P to ts dameter D determnes the shape of the graph descrbed by π -ndex: π = P / D. Table present values of the observed parameters for 3 graph model shown n Fg.. (b) Table Parameters of graphs structures shown n Fg.. Graph (a) (c) S =S 7 = S =S 6 =6 S 3 =S 5 =3 S 4 = * =4 S =5 S = S6=3 S 3 =9, S 4 =4 S 5 = S 7 = *=3 S =S =9 S 3 =S 5 =9 S 6 =S 7 =9 S 4 =6 *=4 К =К 7 =6 К =К 6 =5 К 3 =К 5 =4 К 4 =3 К =К =4 К 4 =К 6 =4 К 5 =К 7 =3 К = К =К = К 3 =К 5 = К6=К 7 = К 4 = Centralty η =η 7 =,8 η =η 6 =,3 η 3 =η 5 =, η 4 = η =,7 η =η 6 =,4 η 3 = η 4 =,6 η 5 =, η 7 =,3 η =η =,5 η 3 =η 5 =,5 η 6 =η 7 =,5 η 4 =,6 γ =γ 7 =3 γ = γ 6 = γ 3= γ 5 = γ 4= γ =γ 4 = γ 6 = γ =γ 5 = γ 7 = γ 4 = γ =γ = γ 3 =γ 5 = γ 6 =γ 7 = γ 4 = S=56 S * = S=43 S * =9 S=3 S * =6 Connectvty (a) α = β =,9 ϕ =,4 (b) α =, β =, ϕ =,5 (c) α =,7 β =,7 ϕ =,8 Shape (a) Q = π = (b) Q = π = (c) Q = 3 π = 6 H=8 H=3 H=8 For attrbuted graph wth weghted edges the dameter of the graph and the length of ts edges can be expressed not only topologcally (as the number of edges), but also metrcally (through the attrbutes of edges). «Штучний інтелект» 4 4 6

5 L. Sarycheva, K. Sergeeva The Examples of Analyss of Subway Graphs Network Characterstcs Consder an example demonstratng the usefulness of analyss of the graph structure for spatally referenced objects []. Gven network of roads s represented as graph n Fg. 3. Each edge of the graph s the road, each node the staton (pont) or a node of roads ntersecton. External road (ndcated by arrows) were not take nto account because of ther secondary mportance. Assume that 7 new roads ndcated by dashed lnes n Fg. 3 were desgned. Whch of these roads wll have the greatest mpact on the avalablty of one pont wth respect to another wthn the terrtory? How wll the appearance of each new road mpact on the relatve avalablty of ndvdual ponts wthn the network? To answer these questons, the parameters of centralty of the orgnal graph (wth no new roads) and graphs (a) (f) n Fg. 3 were calculated. The calculated parameters are presented n Table. It s evdent from the table that each new road mproves communcaton between settlements (reduces average path length n the network of roads), but ther effectveness vares. (a) (b) (c) (d) (e) (f) - proposed roads Fgure 3 The mpact of new roads n the network on the relatve avalablty of settlements (dots hghlghts tems that wll beneft from the constructon of these roads, the sze of dots ndcates the degree of wnnng) 6 «Искусственный интеллект» 4 4

6 Graph Analyss of Underground Transport Networks 3S Table The parameters of graphs modelng road networks Paramerets Roads network graphs (accordng to Fg. 3) Intal а b c d e f The average length of the path n the network S * S The placepreference (accordng to S) H % of decrease of the average path length n the network Nodes v benefcal to the new road wth the degree 9,,7,6, 8,6 8,3 - =7 =58 5 =5 4 =46 3 =46 36 =33 37 =33 35 = 37 =5 =4 36 =74 3 =69 =5 =8 3 =65 =43 8 =4 9 =4 =74 =36 9 =34 8 =34 37 =46 =4 36 =6 = 3 =6 The most effectve road s (f) as t reduces the average length of a path n the network on.6%, the least effectve (b). Smultaneous creaton of new roads (a) - (f), shown by the graph n Fg. 3 (f), reduces the average path length of the network on 8.3%. The beneft taken for settlements from buldng the new roads s determned by changes n the average path length for each localty. For the (a) road constructon the most nterested are the -th, -th and 5-th settlements, n road (b) - the 36-th, 37-th, n road (c) the 37-th, -th, 36-th, n road (d) the -th, 3-th, -th, for the (e) road the -th, -th settlements. The costs gven by settlements for buldng of new roads can be dstrbuted n proporton to the wnnng, defned by parameter ( = S node - S ) n Table. Calculaton of Subway Indexes ntal graph In the other example the subway network graphs structure for some European ctes s analyzed (Table 3). All subway schemes and ctes subway statstcs are avalable through [, 3] (Table 4, 5, 6). The number of statons X 4 ncludes transfer statons taken from [4]. The number of edges and nodes m, n of subway network graphs are calculated by the scheme takng nto account the fact that several transfer statons (from varous subway lnes) create one node of the graph. Correlaton coeffcents for graph connectvty characterstcs ( α,β, π ndexes аlfa, beta, P, PID) and nput ndexes X,X,..., X 6 s the hghest for the π -ndexes and the lowest for ϕ -ndexes f. Ths means that the π - ndces are nformatve characterstc for analyss of subways networks graphs (Table 7). The Pars, Moscow, Prague, Kev and Sant Petersburg are characterzed by the largest values of passenger-to-klometers (klometers) rato, whch s especally evdent from fg.4(b) n auxlary scale. Only Prague and Kev have n common the largest values of passenger / populaton rato (ths fact may be caused by the large number of toursts) (fg.4(a)). «Штучний інтелект»

