Introduction to Social Network Analysis s with Pajekmrvar.fdv.uni-lj.si/sola/dr/slideseng.pdf · 2017. 3. 22. · A. Mrvar: Social Network Analysis 5 ’ & $ % X iRX j)there exists

A. Mrvar: Social Network Analysis 1'

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Introduction toSocial Network Analysiswith Pajek

Andrej MrvarFaculty of Social SciencesLjubljana

Postgraduate study - 3rd cycle

s s y s l s y ss * 6

http://slo-foto.portal.si/4images/details.php?image_id=551


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IntroductionLet U = {X1, X2, . . . , Xn} be a finite set of units. Connections amongunits are described using one or more binary relations Rt ⊆ U× U, t =

1, . . . , r, which determine a network N = (U, R1, R2, . . . , Rr).

Example: A relation can represent friendship, negative relation, kinshiprelation (. . .is a child of. . ., . . .is a daughter of . . ., . . .is married to. . .),citations. . .

In the following we will use mostly one relation R.

XiRXj is read as:unit Xi is in relation R with unit Xj .

Example: if R corresponds to relation ’liking’, then XiRXj , means thatperson Xi likes person Xj .



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Relations can have additional properties:

• reflexive relation: ∀x ∈ U : xRx

• irreflexive relation: ∀x ∈ U : ¬xRx

• symmetric relation: ∀x, y ∈ U : (xRy ⇒ yRx) (marriage)

• asymmetric relation: ∀x, y ∈ U : ¬(xRy ∧ yRx) (is a son of)

• antisymmetric relation: ∀x, y ∈ U : (xRy ∧ yRx⇒ x = y)

• transitive relation: ∀x, y, z ∈ U : (xRy ∧ yRz ⇒ xRz)

• intransitive relation: ∀x, y, z ∈ U : (xRy ∧ yRz ⇒ ¬xRz)

• comparable relation: ∀x, y ∈ U : (x 6= y ⇒ xRy ∨ yRx)

• strictly comparable relation: ∀x, y ∈ U : (xRy ∨ yRx)

• partial ordering: a reflexive, antisymmetric and transitive relation;

• equivalence relation: a reflexive, symmetric and transitive relation.



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A network defined using relation R can be represented in different ways:

• Representation using corresponding binary matrixR = [rij ]n×n, where

rij =

{1 XiRXj

0 otherwise

Sometimes rij is a real number, expressing the strength of relation R betweenunits Xi in Xj .

• list of neighboursA network can be described by specifying the list of all other units with whichthe unit is in relation.

• description by a graph G = (V,L) where V is set of vertices and L set of(directed or undirected) lines. Vertices represent units of a network, linesrepresent the relation. A graph is usually represented by a picture: vertices aredrawn as small circles, directed lines are drawn as arcs and undirected lines asedges connecting the corresponding two vertices.



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XiRXj ⇒ there exists directed line from unit Xi to Xj in correspond-ing graph. Vertex Xi is called initial, vertex Xj is called terminalvertex.

A line whose initial and terminal vertices are the same is called a loop.

If directed lines between two vertices exist in both directions, they aresometimes replaced by a single undirected line.

Sometimes we do the opposite: an undirected line is replaced by twodirected lines in opposite directions.

We will denote the number of vertices in a graph with n, and numberof directed lines with m.



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1 2 3 4 5

1 0 1 0 1 1

2 0 0 1 0 0

3 1 0 0 1 0

4 0 0 0 0 1

5 1 0 0 0 0

Arcslist

1: 2 4 5

2: 3

3: 1 4

4: 5

5: 1

1

2

3

4

5



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Types of networks• undirected network – the relation is symmetric – all lines are undirected –

edges, L = E.

• directed network – the relation is not symmetric – all lines are directed – arcs,L = A.

• mixed network – both arcs and edges exist in a corresponding graph –L = A ∪ E.

• two-mode networkA two-mode network consists of two sets of units (e. g. people and events),relation connects the two sets, e. g. participation of people in social events.



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Small and large networks

Networks with some 10 units and lines are called small networks, while networkswith some 1000 units and lines are called large networks.

Dense and sparse networks

A network is called sparse if the number of lines in the corresponding graph is ofthe same order as the number of vertices (n ≈ km). Large networks that are sparsecan still be efficiently analysed with some algorithms. In real life we often find verylarge but sparse networks.

