Mill Hill East. Finchley Central Bounds Green. East Finchley Wood Green. Highgate Turnpike Lane. Archway. Manor House. Gospel. Tufnell Park.
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1 h Special fares apply for printed single and return North White Holland Notting Lancaster Bond Oxford Chancery Acton City Park Hill Gate Gate Street Circus Holborn Lane West East Shepherd's Queensway Marble Tottenham St. Paul's Acton Acton Bush Arch Court Road 3 2 Bank Acton Kensington Covent Garden High Street bols ity Croxley Moor Park Ruislip Manor Eastcote rs Lane ly line service Rayners Lane rly mornings t ger ane ommon h Ealing s bury Hill ury Hill Harrow 50m ry Town Alperton rk Royal th Ealing Chiswick Park terley w nal 4 Northwood Northwood Hills Opens December 2005 Pinner North Harrow West Harrow Northwick Park South Kenton South Harrow North Wembley Wembley Central Stonebridge Park Harlesden Willesden Junction Acton Town South Acton Turnham Green Also served by Piccadilly line trains early mornings and Harrow & Wealdstone tickets to and from this station Preston Road Wembley Park Maida Vale Kilburn Park Warwick Avenue Paddington Royal Oak Westbourne Park Paddington Ladbroke Grove Latimer Road Central Shepherd's Bush Goldhawk Road Hammersmith Stamford Ravenscourt Brook Park Kenton C8 East Ham Kensal Rise Brondesbury Kensal Green Queen's Park (Olympia) Mondays - Saturdays open Sundays open Barons Court West Kensington Stanmore Canons Park Queensbury Kingsbury Brondesbury Park Earl's Court Neasden Dollis Hill Willesden Green Kilburn Edgware Road Bayswater Kensington Gloucester Road Edgware Edgware Road Closed until May 2006 Burnt Oak Knightsbridge South Kensington Hendon Central West Hampstead Colindale Marylebone Finchley Road Hyde Park Corner Swiss Cottage St. John's Wood Sloane Square Brent Cross Finchley Road & Frognal Victoria Golders Green 200m Hampstead Baker Street Green Park St. James's Park Piccadilly Circus 4 3 Great Portland Street Westminster Belsize Park Hampstead Heath Chalk Farm Camden Town No entry from the street on Sundays (exit and interchange only) Mornington Crescent Warren Street Regent s Park Goodge Street Euston Mill Hill East Euston Square Leicester Square Gospel Oak Kentish Town West Euston 200m Charing Cross Mansion House Embankment Charing Cross 00m Camden Road King's Cross St. Pancras Russell Square Temple Totteridge & Whetstone Southgate Woodside Park West Finchley Arnos Grove Finchley Central Bounds Green East Finchley Wood Green Highgate Turnpike Lane Archway Manor House Tufnell Park Kentish Town Blackfriars Caledonian Road Caledonian Road & Barnsbury A B C D King George V E F C3 Westbourne Park This diagram is an evolution of the original design conceived in 93 by Harry Beck Poster Grid Stations Facilities Zones Grid Stations Facilities Zones Grid Stations Facilities Zones Grid Stations Facilities Zones Grid Stations Facilities Zones Grid Stations Facilities Zones Grid Stations Facilities Zones Grid Stations Facilities Zones Grid Stations Facilities Zones Grid Stations Facilities Zones Grid Stations Facilities Zones