7 L. Sarycheva, K. Sergeeva Table 3 Subway schemes of European ctes Pars Madrd Moscow London Berln Sant Petersburg Prague Athens Budapest Kev Kharkov Mnsk 64 «Искусственный интеллект» 4 4

8 Graph Analyss of Underground Transport Networks 3S Table 4 Input data for subways Cty (subway locaton) Length P, km Populaton (mllon) Rdershp per year (mllon) Number of statons Urban area, km Number of lnes X X X 3 X 4 X 5 X 6 Athens 78,3 3, Berln 46, 3, Budapest 3,3, Warsaw,7, Hamburg,9, Dnepropetrovsk 7,, Kev 65,9, London 47 9, Madrd 4,5 5, Mnsk 35,4, Moscow 33, 5, Pars 7,3, Prague 59,4, Sant Petersburg 3 4, Kharkov 39,3, Table 5 Subway characterstcs Cty (subway locaton) Edges weghts Node weghts Frequency of travels Areal weghts Areal densty of lnes Lnear densty of statons Areal densty of statons Х 3 /Х Х 3 /Х 4 Х 3 /Х Х 3 /Х 5 * 3 Х 5 /Х Х /Х 4 Х 5 /Х 4 Athens 5, 6,4 6,3 698, 7,4, 9, Berln 3,5,6 8,4 376,6 9,,7 6,9 3 Budapest 5,3 4, 98,6 9,7 7,7,8,3 4 Warsaw 6,4 6,6 8,4 55,9 5,, 5,9 5 Hamburg,9,9 6,8 93,5 6,3, 6,6 6 Dnepropetrovsk,,4 8, 5, 45,6, 54, 7 Kev 8,,3 87,8 968, 8,3,3,7 8 London,5 3, 4,6 7,5 3,5, 4,3 9 Madrd,7,,9 455,3 5,9,8 4,6 Mnsk 7,9, 5,3 868, 9,,3,6 Moscow 7,9 3, 59, 559,6 4,,7 3,4 Pars 7, 4, 47, 535,7 3,,6 7,4 3 Prague 9,9,3 465,8 67,4 4,8, 5, 4 Sant Petersburg 6,9,7 6, 658,,5,7 7,8 5 Kharkov 6, 8,3 65, 53,5,9,4 6, «Штучний інтелект»

9 L. Sarycheva, K. Sergeeva Table 6 Subway graphs characterstcs Cty (subway DM P PID m n P D alfa beta f locaton) (D, km) (P/D) (X /DM) Athens ,,,,35,65,4 Berln ,8,3,6,36 4,79 4,6 3 Budapest ,3,,98,34,5,87 4 Warsaw,7,,95,35,, 5 Hamburg ,8,4,7,36,3, 6 Dnepropetrovsk ,,,83,4,, 7 Kev ,9,,,35,8,76 8 London ,,9,7,39 6,7 6,35 9 Madrd ,6,9,6,39 8,63 5,53 Mnsk ,,,96,35,,96 Moscow ,,8,6,39 7,33 6,94 Pars ,3,,,4 9,9 8,94 3 Prague ,6,,,35,35,3 4 Sant Petersburg ,,4,5,36 3,44 3,75 5 Kharkov ,3,,,36,7,7 Table 7 Correlaton coeffcents for subway graphs characterstcs X X X3 X4 X5 X6 alfa beta f P PID X X,8 X 3,7, X 4,8,8,6 X 5,7,,9,7 X 6,8,8,8,,8 alfa,9,8,8,9,8,9 beta,8,8,7,9,7,9,9 f,5,5,4,6,5,6,6,3 P,8,8,8,,8,,9,9,6 PID,8,9,8,9,8,,9,9,5, Does the rdershp per year depend on parameters of the graph? Accordng to Fg. 5, the P parameter of the Pars and Madrd subway network graphs allows to suggest about the possblty to ncrease rdershp n these ctes n comparson wth the observed stuaton. Rdershp per year n Moscow, Sant Petersburg and Prague are optmal wth P respect to P parameter. At the same tme the dfference (Fg. 6) of P = (calculated D X from the graph) and PID = (calculated from the length of subway lnes n km) DM parameters s the largest for Madrd and Pars. Pars and Madrd subway networks may have the largest passenger traffc (larger than n Moscow). If we compare the subway networks to ndcate how the total length of subway lnes ( X ) matches to π -ndex (Fg. 7), we wll conclude that the London network s the longest and possesses the lower π -ndex value than the Pars network (whose length s two tmes shorter). 66 «Искусственный интеллект» 4 4

