Connectivity

V. Batagelj

Connectivity

Condensation

Bow-tie

Otherconnectivities

Importantnodes

Closeness

Betweeness

Hubs andauthorities

Clustering

Introduction to Network Analysis using Pajek

4. Structure of networks 2Connectivity

Vladimir Batagelj

IMFM Ljubljana and IAM Koper

Phd program on StatisticsUniversity of Ljubljana, 2018

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Connectivity

V. Batagelj

Connectivity

Condensation

Bow-tie

Otherconnectivities

Importantnodes

Closeness

Betweeness

Hubs andauthorities

Clustering

Outline

1 Connectivity2 Condensation3 Bow-tie4 Other connectivities5 Important nodes6 Closeness7 Betweeness8 Hubs and authorities9 Clustering

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Quad Royal Tube Map Version 2 Date

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River Thames

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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

Sponsored by

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

e-mail: vladimir.batagelj@fmf.uni-lj.siCurrent version of slides (April 17, 2018 at 02 : 59):http://vladowiki.fmf.uni-lj.si/doku.php?id=pajek:ev:pde

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Connectivity

V. Batagelj

Connectivity

Condensation

Bow-tie

Otherconnectivities

Importantnodes

Closeness

Betweeness

Hubs andauthorities

Clustering

Walks

length |s| of the walk s is the num-ber of links it contains.s = (j , h, l , g , e, f , h, l , e, c , b, a)|s| = 11A walk is closed iff its initial andterminal node coincide.If we don’t consider the directionof the links in the walk we get asemiwalk or chain.trail – walk with all links differentpath – walk with all nodes differ-entcycle – closed walk with all internalnodes different

A graph is acyclic if it doesn’t contain any cycle.

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Connectivity

V. Batagelj

Connectivity

Condensation

Bow-tie

Otherconnectivities

Importantnodes

Closeness

Betweeness

Hubs andauthorities

Clustering

Shortest paths

A shortest path from u to v isalso called a geodesic from uto v . Its length is denoted byd(u, v).If there is no walk from u to vthen 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))is a distance:d(v , v) = 0, d(u, v) = d(v , u),d(u, v) ≤ d(u, t) + d(t, v).

The diameter of a graph equals to the distance between the mostdistant pair of nodes: D = maxu,v∈V d(u, v).

Network/Create New Network/Subnetwork with Paths/

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Connectivity

V. Batagelj

Connectivity

Condensation

Bow-tie

Otherconnectivities

Importantnodes

Closeness

Betweeness

Hubs andauthorities

Clustering

Shortest paths

black

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.

5 / 36

Connectivity

V. Batagelj

Connectivity

Condensation

Bow-tie

Otherconnectivities

Importantnodes

Closeness

Betweeness

Hubs andauthorities

Clustering

Equivalence relations and Partitions

A relation R on V is an equivalence relation iff it isreflexive ∀v ∈ V : vRv , symmetric ∀u, v ∈ V : uRv ⇒ vRu, andtransitive ∀u, v , z ∈ V : uRz ∧ zRv ⇒ uRv .

Each equivalence relation determines a partition into equivalenceclasses [v ] = {u : vRu}.

Each partition C determines an equivalence relationuRv ⇔ ∃C ∈ C : u ∈ C ∧ v ∈ C .

k-neighbors of v is the set of nodes on ’distance’ k from v ,Nk(v) = {u ∈ v : d(v , u) = k}.The set of all k-neighbors, k = 0, 1, ... of v is a partition of V.

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

Network/Create Partition/k-Neighbors

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Connectivity

V. Batagelj

Connectivity

Condensation

Bow-tie

Otherconnectivities

Importantnodes

Closeness

Betweeness

Hubs andauthorities

Clustering

Motorola’s neighborhood

####

AOL

MSFT

Yahoo!

Sun

IBM

Cisco

iPlanet

3COM

Motorola

AvantGo

BaltimoreTechnologies

Unisys

ComputerAssociates

Nextel

MapInfo

GM

MasterCard

Fujitsu

55

1

365

40

46

1

33

19

23

4

22

2

1

4

6

1

1 1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

11

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1

1

1

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1

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####

AOL

MSFT

Yahoo!

Sun

IBM

Cisco

iPlanet

3COM

Motorola

AvantGo

BaltimoreTechnologies

Unisys

ComputerAssociates

Nextel

MapInfo

GM

MasterCard

Fujitsu

The thickness of edges is a square root of its value.