Grid Stations Facilities Zones A4 Colindale Ÿ Á 4 Á µ 3/4 B8 Gants Hill 4 E Heathrow Terminals : µ 6 B3 Kensal Green 2 A8 Loughton Ÿ Á 6 A2 North Harrow Á : 5 D8 Poplar Á : µ 2 C5 Russell Square E5 Southwark Á µ D6 Tower Hill 2 F4 Colliers Wood 3 D8 East India Á µ 2/3 D3 Gloucester Road Á :, 2, 3 B3 Kensal Rise 2 M B Northolt Á : 5 B3 Preston Road : 4 S F4 South Wimbledon 3/4 B5 Tufnell Park 2 D3 West Brompton 2 D5 Covent Garden E3 East Putney Á 2/3 B4 Golders Green Á 3 E Heathrow Terminal 4 : 6 D3 Kensington (Olympia) : µ 2 C3 Maida Vale 2 B3 North Wembley 4 D9 Prince Regent Á µ 3 D4 St. James s Park B8 South Woodford Ÿ Á : 4 D2 Turnham Green 2/3 D7 Westferry µ 2 E8 Crossharbour µ 2 A4 Edgware Ÿ Á 5 D3 Goldhawk Road 2 A4 Hendon Central 3/4 B5 Kentish Town 2 B6 Manor House 2/3 B3 Northwick Park Á : 4 C8 Pudding Mill Lane µ 2/3 C4 St. John s Wood Á : 2 D2 Stamford Brook Á : 2 A6 Turnpike Lane 3 A5 West Finchley : µ 4 & London Arena C4 Edgware Road (Bakerloo) C5 Goodge Street D8 Heron Quays µ 2 B5 Kentish Town West 2 D5 Mansion House A2 Northwood Ÿ Á : 6 E3 Putney Bridge Á 2 C6 St. Paul s A3 Stanmore Ÿ Á : µ 5 U C8 West Ham µ 3 A2 Croxley Ÿ Á : A C4 Edgware Road B5 Gospel Oak 2 A5 High Barnet Ÿ Á : 5 A3 Kenton 4 C4 Marble Arch A2 Northwood Hills : 6 Q B7 Seven Sisters Á : 3 C7 Stepney Green 2 B9 Upminster Ÿ Á : µ 6 B4 West Hampstead : 2 D9 Custom House Á µ 3 (Circle/District/H&C) A9 Grange Hill : 5 B6 Highbury & Islington Á 2 E2 Kew Gardens Á : µ 3/4 C4 Marylebone Á : E9 North Woolwich 3 A3 Queensbury Ÿ Á : 4 D7 Shadwell 2 F5 Stockwell 2 B9 Upminster Bridge 6 B2 West Harrow : 5 F8 Cutty Sark Á µ 2/3 E5 Elephant & Castle /2 C5 Great Portland Street A5 Highgate Ÿ 3 B4 Kilburn Á : µ 2 C7 Mile End 2 C3 Notting Hill Gate /2 B3 Queen s Park 2 C3 Shepherd s Bush 2 B3 Stonebridge Park 3 C9 Upney : µ 4 D8 West India Quay µ 2 D9 Cyprus Á µ 3 B9 Elm Park µ 6 B Greenford Ÿ Á : 4 D3 High Street Kensington C3 Kilburn Park 2 A5 Mill Hill East Ÿ 4 O C3 Queensway (Central) C8 Stratford µ 3 C8 Upton Park 3 D3 West Kensington 2 D F7 Elverson Road Á µ 2/3 F7 Greenwich Á µ 2/3 A Hillingdon Ÿ Á µ 6 B3 Kingsbury Á : 4 D6 Monument A6 Oakwood Ÿ Á : 5 R D3 Shepherd s Bush izpisano: 2 B2 Sudbury Hill December : 4 A5, Uxbridge 2006Á µ ob6 5 D5: Westminster 40 µ B9 Dagenham East Á 5 D5 Embankment D4 Green Park : C5 Holborn C5 King s Cross St. Pancras Á : C6 Moorgate C6 Old Street D3 Ravenscourt Park Á 2 (Hammersmith & City) B2 Sudbury Town Ÿ Á : µ 4 V A West Ruislip Ÿ : 6 B9 Dagenham Heathway : µ 5 A8 Epping Ÿ Á : µ 6 E2 Gunnersbury Á 3 C3 Holland Park Á 2 D4 Knightsbridge A2 Moor Park Ÿ Á : 6/A D3 Olympia : µ 2 B2 Rayners Lane Ÿ Á : 5 C7 Shoreditch 2 E7 Surrey Quays Á 2 E4 Vauxhall /2 C7 Whitechapel : 2 B7 Dalston Kingsland 2 C5 Euston : H B6 Holloway Road 2 L F4 Morden Ÿ : 4 D Osterley Ÿ Á : 4 B8 Redbridge Ÿ Á : 4 E9 Silvertown 3 B4 Swiss Cottage Á 2 D4 Victoria : C3 White City : 2 A8 Debden Ÿ Á 6 C5 Euston Square B7 Hackney Central 2 B7 Homerton 2 C3 Ladbroke Grove Á 2 B5 Mornington Crescent 2 F5 Oval 2 C4 Regent s Park D4 Sloane Square T W B3 Willesden Green Á : 2/3 F7 Deptford