In general, the number of lines can be much higher than the number of vertices.Such networks are called dense.

If every unit is connected to every other unit the number of lines is n2 (number ofelements in matrix).

If every unit is connected to every other unit except to itself (graph without loops),the number of lines is n(n− 1) (number of elements in matrix without diagonal).



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According to that the density of a network can be defined:

For networks with loops:

Density1 =m

n2

For networks without loops:

Density2 =m

n(n− 1)

If at most one line can exist among any two vertices the density is a real numberbetween 0 and 1.

Density of a network is one of the measures by which we can compare differentnetworks.



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Example of large networks• social networks

– connections among people (friendship);

– relation among political parties;

– trade among organizations, countries;

– genealogies;

– citation networks;

– computer networks (local networks, Internet, links among home pages);

– telephone calls;

• flow charts in computer science;

• Petri nets;

• organic molecule in chemistry;

• connections among words in text;

• transportation networks (airlines, streets, electric networks. . . ).



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PajekPajek is a program package for Windows 32 and64, which enables analyses of large networks. Pro-gram is freely available at:http://mrvar.fdv.uni-lj.si/pajek/

de Nooy, Mrvar, Batagelj (2011):Exploratory Social Network Analysis with Pajek:Revised and Expanded Second Edition.Cambridge University Press, New York.


http://www.amazon.com/gp/product/0521602629/104-4818949-5145535?v=glance&n=283155

http://www.amazon.com/gp/product/0521602629/104-4818949-5145535?v=glance&n=283155


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Analyses in Pajek are performed using six data structures:

1. network,

2. partition,

3. cluster,

4. permutation,

5. vector,

6. hierarchy.



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Main goals in designing PajekThe main goals in the design of Pajek are:

• to support abstraction by (recursive) decomposition of a large network intoseveral smaller networks that can be treated further using more sophisticatedmethods;

• to provide the user with some powerful visualization tools;

• to implement a selection of efficient (subquadratic) algorithms for analysis oflarge networks.



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Two-mode networksA two-mode network consists of two sets of units (e. g. people and events), relationconnects the two sets, e. g. participation of people in social events.

There exist several two-mode networks:

• Membership in institutions - people, institutions, is a member, e.g. directorsand commissioners on the boards of corporations.

• Voting for suggestions - polititians, suggestions, votes for.

• ’Buying articles in the shop’, where first set consists of consumers, the secondof articles, the connection tells which article was bought by a consumer.

• Readers and magazines.

• Citation network, where first set consists of authors,the second set consists of articles/papers,connection is a relation author cites a paper.

• Co-autorship networks - authors, papers, is a (co)author.

A corresponding graph is called bipartite graph – lines connect only vertices fromone to vertices from another set – inside sets there are no connections.



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Example: participation of women in social events

EVELYN

LAURA

THERESA

BRENDA

CHARLOTTE

FRANCES

ELEANOR

PEARL

RUTH

VERNE

MYRNA

KATHERINE

SYLVIA

NORA

HELEN

DOROTHY

OLIVIA

FLORA

E1

E2

E3

E4

E5

E6

E7

E8

E9

E10

E11

E12

E13

E14

Davis.net



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Transforming two-mode networks to ordinaryvalued networks

Two-mode network can be transformed to ’ordinary’ network, where units are onlyunits from first or only units from the second set.

The previous two-mode network (women-events) can be transformed to ordinarynetwork, where units are women. Two women are in relation (in the correspondinggraph there exists an undirected line) if they took part in at least one common event.The line value tells the number of events where both of them took place. Loopvalues represent total number of events for each woman.

But if we transform network to ordinary network where units are events, the twoevents are in relation (in the corresponding graph there exists an undirected line) ifthere exists at least one woman who took part in both events. The line value betweentwo events tells the number of women who took place in both events. Loop valuesrepresent total number of women present at each event.



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6

6

6

7

6

4

4

4

4

3

3

6

4

4

3

3

4

3

3

3

4

4

4

3

3

3

3

3

6

6

3

3

3

3 4

3

6

3

4

4

4

3

3

3

4

4

3

6

5

3

53

EVELYN

LAURA

THERESA

BRENDA

CHARLOTTE

FRANCES

ELEANOR

PEARL

RUTH VERNE

MYRNA

KATHERINE

SYLVIA

NORAHELEN

DOROTHY

OLIVIA

FLORA



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Signed networksSigned graph is an ordered pair (G, σ), where:

• G = (V,L) is a graph with a set of vertices V and a set of lines L

• σ : L → {p, n} is a sign function, the lines with the sign p are positive, thelines with the sign n are negative. Positive lines are drawn with solid, negativewith dotted lines.