7 / 36

Connectivity

V. Batagelj

Connectivity

Condensation

Bow-tie

Otherconnectivities

Importantnodes

Closeness

Betweeness

Hubs andauthorities

Clustering

Connectivity

Node u is reachable from node v iffthere exists a walk with initial nodev and terminal node u.

Node v is weakly connected withnode u iff there exists a semiwalkwith v and u as its end-nodes.

Node v is strongly connected withnode u iff they are mutually reach-able.

Weak and strong connectivity are equivalence relations.Equivalence classes induce weak/strong components.

Network/Create Partition/Components/

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Connectivity

V. Batagelj

Connectivity

Condensation

Bow-tie

Otherconnectivities

Importantnodes

Closeness

Betweeness

Hubs andauthorities

Clustering

Weak components

Reordering the nodes of net-work such that the nodes fromthe same class of weak partitionare put together we get a ma-trix representation consisting ofdiagonal blocks – weak compo-nents.

Most problems can be solvedseparately on each componentand afterward these solutionscombined into final solution.

9 / 36

Connectivity

V. Batagelj

Connectivity

Condensation

Bow-tie

Otherconnectivities

Importantnodes

Closeness

Betweeness

Hubs andauthorities

Clustering

Special graphs – bipartite, tree

A graph G = (V,L) is bipartite iff its set of nodes V can bepartitioned into two sets V1 and V2 such that every link from L hasone end-node in V1 and the other in V2.

A weakly connected graph G is a tree iff it doesn’t contain loops andsemicycles of length at least 3.

10 / 36

Connectivity

V. Batagelj

Connectivity

Condensation

Bow-tie

Otherconnectivities

Importantnodes

Closeness

Betweeness

Hubs andauthorities

Clustering

Condensation

If we shrink every strong component of a given graph into a node,delete all loops and identify parallel arcs the obtained reduced graphis acyclic. For every acyclic graph an ordering / level functioni : V → N exists s.t. (u, v) ∈ A ⇒ i(u) < i(v).

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Condensation – Example

Network/Create Partition/Components/Strong [1]Operations/Network+Partition/Shrink Network [1][0]Network/Create New Network/Transform/Remove/Loops [yes]Network/Acyclic Network/Depth Partition/AcyclicPartition/Make PermutationPermutation/Inverse Permutationselect partition [Strong Components]Operations/Partition+Permutation/Functional Composition Partition*PermutationPartition/Make Permutationselect [original network]File/Network/Export Matrix to EPS/Using Permutation

a

b

c

d

e

f

g

h

i

jk

l

#a

#e

f g

i

k

Pajek - shadow [0.00,1.00]

i

e

h

j

l

a

b

c

d

g

f

k

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

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Internal structure of strong components

Let d be the largest common divisor oflengths of closed walks in a strong com-ponent.

The component is said to be simple, iffd = 1; otherwise it is periodical with aperiod d .

The set of nodes V of strongly connecteddirected graph G = (V,R) can be parti-tioned into d clusters V1, V2, . . . , Vd , s.t.for every arc (u, v) ∈ R holds u ∈ Vi ⇒v ∈ V(imod d)+1 .

Network/Create Partition/

Components/Strong-Periodic

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. . . Internal structure of strong components

Bonhoure’s periodical graph. Pajek data

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Bow-tie structure of the Web graph

Kumar &: The Web as a graph

Let S be the largest strong com-ponent in network N ;W the weakcomponent containing S; I theset of nodes from which S canbe reached; O the set of nodesreachable from S; T (tubes) setof nodes (not in S) on paths fromI to O; R =W \ (I ∪S ∪O∪T )(tendrils); and D = V \ W. Thepartition

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

is called the bow-tie partition of V.

Note: chains can exist in the set R.

Network/Create Partition/Bow-Tie

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Biconnectivity

Nodes u and v are biconnected iff they are connected (in bothdirections) by two independent (no common internal node) paths.Biconnectivity determines a partition of the set of links.

A node is an articulation node iff its deletion increases the number ofweak components in a graph.

A link is a bridge iff its deletion increases the number of weakcomponents in a graph.

Network/Create New Network/with Bi-Connected

Components.

The notion of biconnectivity can be generalized do k-connectivity. Novery efficient algorithm for k > 3 exists.

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k-connectivity

Node connectivity κ of graph G is equal to the smallest number ofnodes that, if deleted, induce a disconnected graph or the trivialgraph K1.

Link connectivity λ of graph G is equal to the smallest number oflinks that, if deleted, induce a disconnected graph or the trivial graphK1.

