5. Zgradba omrežij povezanosti. Vladimir Batagelj. Interdisciplinarni doktorski študijski program Statistika Ljubljana, april 2014
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1 5. Zgradba povezanosti Vladimir Batagelj Univerza v Ljubljani, FMF, matematika Interdisciplinarni doktorski študijski program Statistika Ljubljana, april 204
2 Textphone Website Transport for London Grid Stations Facilities Zones Grid Stations Facilities Zones D9 Beckton Park Á µ 3 D2 Acton Central 2 C9 Becontree Á : 5 D2 Acton Town 3 B5 Belsize Park 2 D6 Aldgate D6 Bermondsey Á µ 2 D7 Aldgate East C7 Bethnal Green 2 D8 All Saints µ 2 D5 Blackfriars B2 Alperton : 4 B7 Blackhorse Road Ÿ Á 3 A Amersham Ÿ Á : D D8 Blackwall Á µ 2 C6 Angel C4 Bond Street B5 Archway 2/3 E6 Borough on most trains in the early mornings late evenings tickets to and from this station Closed until May 2006 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 A4 Colindale Ÿ Á 4 C8 East Ham 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 B6 Caledonian Road µ 2 F4 Colliers Wood 3 D8 East India Á µ 2/3 D3 Gloucester Road Á :, 2, 3 B3 Kensal Rise 2 B Northolt Á : 5 B3 Preston Road : B6 Caledonian Road 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 & Barnsbury 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 B5 Camden Road 2 & 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 B5 Camden Town 2 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 D7 Canada Water Á : µ 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 : D8 Canary Wharf : µ 2 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 D8 Canning Town Á : µ 3 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 C3 Queensway D6 Cannon Street 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 B7 Canonbury 2 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 A3 Canons Park Ÿ Á : 5 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 : 200m (exit and interchange only) Euston 200m Charing Cross 00m Waterloo East 00m 00m Open Mondays - Saturdays Saturdays Sundays closed Fenchurch Street 50m 0 Kazalo Tube map Sponsored by µ Step-free access Stations displaying this symbol in the index have step-free access between the street and platforms. This facility is useful for passengers with luggage, shopping or buggies as well as for wheelchair users To plan a journey in a wheelchair, see our leaflet Tube access guide or call A B C D CLatimer North Ealing North White Acton City Ealing Broadway Station facilities West East Acton Acton The index on this map also shows Acton 2 Á Bicycle parking Central Car parks Ealing Common Shepherd's South Bush 2 Stations with toilets on site Acton or nearby Goldhawk Road Travel Information Centres D Acton Town South Ealing Hammersmith Covent Garden station gets 3 very busy at weekends and in Northfields the evenings, but you can avoid the crowds by walking there from Holborn, Leicester Square or Charing Cross. The short walk is clearly signposted above ground and maps are on display at each station. Travel information For journey planning and travel advice call 24 hour travel information E B Key to lines and symbols A Chesham High Barnet Cockfosters Change at Chalfont & Latimer Chalfont & Watford Oakwood Totteridge & Whetstone Amersham Croxley Chorleywood Southgate Mill Hill East Woodside Park Rickmansworth Moor Park Stanmore Edgware Harrow & West Finchley Arnos Grove Northwood Wealdstone West Ruislip Special fares apply for Burnt 4 Finchley Central printed single and return Northwood Canons Park Oak Bounds Green Hills Hillingdon Ruislip Colindale East Finchley Ruislip Manor Queensbury Wood Green Pinner Hendon Central Uxbridge Ickenham Eastcote North Harrow Kenton Highgate Turnpike Lane Harrowon-the-Hill Kingsbury Brent Cross S Preston Ruislip Rayners Lane Road Golders Green 3 Archway Manor House No Piccadilly line service Gardens Uxbridge - Rayners Lane West Harrow Northwick Neasden Hampstead Hampstead Gospel Tufnell Park South Park Wembley Heath Oak Ruislip South Kenton Park Dollis Hill Northolt Finchley Road Arsenal Finsbury South Harrow North