Bridge Á µ 2/3 F B7 Hackney Wick 2 B9 Hornchurch Ÿ Á 6 E5 Lambeth North Á E8 Mudchute µ 2 C4 Oxford Circus E2 Richmond Ÿ Á : µ 4 B8 Snaresbrook Ÿ Á : 4 D5 Temple B7 Walthamstow Central Ÿ Á 3 B3 Willesden Junction Á : µ 3 C8 Devons Road µ 2 B9 Fairlop Ÿ : 5 A9 Hainault Ÿ Á : 5 E Hounslow Central Á 4 C4 Lancaster Gate N P A Rickmansworth Ÿ Á : A D2 South Acton 3 A8 Theydon Bois Ÿ : 6 B8 Wanstead Ÿ 4 E3 Wimbledon Ÿ Á : µ 3 B3 Dollis Hill : 3 C6 Farringdon : D3 Hammersmith Á µ 2 D Hounslow East Ÿ : µ 4 C3 Latimer Road 2 B3 Neasden : 3 C3 Paddington : A8 Roding Valley 5 D2 South Ealing 3 F4 Tooting Bec 3 D7 Wapping 2 E3 Wimbledon Park 3 E A5 Finchley Central Ÿ Á 4 B5 Hampstead 2/3 E Hounslow West Ÿ Á µ 5 D5 Leicester Square B9 Newbury Park Ÿ Á : 4 C2 Park Royal : 3 D7 Rotherhithe Á 2 E3 Southfields 3 F4 Tooting Broadway 3 C5 Warren Street A8 Woodford Ÿ Á : µ 4 C Ealing Broadway Á : 3 B4 Finchley Road 2 B5 Hampstead Heath 3 D4 Hyde Park Corner F7 Lewisham F7 New Cross Á µ 2 E3 Parsons Green Á 2 D9 Royal Albert Á µ 3 A6 Southgate 4 C5 Tottenham C3 Warwick Avenue 2 A6 Wood Green Á 3 Angel Farringdon Barbican Moorgate Cannon Street Holloway Road Arsenal Old Street Finsbury Park Seven Sisters Tottenham Hale Bethnal Green Blackhorse Road Walthamstow Central Mile End Buckhurst Hill Woodford South Woodford Snaresbrook Leytonstone Stratford Loughton Hackney Hackney Canonbury Central Wick Leyton Highbury & Kingsland Dalston Homerton Islington podiplomski študij Pudding Mill Lane Roding Valley Debden Chigwell Grange Hill Newbury Park Redbridge Wanstead East Ham Upton Park Dagenham Heathway Becontree Hornchurch Elm Park Harrowon-the-Hill Mondays - Fridays open Plaistow and Saturdays closed Sundays open Bow Road Bow Bromleyby-Bow West Ham Shoreditch Church Stepney Green Aldgate East Whitechapel m Devons Road Aldgate All Saints Canning Town Analiza omrežij Bus to London City Airport East Shadwell Westferry Poplar India Royal Victoria Monument Tower Limehouse Blackwall Custom House Hill for ExCeL 4. Fenchurch Street 50m Tower Zgradba omrežij: Gateway Wapping West Prince Regent West India Silvertown River Thames Quay Royal Albert Rotherhithe London Bridge Canary Wharf Beckton Park povezanosti North Pontoon Bermondsey Canada Greenwich Cyprus Dock Water Heron Quays West Brompton late evenings Waterloo & City line Waterloo Mondays - Fridays Gunnersbury Fulham Broadway South Quay Saturdays Parsons Green Crossharbour & Pimlico Southwark Borough London Arena Kew Gardens Putney Bridge Vladimir Batagelj Sundays closed Waterloo East Surrey Quays River Thames Lambeth Mudchute East Putney North Richmond Southfields Island Gardens Vauxhall Elephant & Castle Wimbledon Park Kennington Univerza v Ljubljani Wimbledon 3 00m At off-peak times most trains run to/from Morden via the Bank branch. To travel to/from the Charing Cross branch please change at Kennington. New Cross Gate New Cross Cutty Sark London Tube Oval for Maritime Greenwich Stockwell Metropolitan Interchange stations Greenwich Clapham North Improvement work to tracks and stations may Northern Clapham High Street 00m affect