Univerza v Ljubljani podiplomski studijˇ statistikevlado.fmf.uni-lj.si/vlado/podstat/AO/2006/anom04.pdf · B9 Fairlop C6 Farringdon A5 Finchley Central B4 Finchley Road B4 Finchley

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Quad Royal Tube Map Version 2 Date

Size

1/9/2005

4 Colour Process + 4 Self Colours S/S

Pantone®

485 CPantone®

470 CPantone®

235 CPantone®

072 C

Process colours

River Thames

River Thames

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1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

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Tubemap

D2 Acton CentralD2 Acton TownD6 AldgateD7 Aldgate EastD8 All SaintsB2 AlpertonA1 AmershamC6 AngelB5 ArchwayA6 Arnos GroveB6 Arsenal

C4 Baker Street F4 BalhamD6 BankC6 BarbicanC9 BarkingB9 BarkingsideD3 Barons CourtC3 BayswaterE9 Beckton

D9 Beckton ParkC9 BecontreeB5 Belsize ParkD6 BermondseyC7 Bethnal GreenD5 BlackfriarsB7 Blackhorse RoadD8 BlackwallC4 Bond StreetE6 BoroughD1 Boston ManorA6 Bounds GreenC8 Bow ChurchC7 Bow RoadB4 Brent CrossF5 Brixton C8 Bromley-by-BowB3 BrondesburyB3 Brondesbury ParkA8 Buckhurst Hill A4 Burnt Oak

B6 Caledonian RoadB6 Caledonian Road

& BarnsburyB5 Camden RoadB5 Camden TownD7 Canada WaterD8 Canary WharfD8 Canning TownD6 Cannon StreetB7 CanonburyA3 Canons ParkA1 Chalfont & LatimerB5 Chalk FarmC5 Chancery LaneD5 Charing CrossA1 CheshamA9 ChigwellD2 Chiswick ParkA1 ChorleywoodF4 Clapham CommonF4 Clapham North F4 Clapham SouthA6 Cockfosters

A4 ColindaleF4 Colliers WoodD5 Covent GardenE8 Crossharbour

& London ArenaA2 CroxleyD9 Custom HouseF8 Cutty SarkD9 Cyprus

B9 Dagenham EastB9 Dagenham HeathwayB7 Dalston KingslandA8 DebdenF7 Deptford BridgeC8 Devons RoadB3 Dollis Hill

C1 Ealing BroadwayD2 Ealing CommonD3 Earl's CourtC2 East ActonA2 EastcoteA5 East Finchley

C8 East HamD8 East IndiaE3 East PutneyA4 EdgwareC4 Edgware Road (Bakerloo)C4 Edgware Road

(Circle/District/H&C)E5 Elephant & CastleB9 Elm ParkF7 Elverson RoadD5 EmbankmentA8 EppingC5 EustonC5 Euston Square

B9 FairlopC6 FarringdonA5 Finchley CentralB4 Finchley RoadB4 Finchley Road & FrognalB6 Finsbury ParkE3 Fulham Broadway

E9 Gallions Reach

B8 Gants HillD3 Gloucester RoadB4 Golders GreenD3 Goldhawk RoadC5 Goodge StreetB5 Gospel OakA9 Grange Hill C5 Great Portland StreetB1 GreenfordF7 GreenwichD4 Green ParkE2 Gunnersbury

B7 Hackney CentralB7 Hackney WickA9 HainaultD3 HammersmithB5 HampsteadB5 Hampstead HeathC2 Hanger LaneB3 HarlesdenA3 Harrow & Wealdstone B2 Harrow-on-the HillE1 Hatton Cross

E1 Heathrow Terminals 1, 2, 3

E1 Heathrow Terminal 4A4 Hendon CentralD8 Heron QuaysA5 High BarnetB6 Highbury & IslingtonA5 HighgateD3 High Street KensingtonA1 HillingdonC5 Holborn C3 Holland ParkB6 Holloway RoadB7 HomertonB9 HornchurchE1 Hounslow CentralD1 Hounslow EastE1 Hounslow WestD4 Hyde Park Corner

A1 IckenhamE8 Island Gardens

E5 Kennington

B3 Kensal GreenB3 Kensal RiseD3 Kensington (Olympia)B5 Kentish TownB5 Kentish Town WestA3 KentonE2 Kew GardensB4 KilburnC3 Kilburn ParkB3 KingsburyC5 King’s Cross St. PancrasD4 Knightsbridge

C3 Ladbroke GroveE5 Lambeth NorthC4 Lancaster GateC3 Latimer RoadD5 Leicester SquareF7 LewishamB8 LeytonB8 LeytonstoneD7 LimehouseC6 Liverpool StreetD6 London Bridge

A8 Loughton

C3 Maida ValeB6 Manor HouseD5 Mansion HouseC4 Marble ArchC4 MaryleboneC7 Mile EndA5 Mill Hill EastD6 MonumentC6 MoorgateA2 Moor ParkF4 MordenB5 Mornington Crescent E8 Mudchute