Wembley Willesden Green Kentish & Frognal Kentish Park Belsize Park Town West Town Holloway Road Wembley Central Kilburn Sudbury Hill Stonebridge Park Brondesbury West Chalk Farm Caledonian Road Sudbury Hill Harrow 50m Park Harlesden Hampstead Caledonian Greenford Camden Town Camden No entry from the street Sudbury Town Road Road & Willesden Junction Kensal Rise Brondesbury on Sundays Finchley Road Barnsbury Canonbur Kensal Green Swiss Cottage Alperton Mornington Queen's Park Highbury & St. John's Wood Crescent King's Cross Islington St. Pancras Maida Vale Great Perivale Kilburn Park Edgware Baker Portland Euston Angel Warwick Avenue Paddington Road Street Street Royal Oak Hanger Westbourne Park Farringdon Old Street Warren Street Euston Lane Paddington Edgware Marylebone Square Barbican Road Regent s Park Russell Liverpool Ladbroke Grove Square Street Park Royal Latimer Road Bayswater Moorgate Holland Notting Lancaster Goodge Bond Oxford Chancery Park Hill Gate Gate Street Street Circus Holborn Lane Aldg Shepherd's Queensway Marble Tottenham St. Paul's Bush E Arch Court Road Kensington Covent Garden Bank Aldgate (Olympia) High Street Hyde Park Green Park Mondays - Saturdays Kensington Corner Leicester Cannon Street open Sundays Piccadilly Square Mansion open Circus House Heathrow Terminals, 2, 3 Heathrow Terminal 4 Hounslow West Bus service Boston Manor Chiswick Hounslow Park East Osterley Bakerloo Metropolitan Central Northern wiki: Piccadilly Circle Other Railways F Victoria April 4, 204/ marec 203 District Waterloo & City For a map of all Railways in East London Docklands Light Railway Greater London, consult the Opens High Frequency Services Map nearby Hammersmith & City December 2005 Jubilee Hounslow Central Hatton Cross for Heathrow Terminal 4 Turnham Green Also served by Piccadilly line trains early mornings and Gunnersbury Kew Gardens Richmond Stamford Brook National Rail Barons Court Ravenscourt West Park Kensington West Brompton Fulham Broadway Parsons Green Putney Bridge East Putney Southfields Wimbledon Park Wimbledon London Tube Earl's Court Interchange stations Connections with National Rail Connections with riverboat services Airport interchange Connection with Tramlink Knightsbridge Gloucester Road Sloane Square South Victoria Kensington River Thames Pimlico Vauxhall St. James's Park Westminster Colliers Wood Tooting Broadway South Wimbledon Morden 5 Waterloo Kennington Stockwell Clapham North Clapham High Street 00m Clapham South Clapham Common Balham Tooting Bec Oval Charing Cross Embankment Lambeth North Brixton Temple Waterloo & City line Mondays - Fridays Southwark Blackfriars Elephant & Castle 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. Open Mondays - Fridays until 200 only Saturdays Borough Monument Tower Hill Tower Gatewa River Thames Rothe London Bridge Bermondsey C New Cross Gate Surrey Q Improvement work to tracks and stations ma affect your journey, particularly at weekend For help planning your journey look for publicity at stations, call or visit A Index to stations C V. D Batagelj M O Q R
3 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 vozlišči cikel sklenjen sprehod z različnimi notranjimi vozlišči.
4 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 vozliščema: D = max u,v V d(u, v). Network/Create New Network/Subnetwork with Paths/
5 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 DICT28. white
6 Enakovrednosti in razbitja Relacija R na V je enakovrednost (ekvivalenčna) 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 v je množica vozlišč, ki so za k oddaljena 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 v, N (k) (v) = {u v : d(v, u) k}. Network/Create Partition/k-Neighbors/
7 Soseščina Motorole #### AOL MSFT Yahoo! Sun IBM Cisco iplanet 3COM Motorola AvantGo BaltimoreTechnologies Unisys ComputerAssociates Nextel MapInfo GM MasterCard Fujitsu #### AOL MSFT Yahoo! Sun IBM Cisco iplanet 3COM Motorola AvantGo BaltimoreTechnologies Unisys ComputerAssociates Nextel MapInfo GM MasterCard Fujitsu Debelina povezav je koren iz vrednosti.