The question is:Is it possible to partition vertices of a signed graph, so that every line that connectsvertices that belong to the same cluster is positive and every line that connects twovertices that belong to different clusters is negative?



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If it is possible to partition vertices in this way, we call the signed graph partitionableor clusterable. Especially important are signed graphs where vertices can bepartitioned into 2 clusters. Such graphs are called balanced.

In the case of people who are friends and enemies a balanced signed graph meansthat there exist two clusters of people, such that there are only friends inside theclusters and nobody has a friend in the other cluster. This situation is very stable –it cannot happen that there exist a person and two other friends, and the person is afriend of one of them and enemy of the other.

In most real life examples signed graph is not partitionable. In such cases we wouldlike to find partition, which is as close to ideal partition as possible – which has thelowest number of errors. The error of given partition is:

• every negative line among vertices in the same cluster and

• every positive line among vertices in different clusters.



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Example: Sampson monasterySampson studied relations among 18 monks in the New England monastery. Hemeasured several relations:

• friendship (affect)

• esteem

• influence

• sanction

Friendship relation was measured in three time points T2, T3 and T4, all others onlyin T4.

Relations among monks can be represented as valued signed graphs. Each monkselected 3 other monks whom he liked / disliked the most. A citation of +3 goes tothe most liked monk, a citation -3 goes to the most disliked monk.



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Friendship relation in T2:(black boxes for positive values; red diamonds for negative values):

John_BoscoGregoryBasilPeterBonaventurBertholdMarkVictorAmbroseRomualdLouisWinfridAmandHughBonifaceAlbertEliasSimplicius

John

_Bos

coG

rego

ryB

asil

Pet

erB

onav

entu

rB

erth

old

Mar

kV

icto

rA

mbr

ose

Rom

uald

Loui

sW

infr

idA

man

dH

ugh

Bon

iface

Alb

ert

Elia

sS

impl

iciu

s



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ResultsIn the table the total error score for friendship relation in all three time points isgiven.

No. of clusters T2 T3 T4

1 48.5 48.0 47.0

2 21.5 16.0 12.5

3 17.5 11.0 10.5

4 19.0 13.5 12.5

5 20.5 16.0 15.0

Two important facts:

• For any number of clusters the error in T3 is lower than in T2 and error in T4 islower than in T3. Conclusion: balance is improving during the time.

• In all time points the lowest error occurs for 3 clusters. Therefore partition into3 clusters in the most ’natural’.



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Look at the partition into 3 clusters:

• Optimal partition is equal for all three time points.

• Partition is equal to the partition reported by Sampson, which he obtained usingexamining the monks.

Maybe this result is a little surprising. It means that the clusters are not changingbut relations among monks are more and more ’clear’:

• more lines inside clusters are positive, more lines among clusters are negative;

• negative lines inside and positive lines between clusters are weaker.



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Permuted matrix in T2:

John_BoscoGregoryMarkWinfridHughBonifaceAlbertBasilAmandEliasSimpliciusPeterBonaventurBertholdVictorAmbroseRomualdLouis

John

_Bos

coG

rego

ryM

ark

Win

frid

Hug

hB

onifa

ceA

lber

tB

asil

Am

and

Elia

sS

impl

iciu

sP

eter

Bon

aven

tur

Ber

thol

dV

icto

rA

mbr

ose

Rom

uald

Loui

s



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Temporal networksNetworks that are changing over time:

• network of friendship in the class in school over several years;

• changes in signed graphs over time(Sampson monastery data, Newcomb fraternity);

• network of phone calls inside selected set of phone numbers (used by the policein the investigation of organized crime);

• citation networks from a selected scientific area;

• network of transitions of a ball in a football game;

• changes in HIV networks;

• relations among actors in different episodes of movies;

• births, marriages and deaths in genealogies. . .