B3 NeasdenB9 Newbury ParkF7 New CrossF7 New Cross GateC2 North ActonC2 North EalingD1 NorthfieldsD8 North Greenwich

A2 North HarrowB1 NortholtB3 North WembleyB3 Northwick ParkA2 NorthwoodA2 Northwood HillsE9 North WoolwichC3 Notting Hill Gate

A6 OakwoodC6 Old StreetD3 OlympiaD1 OsterleyF5 Oval C4 Oxford Circus

C3 PaddingtonC2 Park RoyalE3 Parsons GreenC1 PerivaleD5 Piccadilly CircusE4 PimlicoA2 PinnerC8 Plaistow

D8 PoplarB3 Preston RoadD9 Prince RegentC8 Pudding Mill LaneE3 Putney Bridge

A3 QueensburyB3 Queen’s ParkC3 Queensway

D3 Ravenscourt ParkB2 Rayners LaneB8 RedbridgeC4 Regent’s ParkE2 RichmondA1 RickmansworthA8 Roding ValleyD7 RotherhitheD9 Royal AlbertC3 Royal OakD9 Royal VictoriaA1 RuislipB1 Ruislip GardensA2 Ruislip Manor

C5 Russell Square

D4 St. James’s ParkC4 St. John’s WoodC6 St. Paul’sB7 Seven SistersD7 ShadwellC3 Shepherd’s Bush

(Central)D3 Shepherd’s Bush

(Hammersmith & City)C7 Shoreditch E9 SilvertownD4 Sloane SquareB8 SnaresbrookD2 South ActonD2 South EalingE3 SouthfieldsA6 SouthgateB2 South HarrowD4 South KensingtonB3 South KentonE8 South QuayB1 South Ruislip

E5 SouthwarkF4 South WimbledonB8 South WoodfordD2 Stamford BrookA3 StanmoreC7 Stepney GreenF5 StockwellB3 Stonebridge ParkC8 StratfordB2 Sudbury HillB2 Sudbury Town E7 Surrey QuaysB4 Swiss Cottage

D5 TempleA8 Theydon BoisF4 Tooting BecF4 Tooting BroadwayC5 Tottenham

Court RoadB7 Tottenham HaleA5 Totteridge &

WhetstoneD7 Tower Gateway

D6 Tower HillB5 Tufnell ParkD2 Turnham Green A6 Turnpike Lane

B9 UpminsterB9 Upminster BridgeC9 UpneyC8 Upton ParkA1 Uxbridge

E4 VauxhallD4 Victoria

B7 Walthamstow CentralB8 WansteadD7 WappingC5 Warren StreetC3 Warwick AvenueE5 WaterlooA2 WatfordB3 Wembley CentralB3 Wembley ParkC2 West Acton

C3 Westbourne ParkD3 West BromptonD7 WestferryA5 West FinchleyC8 West HamB4 West HampsteadB2 West HarrowD8 West India QuayD3 West KensingtonD5 WestminsterA1 West RuislipC7 WhitechapelC3 White CityB3 Willesden GreenB3 Willesden JunctionE3 WimbledonE3 Wimbledon ParkA8 WoodfordA6 Wood GreenA5 Woodside Park

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Grid Stations 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 Grid Stations Facilities Zones Grid Stations Facilities Zones Grid Stations Facilities ZonesIndex to stations

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Travel information

Step-free accessStations displaying this symbolin the index have step-freeaccess between the street andplatforms. This facility is usefulfor passengers with luggage,shopping or buggies as well asfor wheelchair users

To plan a journey in a wheelchair,see our leaflet ‘Tube access guide’ or call

0845 330 9880

For journey planning and travel advice call

µ

Station facilitiesThe index on this map also shows

Other RailwaysFor a map of all Railways in Greater London, consult the High Frequency Services Map nearby

‰ Car parksBicycle parking

Stations with toilets on siteor nearby

Á

∑

Travel Information Centres∏

Facilities

Transport for London

Key to lines and symbols

Central

Bakerloo

District

Circle

East London Docklands Light Railway

Northern

Metropolitan

Piccadilly

Victoria

Waterloo & City

National Rail

Connections withNational Rail

Connection withTramlink

Airport interchange

Connections withriverboat services

Interchange stations

Poster 09.05

OpensDecember 2005

020 7222 123424 hour travel information

020 7918 3015Textphone

www.tfl.gov.ukWebsite

020 7918 3015Textphone

www.tfl.gov.ukWebsite

Jubilee

Hammersmith & City

Covent Garden station gets very busy at weekends and in 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.