8 Vozlišče u je dosegljivo iz v ntk. obstaja sprehod z začetkom v in koncem u. Vozlišče v je šibko povezano z vozliščem u ntk. obstaja veriga s krajiščema v in u. Vozlišče v je krepko povezano z vozliščem u ntk. sta vzajemno dosegljivi. Šibka in krepka povezanost sta enakovrednosti. graphcon.net Razredi porajajo šibke/krepke komponente ali kose grafa. Network/Create Partition/Components/
9 Šibke komponente Če preuredimo omrežja, tako da 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.
10 Posebni grafi dvodelni, drevo Graf G = (V, L) je dvodelen ntk. lahko množico vozlišč 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.
11 Krepka skrčitev grafa (kondenzacija) Če v danem grafu skrčimo vsako krepko komponento v ustrezno vozlišče, 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).
12 i Skrčitev grafa primer Network/Create Partition/Components/Strong [] Operations/Network+Partition/Shrink Network [][0] Network/Create New Network/Transform/Remove/Loops [yes] Network/Acyclic Network/Depth Partition/Acyclic Partition/Make Permutation Permutation/Inverse Permutation select partition [Strong Components] Operations/Partition+Permutation/Functional Composition/Partition*Permutation Partition/Make Permutation select [original network] File/Network/Export Matrix to EPS/Using Permutation Pajek - shadow [0.00,.00] a f b d h c e l i g k f #a g i e h j l a b c d g f k k j #e i e h a b c d g k j l f
13 Zgradba krepkih komponent Notranja zgradba krepke komponente naj bo d največji skupni delitelj dolžin obhodov v krepki komponenti. Komponenta je enostavna, če je d = ; sicer je periodična s periodo d. Množico vozlišč V krepko povezanega usmerjenega grafa G = (V, R) je mogoče razbiti na d skupin V, V 2,..., V d, tako da za vsako povezavo (u, v) R velja u V i v V (imod d)+. Network/Create Partition/ Components/Strong-Periodic
14 ... Zgradba krepkih komponent Bonhourejev periodični graf. Pajek matrike
15 Spletni metuljček (Bow-tie) Naj bo S največja krepka komponenta v omrežju N ; W šibka komponenta, ki vsebuje S; I množica vozlišč, iz katerih je S dosegljiva; O množica vozlišč dosegljivih iz S; T (cevi) (niso v S) na poteh iz I v O; R = W \ (I S O T ) (lovke); in D = V \ W. Razbitje Kumar &: The Web as a graph {I, S, O, T, R, D} imenujemo metuljčno razbitje V. Opomba: za splošna usmerjena omrežja slika ne prikazuje vseh možnosti v množici R so lahko tudi verige. Network/Create Partition/Bow-Tie
16 Dvopovezanost Vozlišči u in v sta dvopovezani ntk. sta povezani (v obe smeri) s po dvema neodvisnima (brez skupnih notranjih vozlišč) potema. Dvopovezanost določa razbitje množice povezav. Vozlišče je stično vozlišče ali stičišče ntk. njegova odstranitev poveča število šibkih komponent grafa. Povezava je most ntk. njena odstranitev poveča število šibkih komponent grafa. Network/Create New Network/with Bi-Connected Components
17 k-povezanost Vozliščna povezanost κ grafa G je enaka najmanjšemu številu vozlišč, ki jih je potrebno odvzeti iz grafa, tako da je graf porojen s prestalimi vozlišči 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 vozliščih) 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 vozliščih/povezavah k povezan ntk. vsak par vozlišč povezuje vsaj k po vozliščih/povezavah ločenih sprehodov.
18 Trikotniška povezanost neusmerjeni grafi V neusmerjenem grafu imenujemo trikotnik podgraf izomorfen K 3. Zaporedje trikotnikov (T, T 2,..., T s) grafa G (vozliščno) trikotniško povezuje vozlišči 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 vozlišči u, v V ntk zadošča še strožji različici zadnjega pogoja E(T i ) E(T i ), i = 2,... s. Vozliščna trikotniška povezanost je enakovrednost na vozliščih; povezavna pa na povezavah. Članek.