your journey, particularly at weekends. Connections with Deptford Bridge Piccadilly National Rail Clapham South Clapham For help planning your journey look for Common Elverson Road Victoria Brixton 00m publicity at stations, call Connections with Waterloo & City riverboat services Balham or visit Tooting Bec Lewisham Docklands Light Railway Airport Ljubljana, interchange Colliers Wood7. Tooting november Broadway 2006 / 0. in 7. november 2003 Connection with National Rail Tramlink South Wimbledon Morden Open Mondays - Saturdays Open Mondays - Fridays until 200 only Saturdays Liverpool Street Univerza v Ljubljani statistike No service between Woodford - Hainault after 2000 hours Gants Hill Upney Opens December 2005 Silvertown Hainault Fairlop Barkingside Dagenham East Barking No Hammersmith & City line service Whitechapel - Barking early mornings, late evenings or all day Sundays. North Woolwich D C B A London City Airport Opens December 2005 Station in Zone D Station in Zone C Station in Zone B Station in Zone A Station in Zone 6 and Zone A Station in Zone 6 Station in Zone 5 Station in Zone 4 Station in both zones Station in Zone 3 Station in Zone 2 Station in both zones Station in Zone Upminster Upminster Bridge Gallions Reach Beckton Explanation of zones
2 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti Kazalo Sprehodi Najkrajše poti Enakovrednosti in razbitja Povezanosti Posebni grafi dvodelni, drevo Skrčitev grafa Dvopovezanost k-povezanost Trikotniška povezanost neusmerjeni grafi Trikotniška povezanost usmerjeni grafi Pomembne točke v omrežju Normalizacija Stopnje Dostopnost Vmesnost Kazala in vsebine Nakopičenost
3 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 2 29 Usredinjenost omrežja Padgett-ove floretinske rodbine
4 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti Sprehodi Graf je acikličen, ntk. ne vsebuje nobenega cikla. dolžina s sprehoda s je število povezav, ki ga sestavljajo. s = (j, h, l, g, e, f, h, l, e, c, b, a) s = Sprehod je sklenjen ali obhod ntk. njegov začetek in konec sovpadata. Če ne upoštevamo smeri povezav v sprehodu, dobimo polsprehod ali verigo. sled sprehod z različnimi povezavami pot sprehod z različnimi točkami cikel sklenjen sprehod z različnimi notranjimi točkami.
5 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 2 Najkrajše poti Dolžino najkrajše poti iz u v v označimo z d(u, v). Če ne obstaja sprehod iz u v v postavimo d(u, v) =. d(j, a) = (j, h, d, c, b, a) = 5 d(a, j) = ˆd(u, v) = max(d(u, v), d(v, u)) je razdalja: ˆd(v, v) = 0, ˆd(u, v) = ˆd(v, u), ˆd(u, v) ˆd(u, t) + ˆd(t, v). Premer grafa je enak razdalji med, glede na d(u, v), najoddaljenejšima točkama: D = max u,v V d(u, v). Net / Paths between 2 vertices /
6 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 3 Najkrajše poti black lack back blank clack rack lace balk wack lick bank lank bask blink clank click rick race late bale walk bilk lice wick bane lane link bast clink chick rice rate hale wale bile wink bine wane line bait wast cline chink chic rite whale wile wine wait chine chit write while whine whit white DICT28.