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Relations among actors in the long-running German soap opera ‘Linden-strasse’ in episodes 5, 6 and 7

Lydia Nolte

Gung Pahm Kien

Gabi Zenker-geb.Skawowski

Helga Beimer Schiller

Hans Beimer

Berta Griese

Marion Beimer

Benny BeimerKlaus Beimer

Franz Schildknecht

Henny Schildknecht

Stefan Nossek

Egon Kling

Else Kling

Tanja Schildknecht-DresslerDr. Ludwig Dressler

Vasily Sarikakis

Chris Barnsteg

Elfie Kronmayr

Sigi Kronmayr

Philo Bennarsch

Joschi Bennarsch

Paul Nolte

Theo Nolte

Lydia Nolte

Gung Pahm Kien



Hans Beimer

Berta Griese

Marion Beimer


Franz Schildknecht

Henny Schildknecht

Stefan Nossek

Egon Kling

Else Kling


Elisabeth Dressler

Carsten Floeter

Vasily Sarikakis

Chris Barnsteg

Elfie Kronmayr

Sigi Kronmayr

Gottlieb Griese

Philo Bennarsch

Joschi Bennarsch

Paul Nolte

Theo Nolte

Herr Floether Der Englaender

Lydia Nolte

Gung Pahm Kien



Hans Beimer

Berta Griese

Marion Beimer


Franz Schildknecht

Henny Schildknecht

Stefan Nossek

Egon Kling

Else Kling


Elisabeth Dressler

Frank Dressler

Carsten Floeter

Vasily Sarikakis

Chris Barnsteg

Elfie Kronmayr

Sigi Kronmayr

Gottlieb Griese

Meike Schildknecht Philo Bennarsch

Joschi Bennarsch

Paul Nolte

Theo Nolte




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Episode 5

Lydia Nolte

Gung Pahm Kien



Hans Beimer

Berta Griese

Marion Beimer


Franz Schildknecht

Henny Schildknecht

Stefan Nossek

Egon Kling

Else Kling


Vasily Sarikakis

Chris Barnsteg

Elfie Kronmayr

Sigi Kronmayr

Philo Bennarsch

Joschi Bennarsch

Paul Nolte

Theo Nolte



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Episode 6

Lydia Nolte

Gung Pahm Kien



Hans Beimer

Berta Griese

Marion Beimer


Franz Schildknecht

Henny Schildknecht

Stefan Nossek

Egon Kling

Else Kling


Elisabeth Dressler

Carsten Floeter

Vasily Sarikakis

Chris Barnsteg

Elfie Kronmayr

Sigi Kronmayr

Gottlieb Griese

Philo Bennarsch

Joschi Bennarsch

Paul Nolte

Theo Nolte




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Episode 7

Lydia Nolte

Gung Pahm Kien



Hans Beimer

Berta Griese

Marion Beimer


Franz Schildknecht

Henny Schildknecht

Stefan Nossek

Egon Kling

Else Kling


Elisabeth Dressler

Frank Dressler

Carsten Floeter

Vasily Sarikakis

Chris Barnsteg

Elfie Kronmayr

Sigi Kronmayr

Gottlieb Griese

Meike Schildknecht Philo Bennarsch

Joschi Bennarsch

Paul Nolte

Theo Nolte




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Example: Internet Industry PartnershipsMost important lines

AOL

AT&T

MSFT

Yahoo!

SunIBM

LookSmart

RealNetworks

Excite

TerraLycos

Apple

EarthLink

Amazon

Dell

RoadRunnerCompaq

HP

EMC

Novell

AkamaiCisco

KPMG

EDS

Sprint

Oracle

InfoSpace

VeriSign

Inktomi

Intel

C|Net

SoftBank

Travelocity

iPlanet

Ariba

3COM

PalmCommerceOne

Nokia

Bertelsmann

Motorola

PwC

Peoplesoft

ExodusComm

Loudcloud

Lucent

SAP

Siebel

KPNQwest

Vignette

Interliant

BMCSoftware

E.piphany

WebMethods

CIBER

FoundryNetworks

divine

GM

UPS D&B



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Example: The Edinburgh Associative Thesaurus• The Edinburgh Associative Thesaurus (EAT) is a set of word association norms

showing the counts of word association as collected from students.

• It is a big network – 23219 vertices (words) and 325624 directed lines.