This diagram is an evolution of the original design conceived in 1931 by Harry Beck

Bermondsey

SouthwarkWaterloo East

Chalfont &Latimer

Moor Park

NorthwoodNorthwoodHills

Pinner

Eastcote North Harrow

Maida Vale

Queen's ParkKensal Green

Neasden

Dollis Hill

Willesden Green

Kilburn

WestHampstead

Swiss CottageSt. John's Wood

Finchley Road

Amersham

Ruislip Manor

Chesham

Chorleywood

Rickmansworth

Watford

Croxley

Harrow-on-the-Hill

PrestonRoad

Hillingdon Ruislip

Rayners Lane

West Harrow NorthwickPark Wembley

Park

Ealing Common

EalingBroadway

GreatPortland

StreetBakerStreet

FarringdonBarbican

Moorgate

Aldgate

EustonSquare

ActonTown

ChiswickPark

TurnhamGreen

WestActon

EastActon

Shepherd'sBush

StamfordBrook

RavenscourtPark

Hammersmith

WestKensington

West Brompton

Fulham Broadway

Parsons Green

Putney Bridge

East Putney

Southfields

Wimbledon Park

Wimbledon

VictoriaSouthKensington

GloucesterRoad

Embankment

Blackfriars

MansionHouse

Temple

Cannon Street

Bank

Monument

BaronsCourt

Fenchurch Street

Whitechapel

TowerGateway

TowerHill

AldgateEast

Stepney Green

Mile End

BowRoad Bow

ChurchBromley-by-Bow

West Ham

Plaistow

Upton Park

East Ham

Becontree

DagenhamHeathway

Elm Park

Upney

DagenhamEast

Hornchurch

UpminsterBridge

Upminster

High StreetKensington

NottingHill Gate

Bayswater

Kensal Rise Brondesbury

EdgwareRoad

St. James'sPark

SloaneSquare

Westminster

Barking

Latimer Road

Westbourne Park

Finchley Road& Frognal

Ladbroke Grove

Royal Oak

Shepherd'sBush

Goldhawk Road

West Ruislip

Greenford

RuislipGardens

SouthRuislip

Northolt

HangerLane

Perivale

NorthActon

WhiteCity

HollandPark

Paddington

Paddington

ChanceryLaneBond

StreetOxfordCircus

TottenhamCourt Road

St. Paul'sMarbleArch

Queensway

LancasterGate

BethnalGreen

Stratford

Leyton

Leytonstone

Snaresbrook

SouthWoodford

Woodford

Epping

Theydon Bois

DebdenLoughton

Buckhurst Hill

Redbridge

ChigwellRodingValley

Hainault

Fairlop

BarkingsideNewbury

Park

GrangeHill

Wanstead GantsHill

South Ealing

Knightsbridge

Hyde ParkCorner

Green Park

PiccadillyCircus

LeicesterSquare

RussellSquare

Caledonian Road

CaledonianRoad &

Barnsbury

DalstonKingsland

Homerton

Holloway Road

Arsenal

Manor House

Turnpike Lane

Wood Green

Bounds Green

Arnos Grove

Southgate

Oakwood

Cockfosters

Uxbridge Ickenham

ActonCentral

Waterloo

Morden

Colliers Wood Tooting Broadway

South Wimbledon

Tooting BecBalham

Clapham South ClaphamCommon

Clapham NorthClapham High Street 100m

StockwellOval

Kennington

Borough

SouthActon

Old Street

Angel

GoodgeStreet

Euston

MorningtonCrescent

Camden Town

Chalk Farm

Regent’s Park

Belsize Park

Hampstead HampsteadHeath

GospelOak

CanonburyHackneyCentral

HackneyWick

KentishTown West

CamdenRoad

Hendon Central

Colindale

BurntOak

Mill Hill East

High Barnet

Totteridge & Whetstone

Woodside Park

West Finchley

Finchley Central

East Finchley

Highgate

Archway

Tufnell Park

KentishTown

CanadaWater

Canary Wharf

Elverson Road

Deptford Bridge

Harrow &Wealdstone

Kenton

Stanmore

Canons Park

Queensbury

Kingsbury

South KentonNorth Wembley

Wembley Central

Stonebridge ParkHarlesden

Willesden Junction

Kilburn ParkWarwick Avenue

EdgwareRoad

BrondesburyPark

Marylebone

LambethNorth

Elephant & Castle

King's CrossSt. Pancras

CharingCross

Covent Garden

Highbury &Islington

BlackhorseRoad

SevenSisters

WalthamstowCentral

TottenhamHale

FinsburyPark

Pimlico

Brixton

Shoreditch

Wapping

Rotherhithe

Surrey Quays

New CrossNew Cross Gate

Vauxhall

Limehouse

Westferry

DevonsRoad

PuddingMill Lane

West IndiaQuay

Cutty Sarkfor Maritime Greenwich

Greenwich

Lewisham

Blackwall

EastIndia

Warren Street

Edgware

All Saints

Heron Quays

South Quay

Crossharbour &London Arena

Mudchute

Island Gardens

Shadwell

No service between Woodford - Hainault

after 2000 hours

Gunnersbury

Richmond

Kew Gardens

Poplar

London Bridge

Change atChalfont & Latimer

on most trains

No Piccadilly line serviceUxbridge - Rayners Lane

in the early mornings

Special fares apply forprinted single and return

tickets to and from this station

Also served byPiccadilly linetrains early

mornings andlate evenings

100m

100m

Euston 200m

150m

Charing Cross 100m

200m

HeathrowTerminals

1, 2, 3

Hounslow Central

Osterley

Northfields

Boston ManorHounslow

East

HounslowWest

LiverpoolStreet

No Hammersmith & City line serviceWhitechapel - Barking early mornings,late evenings or all day Sundays.