19 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. Network/Create New Network/with Ring Counts/3-Rings/Undirected
20 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. CHUNG, FAN RONG K. GRAHAM, RONALD L. SAKS, MICHAEL E. 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.
21 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
22 ... 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 Network/Create New Network/with Ring Counts/3-Rings/Directed 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; lahko pa tudi na povezanost z majhnimi podgrafi (npr. klikami, glejte Palla).
23 Povezavni prerez na ravni tranzitivnega trikotniškega omrežja slovarja ODLIS serial publication American Library Directory transaction log periodical suggestion box review frequency charge series issue library call number Library Literature 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
24 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 pomembnega iz omrežja povzroči občutno spremembo v zgradbi/delovanju omrežja.
25 Normalizacija Naj bo p : V R neka mera pomembnosti vozlišč 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 nad isto množico V, p max = max max N N(V) v V pn (v) in p min = min Tedaj je normalizirana mera enaka p (v) = p(v) p min p max p min [0, ] min N N(V) v V pn (v)
26 Stopnje Najpreprostejšo mero pomembnosti predstavljajo stopnje vozlišč. 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 vpliv indeg (v) = indeg(v) n outdeg (v) = outdeg(v) n Namesto stopenj glede na osnovno omrežje lahko vzamemo tudi stopnje glede na relacijo dosegljivosti (tranzitivna ovojnica). Network/Create Partition/Degree Network/Create Vector/Centrality/Degree Network/Create Vector/Centrality/Proximity Prestige
27 Dostopnost Če upoštevamo razdalje d(u, v) med vozlišči 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 n dostopnost cl(v) = d(v, u) u V Network/Create Vector/Centrality/Closeness Network/Create New Network/Subnetwork with Paths/Info on Diameter
28 Vmesnost so tudi, 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:gu,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 vozlišče v. Hiter postopek za izračun vmesnosti je razvil Brandes. Network/Create Vector/Centrality/Betweenness
29 Padgett-ove floretinske rodbine Ginori Lamberteschi Albizzi Guadagni Pazzi Acciaiuoli Salviati Medici Barbadori Tornabuoni Ridolfi Castellani Strozzi Bischeri close between. Acciaiuoli Albizzi Barbadori Bischeri Castellani Ginori Guadagni Lamberteschi Medici Pazzi Peruzzi Ridolfi Salviati Strozzi Tornabuoni Peruzzi
30 Kazala in viri Vozliščem povezanega usmerjenega omrežja N = (V, L) priredimo dve vrednosti: kakovost vira (vsebine) x v in kakovost kazala y v (Kleinberg, 998). Na dober vir kažejo dobra kazala in dobro kazalo kaže na dobre vire x v = y u in y v = u:(u,v) L u:(v,u) L x u 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.
31 ... Kazala in viri Podobni postopki se uporabljajo v spletnih iskalnikih za ocenjevanje pomembnosti posameznih strani. PageRank, PageRank / Google, HITS / AltaVista, SALSA, teorija. Network/Create Vector/Centrality/Hubs-Authorities Na naslednji prosojnici: 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
32 ... Kazala in viri: nogometaši IRN IRN KOR TUN AUT KOR TUN AUT FRA DEU CHE FRA DEU CHE JPN CMR PRT MAR BGR ZAF BEL NGA TUR GRE DNK NOR JPN CMR PRT MAR BGR ZAF BEL NGA TUR 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 (viri/authorities)
33 Nakopičenost Nakopičenost v vozlišču v je določena kot razmerje med številom vseh povezav v podgrafu G (v) porojenim s soseščino danega in številom povezav v polnem grafu na teh vozliščih C(v) = za deg(v) > ; in C(v) = 0 sicer. 2 L(G (v)) deg(v)(deg(v) ) 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 vozliščih, ki pripadajo osamljeni kliki reda. Network/Create Vector/Clustering Coefficients/CC2 članek
34 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 N N(V) D(pN ) C(p) = D(p) D Za večino mer je najbolj usredinjena zvezda S n in najmanj polni graf K n.
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