7 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 4 Enakovrednosti in razbitja Relacija R na V je enakovrednost ntk. je refleksivna v V : vrv, simetrična u, v V : (urv vru) in tranzitivna u, v, z V : urz zrv urv. Vsaka enakovrednost R določa neko razbitje v razrede [v] = {u : vru}. Vsako razbitje C določa neko enakovrednost urv C C : u C v C. k-sosedi točke v je množica točk, ki so za k oddaljene od v, N k (v) = {u v : d(v, u) = k}. Množica vseh množic k-sosedov, k = 0,,... of v je razbitje množice V. k-soseščina točke v, N (k) (v) = {u v : d(v, u) k}. Net / k-neighbors /
8 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 5 Soseščina Motorole MasterCard MasterCard Fujitsu Fujitsu MapInfo MapInfo ComputerAssociates ComputerAssociates Nextel Nextel iplanet 3 3 iplanet #### Sun Cisco AOL IBM GM Motorola 365 #### Sun Cisco AOL IBM GM Motorola 3COM 3COM 2 MSFT 6 MSFT 22 BaltimoreTechnologies BaltimoreTechnologies Unisys Unisys AvantGo AvantGo Yahoo! Yahoo! Debelina povezav je koren iz vrednosti.
9 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 6 Povezanosti Točka u je dosegljiva iz točke v ntk. obstaja sprehod z začetkom v in koncem u. Točka v je šibko povezana s točko u ntk. obstaja veriga s krajiščema v in u. Točka v je krepko povezana s točko u ntk. sta vzajemno dosegljivi. Šibka in krepka povezanost sta enakovrednosti. graphcon.net Razredi porajajo šibke/krepke komponente ali kose grafa. Net / Components /
10 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 7 Šibke komponente Če preuredimo točke omrežja, tako da točke iz iste skupine šibkega razbitja postavimo skupaj, dobimo matrični prikaz sestavljen iz diagonalnih blokov šibkih komponent. Za večino problemov velja, da jih lahko ločeno rešimo za vsako šibko komponento posebej in nato dobljene rešitve združimo v rešitev za celotno omrežje.
11 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 8 Posebni grafi dvodelni, drevo Graf G = (V, L) je dvodelen ntk. lahko množico točk V razbijemo na podmnožici V in V 2, tako da ima vsaka povezava iz L eno krajišče v V drugo pa v V 2. Šibko povezan graf G je drevo ntk. ne vsebuje (zank in) polciklov dolžine vsaj 3.
12 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 9 Skrčitev grafa Če v danem grafu skrčimo vsako krepko komponento v ustrezno točko, odstranimo zanke in združimo vzporedne povezave, je tako dobljeni skrčeni graf acikličen. Za vsak aciklični graf obstaja urejenost / oštevilčenje i : V N, tako da velja (u, v) A i(u) < i(v).
13 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 0 Skrčitev grafa primer Net / Components / Strong [] Operations / Shrink Network / Partition [][0] Net / Transform / Remove / Loops [yes] Net / Partitions / Depth / Acyclic Partition / Make Permutation Permutation / Inverse select partition [Strong Components] Operations / Functional Composition / Partition*Permutation Partition / Make Permutation select [original network] File / Network / Export Matrix to EPS / Using Permutation Pajek - shadow [0.00,.00] b k i e c h a d e g f #a g j l a b c f h l d g i f k k j #e i e h a b c d g i j l f k
14 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti Kumar &: The Web as a graph Spletni metuljček (Bow-tie) Net / Partitions / Bow-Tie Naj bo S največja krepka komponenta v omrežju N ; W šibka komponenta, ki vsebuje S; I množica točk, iz katerih je S dosegljiva; O množica točk dosegljivih iz S; T (cevi) točke (niso v S) na poteh iz I v O; R = W \(I S O T ) (lovke); in D = V \ W. Razbitje {I, S, O, T, R, D} imenujemo metuljčno razbitje V.