Mondays - Fridays open0700 - 1030 and 1530 - 2030

Saturdays closedSundays open 0700 - 1500

No entry from the streeton Sundays 1300 - 1730

(exit and interchange only)

Waterloo & City lineMondays - Fridays 0615 - 2130

Saturdays 0800 - 1830Sundays closed

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 -Saturdays

200m

South Harrow

Sudbury Hill

North Ealing

Park Royal

Alperton

Sudbury Town

Mondays - Saturdaysopen 0700-2345

Sundaysopen 0800-2345

Kensington(Olympia)

Earl'sCourt

Holborn

King George V

LondonCityAirport

WestSilvertown

PontoonDock

Opens

December 2005

Opens December 2005

Silvertown

North Woolwich

Royal Victoria

Custom Housefor ExCeL

Prince Regent

Royal Albert

Beckton Park

Cyprus

GallionsReach

Beckton

Canning TownBus to London City Airport

OpenMondays - Fridays

until 2100 onlySaturdays 0730 - 1930

Golders Green

Brent Cross

HeathrowTerminal 4

Hatton Crossfor Heathrow Terminal 4

Bus ser

vice

Improvement work to tracks and stations mayaffect your journey, particularly at weekends.

For help planning your journey look forpublicity at stations, call 020 7222 1234

or visit www.tfl.gov.uk

Closed until May 2006

Sudbury Hill Harrow150m

Station in Zone 11

Station in Zone 2

3

4

5

6

Station in Zone 3

Station in Zone 5

Station in Zone 6

2

Station in Zone 4

A Station in Zone A

B Station in Zone B

C Station in Zone C

D Station in Zone D

Station in Zone 6 and Zone A

Station in both zones

Station in both zones

Explanation of zones

TM Quad 2n Version 2 1/9/05 4 Colour Process + 4 Self Colours

NorthGreenwich

London Tube

Univerza v Ljubljanipodiplomski studijstatistike

Analiza omrezij4. Zgradba omrezij:povezanosti

Vladimir Batagelj

Univerza v Ljubljani

Ljubljana, 17. november 2006 / 10. in 17. november 2003

izpisano: December 15, 2006 ob 15 : 40

V. Batagelj: Analiza omrezij, 4. Zgradba omrezij – povezanosti 1'

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Kazalo1 Sprehodi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Najkrajse poti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

4 Enakovrednosti in razbitja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

6 Povezanosti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Posebni grafi – dvodelni, drevo . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

9 Skrcitev grafa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

12 Dvopovezanost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

13 k-povezanost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

14 Trikotniska povezanost – neusmerjeni grafi . . . . . . . . . . . . . . . . . . . . 14

17 Trikotniska povezanost – usmerjeni grafi . . . . . . . . . . . . . . . . . . . . . . 17

20 Pomembne tocke v omrezju . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

21 Normalizacija . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

22 Stopnje . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

23 Dostopnost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

24 Vmesnost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2425 Kazala in vsebine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2528 Nakopicenost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Univerza v Ljubljani, Podiplomski studij statistike s s y s l s y ss * 6


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29 Usredinjenost omrezja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

30 Padgett-ove floretinske rodbine . . . . . . . . . . . . . . . . . . . . . . . . . . . 30



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Sprehodidolzina |s| sprehoda s je stevilopovezav, ki ga sestavljajo.s = (j, h, l, g, e, f, h, l, e, c, b, a)|s| = 11Sprehod je sklenjen ali obhod ntk. nje-gov zacetek in konec sovpadata.Ce ne upostevamo smeri povezav v’sprehodu’, dobimo polsprehod aliverigo.sled – sprehod z razlicnimi povezavamipot – sprehod z razlicnimi tockamicikel – sklenjen sprehod z razlicniminotranjimi tockami.

Graf je aciklicen, ntk. ne vsebuje nobenega cikla.



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Najkrajse potiDolzino najkrajse poti iz u v v

oznacimo z d(u, v).Ce ne obstaja sprehod iz u v v

postavimo d(u, v) = ∞.d(j, a) = |(j, h, d, c, b, a)| = 5d(a, j) = ∞d(u, v) = max(d(u, v), d(v, u))je razdalja:d(v, v) = 0, d(u, v) = d(v, u),d(u, v) ≤ d(u, t) + d(t, v).

Premer grafa je enak razdalji med, glede na d(u, v), najoddaljenejsimatockama: D = maxu,v∈V d(u, v).