Whitney’s inequality: κ(G) ≤ λ(G) ≤ δ(G) .

Graph G is (node) k−connected, if κ(G) ≥ k and is linkk−connected, if λ(G) ≥ k .

Whitney / Menger theorem: Graph G is node/link k−connected iffevery pair of nodes can be connected with k node/link internallydisjoint (semi)walks.

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Triangular and short cycle connectivities

In an undirected graph we call a triangle a subgraph isomorphic toK3.

A sequence (T1,T2, . . . ,Ts) of triangles of G (node) triangularlyconnects vertices u, v ∈ V iff u ∈ T1 and v ∈ Ts or u ∈ Ts andv ∈ T1 and V(Ti−1) ∩ V(Ti ) 6= ∅, i = 2, . . . s. It edge triangularlyconnects vertices u, v ∈ V iff a stronger version of the secondcondition holds E(Ti−1) ∩ E(Ti ) 6= ∅, i = 2, . . . s.

Node triangular connectivity is an equivalence on V; and edgetriangular connectivity is an equivalence on E . See the paper.

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Triangular connectivity in directed graphs

If a graph G is mixed we replaceedges with pairs of opposite arcs.In the following let G = (V,A)be a simple directed graph with-out loops. For a selected arc(u, v) ∈ A there are only two dif-ferent types of directed triangles:cyclic and transitive.

cyc tra

For each type we get the corresponding triangular network Ncyc andNtra by determining the corresponding weight wcyc or wtra to its arcs,counting the number of cyclic/transitive triangles that contain thearc. We remove arcs with weight zero. The notion of triangularconnectivity can be extended to the notion of short (semi) cycleconnectivity.

Network/Create New Network/with Ring Counts/3-Rings/Directed

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Edge-cut at level 16 of triangular networkof Erdos collaboration graph

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.

without Erdos,n = 6926,m = 11343

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Arc-cut at level 11 of transitive triangular networkof ODLIS dictionary

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

Pajek

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Important vertices in network

To identify important / interesting elements (nodes, links) in anetwork we often try to express our intuition about important /interesting element using an appropriate measure (index, weight)following the scheme

larger is the measure value of an element,more important / interesting is this element

Too often, in analysis of networks, researchers uncritically pick somemeasure from the literature. For formal approach see Roberts.

It seems that the most important distinction between different nodeindices is based on the view/decision whether the network isconsidered directed or undirected.

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. . . Important nodes in network

This gives us two main types of indices:

• networks containing directed links (we replace edges by pairs ofopposite arcs): measures of importance; with two subgroups:measures of influence, based on out-going arcs; and measures ofsupport, based on incoming arcs;

• measures of centrality, based on all links.

For undirected networks all three types of measures coincide.If we change the direction of all arcs (replace the relation with itsinverse relation) the measure of influence becomes a measure ofsupport, and vice versa.

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. . . Important nodes in network

The real meaning of measure of importance depends on the relationdescribed by a network. For example the most ’important’ person forthe relation ’ doesn’t like to work with ’ is in fact the leastpopular person.

Removal of an important node/link from a network produces asubstantial change in structure/functioning of the network.

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Normalization

Let p : V → R be an index in network N = (V,L, p). If we want tocompare indices p over different networks we have to make themcomparable. Usually we try to achieve this by normalization of p.Let N ∈ N(V), where N(V) is a selected family of networks over thesame set of nodes V ,

pmax = maxN∈N(V)

maxv∈V

pN (v) and pmin = minN∈N(V)

minv∈V

pN (v)

then we define the normalized index as

p′(v) =p(v)− pmin

pmax − pmin∈ [0, 1]

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Degrees

The simplest index are the degrees of nodes. Since for simplenetworks degmin = 0 and degmax = n − 1, the correspondingnormalized indices are

centrality deg′(v) =deg(v)

n − 1and similary

support indeg′(v) =indeg(v)

n

influence outdeg′(v) =outdeg(v)

nInstead of degrees in original network we can consider also thedegrees with respect to the reachability relation (transitive closure).

Network/Create Partition/Degree

Network/Create Vector/Centrality/Degree

Network/Create Vector/Centrality/Proximity Prestige

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Closeness

Most indices are based on the distance d(u, v) between vertices in anetwork N = (V,L). Two such indices areradius r(v) = maxu∈V d(v , u)total closeness S(v) =

∑u∈V d(v , u)

These two indices are measures of influence – to get measures ofsupport we have to replace in definitions d(u, v) with d(v , u).If the network is not strongly connected rmax and Smax are equal to∞. Sabidussi (1966) introduced a related measure 1/S(v); or in itsnormalized form

closeness cl(v) =n − 1∑

u∈V d(v , u)D = maxu,v∈V d(v , u) is called the diameter of network.