15 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 2 Dvopovezanost Točki u in v sta dvopovezani ntk. sta povezani (v obe smeri) s po dvema neodvisnima (brez skupnih notranjih točk) potema. Dvopovezanost določa razbitje množice povezav. Točka je stična točka ali stičišče ntk. njena odstranitev poveča število šibkih komponent grafa. Povezava je most ntk. njena odstranitev poveča število šibkih komponent grafa. Net / Components / Bi-Components
16 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 3 k-povezanost Točkovna povezanost κ grafa G je enaka najmanjšemu številu točk, ki jih je potrebno odvzeti iz grafa, tako da je graf porojen s prestalimi točkami nepovezan ali trivialen (enak K ). Povezavna povezanost λ grafa G je enaka najmanjšemu številu povezav, ki jih je potrebno odvzeti iz grafa, tako da je graf porojen s prestalimi povezavami nepovezan ali trivialen. Velja Whitneyeva neenakost: κ(g) λ(g) δ(g). Graf G je (po točkah) k povezan, če je κ(g) k in je po povezavah k povezan, če je λ(g) k. Velja Whitneyeva različica Mengerjevega izreka: Graf G je po točkah/povezavah k povezan ntk. vsak par točk povezuje vsaj k po točkah/povezavah ločenih sprehodov.
17 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 4 Trikotniška povezanost neusmerjeni grafi V neusmerjenem grafu imenujemo trikotnik podgraf izomorfen K 3. Zaporedje trikotnikov (T, T 2,..., T s ) grafa G (točkovno) trikotniško povezuje točki u, v V ntk. u T in v T s ali u T s in v T ter V(T i ) V(T i ), i = 2,... s; in povezavno trikotniško povezuje točki u, v V ntk zadošča še strožji različici zadnjega pogoja E(T i ) E(T i ), i = 2,... s. Točkovna trikotniška povezanost je enakovrednost na točkah; povezavna pa na povezavah. Članek.
18 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 5 Trikotniško omrežje Naj bo G = (V, E) enostaven neusmerjen graf. Prirejeno trikotniško omrežje N T (G) = (V, E T, w) določeno z G je podgraf G T = (V, E T ) grafa G, kjer je E T množica tistih povezav iz E, ki leže na vsaj enem trikotniku. Utež w(e) povezave e E T je enaka številu različnih trikotnikov, ki jim povezava e pripada. Trikotniška omrežja omogočajo učinkovito razkrivanje gostih delov omrežja. Če povezava e pripada k-kliki podgrafu izomorfnemu K k v G, je w(e) k 2. Net / Count / 3-Rings
19 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 6 Povezavni prerez na ravni 6 trikotniškega omrežja Erdős-ovega grafa sodelovanj WORMALD, NICHOLAS C. LASKAR, RENU C. SHELAH, SAHARON MCKAY, BRENDAN D. HEDETNIEMI, STEPHEN T. MAGIDOR, MENACHEM KLEITMAN, DANIEL J. SAKS, MICHAEL E. CHUNG, FAN RONG K. GRAHAM, RONALD L. ARONOV, BORIS LINIAL, NATHAN PACH, JANOS POLLACK, RICHARD M. HENNING, MICHAEL A. FRANKL, PETER SPENCER, JOEL H. ALON, NOGA OELLERMANN, ORTRUD R. LOVASZ, LASZLO KOMLOS, JANOS GODDARD, WAYNE D. FUREDI, ZOLTAN TUZA, ZSOLT ALAVI, YOUSEF BABAI, LASZLO SZEMEREDI, ENDRE CHARTRAND, GARY BOLLOBAS, BELA HARARY, FRANK AJTAI, MIKLOS KUBICKI, GRZEGORZ SCHWENK, ALLEN JOHN brez Erdősa, n = 6926, m = 343 RODL, VOJTECH NESETRIL, JAROSLAV ROSA, ALEXANDER GYARFAS, ANDRAS SCHELP, RICHARD H. STINSON, DOUGLAS ROBERT LEHEL, JENO CHEN, GUANTAO MULLIN, RONALD C. FAUDREE, RALPH J. COLBOURN, CHARLES J. JACOBSON, MICHAEL S. PHELPS, KEVIN T.
20 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 7 Trikotniška povezanost usmerjeni grafi Če je graf G mešan, zamenjamo neusmerjene povezave s pari nasprotno usmerjenih. Naj bo G = (V, A) enostaven usmerjen graf brez zank. Za izbrano usmerjeno povezavo (u, v) A obstajajo štiri vrste usmerjenih trikotnikov: cyclic, transitive, input in output. cyc tra in out
21 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 8... Trikotniška povezanost usmerjeni grafi Če pozabimo na vlogo izbrane povezave, imamo le dve vrsti trikotnikov, ki jim povezava lahko pripada: ciklične (cyc) in tranzitivne (tra, in, out). V programu Pajek ukaz Net / Count / 3-Rings omogoča določiti ustrezna omrežja (N cyc ciklične uteži, N tra tranzitivnostne uteži, N sc tranzitivne bližnjice). Pojem trikotniške povezanosti lahko posplošimo na povezanost s kratkimi (pol)cikli obroči in ustrezna omrežja.