Net / Paths between 2 vertices /



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Najkrajse potiblack

backlack clackblank

rack wackbalk bank basklick lanklace clickclankblink

rick race wickwalk bilkbale bane bastlinklice lanelate clink chick

rice rate winkwale bilehale wanebine wastbaitline chinkcline chic

rite winewilewhale wait chine chit

write whinewhile whit

white

DICT28.


http://vlado.fmf.uni-lj.si/pub/networks/data/dic/dic28.zip


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Enakovrednosti in razbitjaRelacija R na V je enakovrednost ntk. jerefleksivna ∀v ∈ V : vRv, simetricna ∀u, v ∈ V : (uRv ⇒ vRu) intranzitivna ∀u, v, z ∈ V : uRz ∧ zRv ⇒ uRv.

Vsaka enakovrednost R doloca neko razbitje v razrede [v] = {u : vRu}.

Vsako razbitje C doloca neko enakovrednostuRv ⇔ ∃C ∈ C : u ∈ C ∧ v ∈ C.

k-sosedi tocke v je mnozica tock, ki so za k oddaljene od v,Nk(v) = {u ∈ v : d(v, u) = k}.

Mnozica vseh mnozic k-sosedov, k = 0, 1, ... of v je razbitje mnozice V .

k-sosescina tocke v, N (k)(v) = {u ∈ v : d(v, u) ≤ k}.

Net / k-Neighbors /



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Sosescina Motorole

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IBM

Cisco

iPlanet

3COM

Motorola

AvantGo

BaltimoreTechnologies

Unisys

ComputerAssociates

Nextel

MapInfo

GM

MasterCard

Fujitsu

Debelina povezav je koren iz vrednosti.



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PovezanostiTocka u je dosegljiva iz tocke v ntk.obstaja sprehod z zacetkom v in kon-cem u.Tocka v je sibko povezana s tocko u

ntk. obstaja veriga s krajiscema v in u.Tocka v je krepko povezana s tocko u

ntk. sta vzajemno dosegljivi.Sibka in krepka povezanost staenakovrednosti.graphCon.net

Razredi porajajo sibke/krepke komponente ali kose grafa.

Net / Components /


http://vlado.fmf.uni-lj.si/vlado/podstat/AO/net/graphCon.net


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Sibke komponente

Ce preuredimo tocke omrezja, takoda tocke iz iste skupine sibkegarazbitja postavimo skupaj, dobimomatricni prikaz sestavljen iz diago-nalnih blokov – sibkih komponent.Za vecino problemov velja, dajih lahko loceno resimo za vsakosibko komponento posebej in natodobljene resitve zdruzimo v resitevza celotno omrezje.



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Posebni grafi – dvodelni, drevo

Graf G = (V,L) je dvodelen ntk. lahko mnozico tock V razbijemo napodmnozici V1 in V2, tako da ima vsaka povezava iz L eno krajisce v V1

drugo pa v V2.

Sibko povezan graf G je drevo ntk. ne vsebuje (zank in) polciklov dolzinevsaj 3.



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Skrcitev grafa

Ce v danem grafu skrcimo vsako krepko komponento v ustrezno tocko,odstranimo zanke in zdruzimo vzporedne povezave, je tako dobljeni skrcenigraf aciklicen. Za vsak aciklicni graf obstaja urejenost / ostevilcenjei : V → N, tako da velja (u, v) ∈ A ⇒ i(u) < i(v).



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Skrcitev grafa – primerNet / Components / Strong [1]Operations / Shrink Network / Partition [1][0]Net / Transform / Remove / Loops [yes]Net / Partitions / Depth / AcyclicPartition / Make PermutationPermutation / Inverseselect partition [Strong Components]Operations / Functional Composition / Partition*PermutationPartition / Make Permutationselect [original network]File / Network / Export Matrix to EPS / Using Permutation

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jk

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#a

#e

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Pajek - shadow [0.00,1.00]

i

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j

l

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g

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i e h j l a b c d g f k



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Spletni metuljcek (Bow-tie)

Kumar &: The Web as a graph

Naj bo S najvecja krepka komponentav omrezju N ; W sibka komponenta,ki vsebuje S; I mnozica tock, iz ka-terih je S dosegljiva; O mnozica tockdosegljivih iz S; T (cevi) tocke (nisov S) na poteh iz I vO;R = W\(I∪S ∪ O ∪ T ) (lovke); in D = V \ W .Razbitje

{I,S,O, T ,R,D}

imenujemo metuljcno razbitje V .Net / Partitions / Bow-Tie


http://www.almaden.ibm.com/cs/k53/algopapers/pods00graph.ps


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DvopovezanostTocki u in v sta dvopovezani ntk. sta povezani (v obe smeri) s po dvemaneodvisnima (brez skupnih notranjih tock) potema.

Dvopovezanost doloca razbitje mnozice povezav.

Tocka je sticna tocka ali sticisce ntk. njena odstranitev poveca stevilo sibkihkomponent grafa.

Povezava je most ntk. njena odstranitev poveca stevilo sibkih komponentgrafa.

Net / Components / Bi-Components



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k-povezanostTockovna povezanost κ grafa G je enaka najmanjsemu stevilu tock, ki jihje potrebno odvzeti iz grafa, tako da je graf porojen s prestalimi tockaminepovezan ali trivialen (enak K1).