Network/Create Vector/Centrality/Closeness

Network/Create New Network/Subnetwork with Paths/Info

on Diameter

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Betweeness

Important are also the nodes that can control the information flow inthe network. If we assume that this flow uses only the shortest paths(geodesics) we get a measure of betweeness (Anthonisse 1971,Freeman 1977)

b(v) =1

(n − 1)(n − 2)

∑u,t∈V:gu,t>0

u 6=v,t 6=v,u 6=t

gu,t(v)

gu,t

where gu,t is the number of geodesics from u to t; and gu,t(v) is thenumber of those among them that pass through node v .

For computation of geodesic matrix see Brandes.

Network/Create Vector/Centrality/Betweenness

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Padgett’s Florentine families

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

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Hubs and authorities

To each node v of a network N = (V,L) we assign two values:quality of its content (authority) xv and quality of its references(hub) yv .A good authority is selected by good hubs; and good hub points togood authorities (see Kleinberg).

xv =∑

u:(u,v)∈L

yu and yv =∑

u:(v ,u)∈L

xu

Let W be a matrix of network N and x and y authority and hubvectors. Then we can write these two relations as x = WTy andy = Wx.We start with y = [1, 1, . . . , 1] and then compute new vectors x andy. After each step we normalize both vectors. We repeat this untilthey stabilize.We can show that this procedure converges. The limit vector x∗ isthe principal eigen vector of matrix WTW; and y∗ of matrix WWT .

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. . . Hubs and authorities

Similar procedures are used in search engines on the web to evaluatethe importance of web pages.PageRank, PageRank / Google, HITS / AltaVista, SALSA, theory.

Network/Create New Network/Subnetwork with Paths/Info on DiameterNetwork/Create Vector/Centrality/ClosenessNetwork/Create Vector/Centrality/BetweenessNetwork/Create Vector/Centrality/Hubs-AuthoritiesNetwork/Create Vector/Centrality/Clustering Coefficients

Examples: Krebs, Krempl. World Cup 1998 in Paris, 22 nationalteams. A player from first country is playing in the second country.

There exist other measures based on eigen-values and eigen-vectors

such as Katz, Bonachich and Brandes. See also Borgatti.

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. . . Hubs and authorities: football

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

Exporters (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

Importers (authorities)

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Clustering

Let G = (V, E) be simple undirected graph. Clustering in node v isusually measured as a quotient between the number of links insubgraph G1(v) = G(N1(v)) induced by the neighbors of node v andthe number of links in the complete graph on these nodes:

C (v) =

2|L(G1(v))|

deg(v)(deg(v)− 1)deg(v) > 1

0 otherwise

We can consider also the size of node neighborhood by the followingcorrection

C1(v) =deg(v)

∆C (v)

where ∆ is the maximum degree in graph G. This measure attains itslargest value in nodes that belong to an isolated clique of size ∆.

Network/Create Vector/Clustering Coefficients/CC2

paper33 / 36

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User defined indices

Xingqin Qi et al. defined in their paper Terrorist Networks, NetworkEnergy and Node Removal a new measure of centrality based onLaplacian energy – Laplacian centrality

L(v) = deg(v)(deg(v) + 1) + 2∑

u∈N(v)

deg(u)

select the networkNetwork/Create Vector/Centrality/Degree/AllOperations/Network+Vector/Neighbours/Sum/All [False]Vector/Transform/Multiply by [2]select the degree vector as Firstselect the degree vector as SecondVectors/Multiply (First*Second)Vectors/Add (First+Second)select the 2*sum on neighbors as SecondVectors/Add (First+Second)dispose auxiliary vectorsFile/Vector/Change Label [Laplace All centrality]

macro

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Network centralization measures

Extremal approach: Let p : V → R be an index in networkN = (V,L). We introduce the quantities

p? = maxv∈V

p(v)

D =∑v∈V

(p? − p(v))

D? = max{D(N ) : N is a network on V}

Then we can define centralization with respect to p

Cp(N ) =D(N )

D?

Usually the most centralized graph is the star Sn and the leastcentralized is the complete graph Kn.

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. . . Network centralization measures

Variational approach: The other approach is based on the variance.First we compute the average node centrality

p =1

n

∑v∈V

p(v)

and then define

Vp(N ) =1

n

∑v∈V

(p(v)− p)2

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