22 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 9 Povezavni prerez na ravni tranzitivnega trikotniškega omrežja slovarja ODLIS American Library Directory transaction log publication periodical serial suggestion box charge library call number review series Library Literature issue frequency colophon journal layout fixed location publishing printing blanket order American Library Association /ALA/ title page Books in Print /BIP/ vendor homepage International Standard Book Number /ISBN/ entry round table published price dummy librarian condition edition catalog plate fiction Oak Knoll bibliographic record imprint abstract dust jacket work book bibliography half-title editor library binding title table of contents /TOC/ index invoice new book text endpaper copyright book size parts of a book front matter collation publisher binding folio cover page Pajek
23 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 20 Pomembne točke v omrežju Pri izgradnji mer pomembnosti moramo najprej upoštevati ali je omrežje usmerjeno ali neusmerjeno. Meram pomembnosti na neusmerjenih omrežjih pravimo mere središčnosti; na usmerjenih omrežjih pa mere veljave. Slednje se naprej delijo na mere ugleda ali podpore (upoštevamo vstopajoče povezave) in mere vpliva (upoštavamo izstopajoče povezave). Če zamenjamo dano usmerjeno omrežje z njemu nasprotnim (obrnemo smeri povezav) preidejo mere ugleda v mere vpliva, in obratno. Dejanski pomen mere pomembnosti je odvisen od relacije (omrežja). Tako npr. je najuglednejša oseba glede na relacijo ne mara sodelovati z dejansko najmanj priljubljena oseba. Odstranitev pomembne točke iz omrežja povzroči občutno spremembo v zgradbi/delovanju omrežja.
24 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 2 Normalizacija Naj bo p : V R neka mera pomembnosti točk omrežja N = (V, L). Če želimo vrednosti mere p primerjati med različnimi omrežji, moramo poskrbeti za primerljivost. Pogosto jo poskušamo zagotoviti tako, da mero p normaliziramo. Naj bo N N(V), kjer je N(V) izbrana množica omrežij nad isto množico V, p max = max max p N (v) in p min = min min p N (v) N N(V) v V N N(V) v V Tedaj je normalizirana mera enaka p (v) = p(v) p min p max p min [0, ]
25 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 22 Stopnje Najpreprostejšo mero pomembnosti predstavljajo stopnje točk. Ker sta za enostavna omrežja deg min = 0 in deg max = n, je ustrezna normalizirana mera središčnost deg (v) = deg(v) n in podobno ugled indeg (v) = indeg(v) n vpliv outdeg (v) = outdeg(v) n Namesto stopenj glede na osnovno omrežje lahko vzamemo tudi stopnje glede na relacijo dosegljivosti (tranzitivna ovojnica). Net / Partitions / Degree Net / Partitions / Domain
26 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 23 Dostopnost Če upoštevamo razdalje d(u, v) med točkami v omrežju N = (V, L) lahko vpeljemo polmer r(v) = max u V d(v, u) Količino D = max u,v V d(v, u) imenujemo premer omrežja. skupna dostopnost S(v) = u V d(v, u) Za usmerjeno omrežje sta vpeljani meri meri vpliva. Meri ugleda dobimo, če v obrazcih d(u, v) zamenjamo z d(v, u). Če omrežje ni krepko povezano, sta r max in S max enaki. Sabidussi (966) je zato kot mero dostopnosti vpeljal /S(v) oziroma v normalizirani obliki dostopnost cl(v) = u V n d(v, u) Net / Vector / Centrality / Closeness