Povezavna povezanost λ grafa G je enaka najmanjsemu stevilu povezav,ki jih je potrebno odvzeti iz grafa, tako da je graf porojen s prestalimipovezavami nepovezan ali trivialen.

Velja Whitneyeva neenakost: κ(G) ≤ λ(G) ≤ δ(G) .

Graf G je (po tockah) k−povezan, ce je κ(G) ≥ k in je po povezavahk−povezan, ce je λ(G) ≥ k.

Velja Whitneyeva razlicica Mengerjevega izreka: Graf G je po tockah/pove-zavah k−povezan ntk. vsak par tock povezuje vsaj k po tockah/povezavahlocenih sprehodov.



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Trikotniska povezanost – neusmerjeni grafiV neusmerjenem grafu imenujemo trikotnik podgraf izomorfen K3.

Zaporedje trikotnikov (T1, T2, . . . , Ts) grafa G (tockovno) trikotniskopovezuje tocki u, v ∈ V ntk. u ∈ T1 in v ∈ Ts ali u ∈ Ts in v ∈ T1 terV(Ti−1) ∩ V(Ti) 6= ∅, i = 2, . . . s; in povezavno trikotnisko povezuje tockiu, v ∈ V ntk zadosca se strozji razlicici zadnjega pogoja E(Ti−1)∩E(Ti) 6=∅, i = 2, . . . s.

Tockovna trikotniska povezanost je enakovrednost na tockah; povezavna pana povezavah. Clanek.


http://vlado.fmf.uni-lj.si/pub/networks/doc/preprint/shortcy.pdf


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Trikotnisko omrezjeNaj bo G = (V, E) enostaven neusmerjen graf.Prirejeno trikotnisko omrezje NT (G) = (V, ET , w)doloceno z G je podgraf GT = (V, ET ) grafa G, kjer jeET mnozica tistih povezav iz E , ki leze na vsaj enemtrikotniku. Utez w(e) povezave e ∈ ET je enakastevilu razlicnih trikotnikov, ki jim povezava e pri-pada.

Trikotniska omrezja omogocajo ucinkovito razkrivanje gostih delovomrezja. Ce povezava e pripada k-kliki – podgrafu izomorfnemu Kk –v G, je w(e) ≥ k − 2.

Net / Count / 3-Rings



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Povezavni prerez na ravni 16 trikotniskega omrezjaErdos-ovega grafa sodelovanj

AJTAI, MIKLOS

ALAVI, YOUSEF

ALON, NOGA

ARONOV, BORIS

BABAI, LASZLO

BOLLOBAS, BELA

CHARTRAND, GARY

CHEN, GUANTAO

CHUNG, FAN RONG K.

COLBOURN, CHARLES J.FAUDREE, RALPH J.

FRANKL, PETER

FUREDI, ZOLTANGODDARD, WAYNE D.

GRAHAM, RONALD L.

GYARFAS, ANDRAS

HARARY, FRANK

HEDETNIEMI, STEPHEN T.

HENNING, MICHAEL A.

JACOBSON, MICHAEL S.

KLEITMAN, DANIEL J.

KOMLOS, JANOS

KUBICKI, GRZEGORZ

LASKAR, RENU C.

LEHEL, JENO

LINIAL, NATHAN

LOVASZ, LASZLO

MAGIDOR, MENACHEMMCKAY, BRENDAN D.

MULLIN, RONALD C.

NESETRIL, JAROSLAV

OELLERMANN, ORTRUD R.

PACH, JANOS

PHELPS, KEVIN T.

POLLACK, RICHARD M.

RODL, VOJTECHROSA, ALEXANDER

SAKS, MICHAEL E.

SCHELP, RICHARD H.

SCHWENK, ALLEN JOHN

SHELAH, SAHARON

SPENCER, JOEL H.

STINSON, DOUGLAS ROBERT

SZEMEREDI, ENDRE

TUZA, ZSOLT

WORMALD, NICHOLAS C.

brez Erdosa,n = 6926,m = 11343



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Trikotniska povezanost – usmerjeni grafiCe je graf G mesan, zamenjamo neusmerjene povezave s pari nasprotnousmerjenih. Naj bo G = (V,A) enostaven usmerjen graf brez zank. Zaizbrano usmerjeno povezavo (u, v) ∈ A obstajajo stiri vrste usmerjenihtrikotnikov: cyclic, transitive, input in output.

cyc tra in out



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. . . Trikotniska povezanost – usmerjeni grafi

Ce pozabimo na vlogo izbrane povezave, imamo le dve vrsti trikotnikov, kijim povezava lahko pripada: ciklicne (cyc) in tranzitivne (tra, in, out). Vprogramu Pajek ukaz

Net / Count / 3-Rings

omogoca dolociti ustrezna omrezja (Ncyc – ciklicne utezi, Ntra – tranzi-tivnostne utezi, Nsc – tranzitivne bliznjice).

Pojem trikotniske povezanosti lahko posplosimo na povezanost s kratkimi(pol)cikli – obroci in ustrezna omrezja.