10 Graph Analyss of Underground Transport Networks 3S Rdershp per year (mllon), Х3 Rdershp per year (mllon), Х M o s c o w M o s c o w Pa rs Lon don S a Pe n t t er s b u r g P a r s Lo n dn o S a P n et st r bu rg X3 X M a d r d P rg aue Ma dr d Pr agu e K e v B e r l n At h e n s M ns k K e v B eln r X3 X A ht es n M ns k Kha kr o v H abmu g r K h a r k o v Ha m rgb u B uda pes t Wa sr aw Dne pro pet r ovs k B u d a p e s t op e t r o v s k War s a w D n e p r Populaton (mllon), Х Length (km), X a) b) Fgure 4 Plots for subway characterstcs of European ctes: a) rdershp per year (mllons) and populaton (mllons); b) rdershp per year (mllons) and length (km) Rdershp per year (mllon), X M oco s w P a r s Londo n S a n t P e t e r s b u r g Mad d r Pr ague K e v B e r l n At hens Mns k X3 P Kha rko v Hambur g B u d a p e s t Wars aw Dnepr ope ro t vsk P Fgure 5 Plots of rdershp and π -ndex values «Штучний інтелект»

11 L. Sarycheva, K. Sergeeva P - PID Mo s c o w P as r L o no dn Sa n tp ete r s b gu r M a d r d P r a g u e K e v Be l rn A t h e n s Mn s k K h a r k o v Ha mb ur g Fgure 6 Dfferences of P and PID ndexes Wars a w Bu d a p e s t D n e p ro r ovpsek t Length P, km, X L o n d o n Madrd M o s c o w P a r s X P Kev B re n l S a n t P e t e r s b u r g Ha m b u r g Ath ens Pr ag ue Mns k K h a r k o v B u d a p e s t W ras a w Fgure 7 Plots of Length P (km) and π -ndex values Dn epr op e t r o v s k P Clusterng of examned networks based on graph characterstcs ( α,β, π ndexes) nto K clusters ( K =, 3, 4 ) usng the k -means method hghlghts the next clusters (Table 8). Table 8 Clusterng of subway networks based on graph characterstcs Cty (subway Clusterng on the bass of α, β, π Clusterng on the bass of Х, locaton) ndexes Х, Х 3, Х 4, Х 5 characterstcs K= K=3 K=4 K= K=3 K=4 Athens 3 Berln Budapest Warsaw 3 3 Hamburg Dnepropetrovsk 3 3 Kev London Madrd 3 3 Mnsk Moscow 4 Pars Prague Sant Petersburg 4 3 Kharkov 68 «Искусственный интеллект» 4 4