27 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 24 Vmesnost Pomembne so tudi točke, ki lahko nadzirajo pretok podatkov po omrežju. Če privzamemo, da so za prenos pomembne le najkrajše poti, dobimo kot mero vmesnosti (Anthonisse 97, Freeman 977) b(v) = (n )(n 2) u,t V:g u,t >0 u v,t v,u t g u,t (v) g u,t kjer je g u,t število najkrajših poti iz u v t; in g u,t (v) število takih med njimi, ki gredo skozi točko v. Net / Vector / Centrality / Betweenness
28 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 25 Kazala in vsebine Točkam povezanega usmerjenega omrežja N = (V, L) priredimo dve vrednosti: kakovost vsebine x v in kakovost kazala y v (Kleinberg, 998). Na dobro vsebino kažejo dobra kazala in dobro kazalo kaže na dobre vsebine x v = y u in y v = u:(u,v) L u:(v,u) L Naj bo W matrika omrežja N in x ter y vektorja obeh lastnosti. Tedaj lahko zvezi zapišemo x = W T y oziroma y = Wx. Začnimo z y = [,,..., ] in nato zaporedoma izračunamo po obeh zvezah nove približke za x in y. Oba vektorja po vsakem koraku normaliziramo. To ponavljamo dokler se vektorja ne ustalita. Pokazati je mogoče, da opisani postopek konvergira. Limitni vektor x je glavni lastni vektor matrike W T W; y pa matrike WW T. x u
29 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti Kazala in vsebine Podobni postopki se uporabljajo v spletnih iskalnikih za ocenjevanje pomembnosti posameznih strani. PageRank, PageRank / Google, HITS / AltaVista, SALSA, teorija. Net / Vector / Important Vertices / -Mode: & Authorities Hubs Na svetovnem nogometnem prvenstvu v Parizu leta 998 je sodelovalo 22 nogometnih reprezentanc. V omrežju so vse države, iz katerih so nogometaši igrali v ligah teh 22 držav, in vse države, v katerih ligah so igrali nogometaši iz teh 22 držav. Relacija je igralec iz države x igra v državi y; utež je število takih igralcev. Podatke je zbral Lothar Krempel. football.net
30 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti Kazala in vsebine: nogometaši IRN IRN KOR TUN AUT KOR TUN AUT FRA DEU CHE FRA DEU CHE JPN CMR BGR ZAF JPN CMR BGR ZAF PRT MAR BEL TUR NGA GRE DNK NOR PRT MAR BEL TUR NGA GRE DNK NOR NLD NLD PRY ESP YUG ROM HRV GBR SCO PRY ESP YUG ROM HRV GBR SCO ITA JAM ITA JAM MEX BRA ARG CHL USA MEX BRA ARG CHL USA COL COL Izvozniki (kazala/hubs) Uvozniki (vsebine/authorities)
31 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 28 Nakopičenost Nakopičenost v točki v je določena kot razmerje med številom vseh povezav v podgrafu G (v) porojenim s soseščino dane točke in številom povezav v polnem grafu na teh točkah C(v) = 2 L(G (v)) deg(v)(deg(v) ) za deg(v) > ; in C(v) = 0 sicer. Vpliv velikosti soseščine lahko zagotovimo z naslednjim popravkom C (v) = deg(v) C(v) kjer je največja stopnja v grafu G. Ta doseže največjo možno vrednost le na točkah, ki pripadajo osamljeni kliki reda. Net / Vector / Clustering Coefficients / CC
32 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 29 Usredinjenost omrežja Mero pomembnosti p : V R lahko povzamemo na celotnem omrežju kot njegovo usredinjenost C(p): p = max v V p(v) D(p) = v V(p p(v)) D = Tedaj je usredinjenost glede na p max D(p N ) N N(V) C(p) = D(p) D Za večino mer je najbolj usredinjena zvezda S n in najmanj polni graf K n.
33 V. Batagelj: Analiza omrežij, 4. Zgradba omrežij povezanosti 30 Ginori Lamberteschi Albizzi Guadagni Padgett-ove floretinske rodbine Acciaiuoli Tornabuoni Bischeri Medici Ridolfi Strozzi Salviati Peruzzi Pazzi Barbadori Castellani close between. Acciaiuoli Albizzi Barbadori Bischeri Castellani Ginori Guadagni Lamberteschi Medici Pazzi Peruzzi Ridolfi Salviati Strozzi Tornabuoni
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