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Povezavni prerez na ravni 11 tranzitivnega trikotniskegaomrezja slovarja ODLIS

abstract

American Library Association /ALA/

American Library Directory

bibliographic record

bibliography

binding

blanket order

book

book size

Books in Print /BIP/

call number

catalog

charge

collation

colophon

condition

copyright

cover

dummy

dust jacket

edition

editor

endpaper

entry

fiction

fixed location

folio

frequency

front matter

half-title

homepage

imprint

index

International Standard Book Number /ISBN/

invoice

issue

journal layout

librarian

library

library binding

Library Literature

new book

Oak Knoll

page

parts of a book

periodical

plate

printing

publication

published price

publisher

publishing

review

round table

serial

series

suggestion box

table of contents /TOC/text

title

title page

transaction log

vendor

work

PajekUniverza v Ljubljani, Podiplomski studij statistike s s y s l s y ss * 6


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Pomembne tocke v omrezjuPri izgradnji mer pomembnosti moramo najprej upostevati ali je omrezjeusmerjeno ali neusmerjeno. Meram pomembnosti na neusmerjenih omrezjihpravimo mere srediscnosti; na usmerjenih omrezjih pa mere veljave.Slednje se naprej delijo na mere ugleda ali podpore (upostevamo vstopajocepovezave) in mere vpliva (upostavamo izstopajoce povezave).

Ce zamenjamo dano usmerjeno omrezje z njemu nasprotnim (obrnemosmeri povezav) preidejo mere ugleda v mere vpliva, in obratno.

Dejanski pomen mere pomembnosti je odvisen od relacije (omrezja). Takonpr. je ’najuglednejsa’ oseba glede na relacijo ’ ne mara sodelovati z ’dejansko najmanj priljubljena oseba.

Odstranitev pomembne tocke iz omrezja povzroci obcutno spremembo vzgradbi/delovanju omrezja.



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NormalizacijaNaj bo p : V → R neka mera pomembnosti tock omrezja N = (V,L).Ce zelimo vrednosti mere p primerjati med razlicnimi omrezji, moramoposkrbeti za primerljivost. Pogosto jo poskusamo zagotoviti tako, da merop normaliziramo.

Naj boN ∈ N(V), kjer je N(V) izbrana mnozica omrezij nad isto mnozicoV ,

pmax = maxN∈N(V)

maxv∈V

pN (v) in pmin = minN∈N(V)

minv∈V

pN (v)

Tedaj je normalizirana mera enaka

p′(v) =p(v)− pmin

pmax − pmin∈ [0, 1]



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StopnjeNajpreprostejso mero pomembnosti predstavljajo stopnje tock. Ker staza enostavna omrezja degmin = 0 in degmax = n − 1, je ustreznanormalizirana mera

srediscnost deg′(v) =deg(v)n− 1

in podobno

ugled indeg′(v) =indeg(v)

n

vpliv outdeg′(v) =outdeg(v)

n

Namesto stopenj glede na osnovno omrezje lahko vzamemo tudi stopnjeglede na relacijo dosegljivosti (tranzitivna ovojnica).

Net / Partitions / Degree

Net / Partitions / Domain



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DostopnostCe upostevamo razdalje d(u, v) med tockami v omrezju N = (V,L) lahkovpeljemo

polmer r(v) = maxu∈V d(v, u)

Kolicino D = maxu,v∈V d(v, u) imenujemo premer omrezja.

skupna dostopnost S(v) =∑

u∈V d(v, u)

Za usmerjeno omrezje sta vpeljani meri meri vpliva. Meri ugleda dobimo,ce v obrazcih d(u, v) zamenjamo z d(v, u).

Ce omrezje ni krepko povezano, sta rmax in Smax enaki ∞. Sabidussi(1966) je zato kot mero dostopnosti vpeljal 1/S(v) oziroma v normaliziraniobliki

dostopnost cl(v) =n− 1∑

u∈V d(v, u)

Net / Vector / Centrality / Closeness



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VmesnostPomembne so tudi tocke, ki lahko nadzirajo pretok podatkov po omrezju.Ce privzamemo, da so za prenos pomembne le najkrajse poti, dobimo kotmero vmesnosti (Anthonisse 1971, Freeman 1977)

b(v) =1

(n− 1)(n− 2)

∑u,t∈V:gu,t>0u 6=v,t6=v,u6=t

gu,t(v)gu,t

kjer je gu,t stevilo najkrajsih poti iz u v t; in gu,t(v) stevilo takih med njimi,ki gredo skozi tocko v.

Net / Vector / Centrality / Betweenness



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Kazala in vsebineTockam povezanega usmerjenega omrezja N = (V,L) priredimo dvevrednosti: kakovost vsebine xv in kakovost kazala yv (Kleinberg, 1998).

Na dobro vsebino kazejo dobra kazala in dobro kazalo kaze na dobrevsebine

xv =∑

u:(u,v)∈L

yu in yv =∑

u:(v,u)∈L

xu

Naj bo W matrika omrezja N in x ter y vektorja obeh lastnosti. Tedajlahko zvezi zapisemo x = WT y oziroma y = Wx.