12 Graph Analyss of Underground Transport Networks 3S The peculartes of obtaned clusters: (London, Madrd, Moscow, Pars) are characterzed by the hghest values of the α,β, π parameters; for (Berln, Sant Petersburg) the values of parameters are above average; for (Athens, Budapest, Hamburg, Kev, Mnsk, Prague, Kharkov) the values of α,β, π parameters are close to average; (Warsaw, Dnepropetrovsk) are characterzed by the lowest values of α,β,π parameters. Ctes clusterng on the bass of nput characterstcs ( X, X, X 4, X 5 ) does not nclude Madrd, Moscow or Warsaw, Dnepropetrovsk nto the same cluster. So the structure of cluster of subway network graphs based on α,β, π ndexes s not the same as the structure of clusters based on nput data X, X, X 3, X 4, X 5. Conclusons The methodc for cty subway networks analyss on the bass of graph characterstcs (centralty, connectvty and shape) s proposed. The subways characterstcs are calculated from values of subways ndexes (number of lnes, number of statons, length n klometers, rdershp per year) and from ndcators of ctes urbanzaton (area and populaton). A relatonshp between graphs (roads) structure and weghts of ther edges, between π -ndex descrbng the shape of the graph and the number of passengers s demonstrated. It s shown on a practcal example that the analyss of structure of proposed road network graphs can be useful n determnng the sequence of new roads constructon. Clusterng of underground transport networks based on characterstcs of network graph structure was performed for the frst tme. References. Levnson D.M. Plannng for Place and Plexus: Metropoltan Land Use and Transport / D.M. Levnson, K.J. Krzek // Routledge, ISBN-3: p.. Levnson D. Forecastng and Evaluatng Network Growth / D. Levnson, X. Feng, M.O. Norah // Networks and Spatal Economcs. ().. p Pavthra P. Network Structure and Spatal Separaton / P. Pavthra, H. Hochmar, D. Levnson // Envronment and Plannng: Plannng and Desgn. 39().. P Levnson D. Network Structure and Cty Sze / D. Levnson // 5. Batty M. Modelng urban dynamcs through GIS-based cellular automata /M. Batty, Y. Xe, Z. Sun // Computers, Envronment and Urban Systems p Batty M. Ctes and Complexty: Understandng Ctes wth Cellular Automata, Agent-Based Models, and Fractals / M. Batty // The MIT Press, ISBN: p. 7. Jn Y. Appled Urban Modelng: New Types of Spatal Data Provde a Catalyst for New Models / Y. Jn, M. Batty // Transactons n GIS. 7(5). 3. p Берж К. Теория графов и ее применения / К. Берж. M.: Госиноиздат, с. 9. Оре О. Теория графов / О. Оре. M.: Наука, с.. Хаггет П. География: синтез современных знаний / П. ХаггетZnatne. M.: Прогресс, с.. Сарычева Л.В. Компьютерный эколого-социально-экономический мониторинг регионов. Математическое обеспечение.нгу / Л.В. Сарычева. Днепропетровск: НГУ, 3. с.. Metro systems by annual passenger rdes. Wkpeda, the free encyclopeda, Demographa World Urban Areas (World Agglomeratons): 9th Annual Edton (March 3) Europe. UrbanRal.net, 3. «Штучний інтелект»

13 L. Sarycheva, K. Sergeeva RESUME L. Sarycheva, K. Sergeeva Graph Analyss of Underground Transport Networks Background: Researches on applyng graph theory to analyze transportaton networks have been carryng out snce the 96-s tll the present days. The most famous works were publshed by Davd Levnson, Mke Batty, Paul Longley etc. Analyss of transport networks graphs nvolves solvng problems: forecastng and evaluatng transport network growth; studyng the nfluence of transport network structure and topology on quanttatve ndcators of traffc flows; nvestgaton of dependence of network structure from the sze of ctes ground transportaton and urban structure; constructon of dynamc models of urban systems usng GIS technologes based on cellular automata, agent-based modelng and fractal analyss; nvestgaton relatonshps between quanttatve ndcators of transport network structure and ts performance, densty and urban spatal pattern, and the trps dstance for early soluton of transport problems etc. The up-to-date researches don t pay enough attenton to network bandwdth analyss dependng on the parameters of structure of the network graph. Ths aspect s nvestgated n the paper. Materals and methods: The subway network graphs structure for some European ctes s analyzed usng the methods of graph theory and clusterng. The number of edges and nodes of subway network graphs were calculated by the scheme takng nto account the fact that several transfer statons (from varous subway lnes) create one node of the graph. All subway schemes and ctes subway statstcs are avalable through Internet. Results: It s observer that the rdershp per year depends on parameters of the graph. In Pars and Madrd subway network graphs structure ndexes allow to suggest about the possblty to ncrease rdershp n these ctes n comparson wth the observed stuaton. Rdershp per year n Moscow, Sant Petersburg and Prague are optmal. At the same tme Pars and Madrd subway networks may have the largest passenger traffc (larger than n Moscow). Clusterng of examned networks based on graph characterstcs) nto clusters hghlghts the next clusters: London, Madrd, Moscow, Pars are characterzed by the hghest values of the graph connectvty parameters; for Berln, Sant Petersburg the values of parameters are above average; for Athens, Budapest, Hamburg, Kev, Mnsk, Prague, Kharkov the values of graph connectvty parameters are close to average; Warsaw, Dnepropetrovsk are characterzed by the lowest values of parameters. The structure of cluster of subway network graphs based on ndexes s not the same as the structure of clusters based on nput data. Concluson: The methodc for cty subway networks analyss on the bass of graph characterstcs (centralty, connectvty and shape) s proposed. The subways characterstcs are calculated from values of subways ndexes (number of lnes, number of statons, length n klometers, rdershp per year) and from ndcators of ctes urbanzaton (area and populaton). The relatonshp between graphs (roads) structure and weghts of ther edges, between π -ndex descrbng the shape of the graph and the number of passengers s demonstrated. It s shown on a practcal example that the analyss of structure of proposed road network graphs can be useful n determnng the sequence of new roads constructon. Clusterng of underground transport networks based on characterstcs of network graph structure was performed for the frst tme. The artcle entered release «Искусственный интеллект» 4 4

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