Zacnimo z y = [1, 1, . . . , 1] in nato zaporedoma izracunamo po obeh zvezahnove priblizke za x in y. Oba vektorja po vsakem koraku normaliziramo.To ponavljamo dokler se vektorja ne ustalita.

Pokazati je mogoce, da opisani postopek konvergira. Limitni vektor x∗ jeglavni lastni vektor matrike WT W; y∗ pa matrike WWT .



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. . . Kazala in vsebine

Podobni postopki se uporabljajo v spletnih iskalnikih za ocenjevanjepomembnosti posameznih strani.

PageRank, PageRank / Google, HITS / AltaVista, SALSA, teorija.

Net / Vector / Important Vertices / 1-Mode: Hubs

& Authorities

Na svetovnem nogometnem prvenstvu v Parizu leta 1998 je sodelovalo 22 nogometnihreprezentanc. V omrezju so vse drzave, iz katerih so nogometasi igrali v ligah teh 22 drzav,in vse drzave, v katerih ligah so igrali nogometasi iz teh 22 drzav. Relacija je igralec izdrzave x igra v drzavi y; utez je stevilo takih igralcev. Podatke je zbral Lothar Krempel.football.net


http://dbpubs.stanford.edu:8090/pub/1999-66

http://dbpubs.stanford.edu:8090/pub/1998-8

http://www.cs.cornell.edu/home/kleinber/auth.pdf

http://www.cs.technion.ac.il/~moran/r/PS/lm-feb01.ps

http://www.cs.technion.ac.il/~moran/r/PS/stab_journal31-1-03.pdf

http://vlado.fmf.uni-lj.si/vlado/podstat/AO/net/football.net


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. . . Kazala in vsebine: nogometasi

ARG

AUT

BEL

BGR

BRA

CHE

CHL

CMR

COL

DEU

DNK

ESP

FRA

GBR

GRE

HRV

IRN

ITA JAM

JPN

KOR

MAR

MEX

NGA

NLD

NORPRT

PRY

ROM

SCO

TUN

TUR

USA

YUG

ZAF

Izvozniki (kazala/hubs)

ARG

AUT

BEL

BGR

BRA

CHE

CHL

CMR

COL

DEU

DNK

ESP

FRA

GBR

GRE

HRV

IRN

ITA JAM

JPN

KOR

MAR

MEX

NGA

NLD

NORPRT

PRY

ROM

SCO

TUN

TUR

USA

YUG

ZAF

Uvozniki (vsebine/authorities)



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NakopicenostNakopicenost v tocki v je dolocena kot razmerje med stevilom vseh povezavv podgrafu G1(v) porojenim s sosescino dane tocke in stevilom povezav vpolnem grafu na teh tockah

C(v) =2|L(G1(v))|

deg(v)(deg(v)− 1)

za deg(v) > 1; in C(v) = 0 sicer.

Vpliv velikosti sosescine lahko zagotovimo z naslednjim popravkom

C1(v) =deg(v)

∆C(v)

kjer je ∆ najvecja stopnja v grafu G. Ta doseze najvecjo mozno vrednost lena tockah, ki pripadajo osamljeni kliki reda ∆.

Net / Vector / Clustering Coefficients / CC1



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Usredinjenost omrezjaMero pomembnosti p : V → R lahko povzamemo na celotnem omrezju kotnjegovo usredinjenost C(p):

p∗ = maxv∈V

p(v)

D(p) =∑v∈V

(p∗ − p(v))

D∗ = maxN∈N(V)

D(pN )

Tedaj je usredinjenost glede na p

C(p) =D(p)D∗

Za vecino mer je najbolj usredinjena zvezda Sn in najmanj polni graf Kn.



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Padgett-ove floretinske rodbine

Acciaiuoli

Albizzi

Barbadori

Bischeri

Castellani

Ginori

Guadagni

Lamberteschi

Medici

Pazzi

Peruzzi

Ridolfi

Salviati

Strozzi

Tornabuoni

close between

1. Acciaiuoli 0.368421 0.000000

2. Albizzi 0.482759 0.212454

3. Barbadori 0.437500 0.093407

4. Bischeri 0.400000 0.104396

5. Castellani 0.388889 0.054945

6. Ginori 0.333333 0.000000

7. Guadagni 0.466667 0.254579

8. Lamberteschi 0.325581 0.000000

9. Medici 0.560000 0.521978

10. Pazzi 0.285714 0.000000

11. Peruzzi 0.368421 0.021978

12. Ridolfi 0.500000 0.113553

13. Salviati 0.388889 0.142857

14. Strozzi 0.437500 0.102564

15. Tornabuoni 0.482759 0.091575


Documents

Univerza v Ljubljani podiplomski studijˇ statistikevlado.fmf.uni-lj.si/vlado/podstat/AO/2006/anom04.pdf · B9 Fairlop C6 Farringdon A5 Finchley Central B4 Finchley Road B4 Finchley