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Selected themes from EAT

BIG

ENORMOUS

EXPANSE

HUGE

LARGE

MASSIVE

SCOPE

SIZE

SPACIOUS

SWELL

VAST

WIDE

BADMINTON

BALL

BALLSBASKETBALL

CHAMPIONSHIP

FIVES

FOOTBALL

GAME

GAMES

GOALIE

GYM

HOCKEY

NETBALLPLAYER

PLAYING

REFEREE

RUGBY

RUGGER

SHORTS

SOCCER

SPORT

TEAM

TENNIS

BACHBEETHOVEN

CONCERTO

DANCE

FESTIVAL

JAZZ

MONOTONE

MUSIC

MUSICAL BOX

MUSICIAN

ORCHESTRA

PIANO

POP GROUP

RECITAL

RECORDS

SAXOPHONE

SING

SINGING

SONATA

SONG

SONGS

SOUNDS

TONE

TRUMPET

TUNE

VIOLIN

VOICES BICYCLE

BICYCLES

BIKE

BIKES

CAR

COACH

DRIVERS

LANE

LORRIES

MACHINES

MOBILE

MOTOR CYCLE

MOVING

PATH

RAILWAY

RAILWAYS

ROAD

ROADS

STOP

STOPPING

STREETTRAIN

TRUCK

VAN



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Selected themes from EAT

AGAIN ALREADY

ANYWAY

AS

BUT

HAPPEN

HAPPENED

JUST

MEANWHILE

NEVERTHELESS

NOTWITHSTANDING

NOW

OFTEN

SOMETIME

SOON

THEREFORE

WHY

YET

BELIEVE

COINS

COULD

DEALER

DEFICIT

INCREASE

LOOT

MONEY

MONIES

MORE MONEY

NO

NOT

OFFER

PAID

PAY

PAYMENT

PLEASE

PROBABLY

PROPERTY

PROVIDE

RECEIPT

REFUSE

REPAY

STOLE

THRIFT

THRIFTY

UNPAID

ACTIVITY

EDUCATION

ENGINEERING

HOMEWORK

LEARNING

LECTURER

LECTURES

LESSONS

MATHS

RESEARCH

SCHOOL

SCIENCE

SCIENTIFIC STUDY

STUDYING

TEACHER

TEACHING

TRAINING

WORK

ADORABLE

ATTRACTIVE

BEAUTIFUL

BELOVED

BOSS

CHAIRMAN

CHARM

DELIGHTFUL

ELEGANCE

FLIRT

GIRL

HAIRY

INHUMAN

KINDNESS

LOVE

LOVELY

MAN

MYSTERIOUS NICE

POWERFUL

PROFESSION

RESPONSIBLE

SHAPELY



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Example: The Knuth’s English DictionaryA large network can be generated from words of dictionary. Two words areconnected using an undirected line if we can reach one from the other by

• changing a single character (e. g., work – word)

• adding / removing a single character (e. g., ever – fever).

Knuth’s dictionary was used. There exist 52,652 words having 2 to 8 characters. Theobtained network has 89,038 edges. The network is sparse: density is 0.0000642.



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Selected clusters in Knuth’s Dictionary

back

bail

bale

bane

bang

bate

bear

bell

bent

best

bill

bine

buck

cade

cage

cane

care

casecate

cave

cell

cent

dale

Dane

dang

date

dear

dell

dent

dill

dine

duck

ears

eave

eine

fail

fane

fang

fate

fear

fell

fest

fill

fine

fuck

Gail

gale

gang

gate

gave

gear

gent

gill

guck

hack

hail

hale

hang

hare

hate

have

hear

hell

hent

hest

hick

hill

huck

jail

Jane

jell

jest

Jill

kale

Kane

Kate

Kent

kick

kill

kine

lack

lane

Lang

late

lave

Learlent

lest

lick

line

luck

mack

mail

male

mane

mate

mell

mill

mine

muck

nail

Nate

near

Nell

nest

nick

nill

nine

pail

pale

pane

pang

patepear

pent

pest

pick

pill

pine

puck

rack

railrale

rang

rate

rave

rear

rent

rest

rick

rill

ruck

sack

sail

sale

sane

sang

sate

save

sear

sell

sent

sick

sill

sine

suck

tack

tail tale

tangtate

tear

tell

tent

test

tick

till

tine

tuck

Vail vale

vane

vent

vest

vine

wack

wail wale

wane

Wang

wave

wear

well

went

west

will

wine

Yale

yangyear

yell

yuck

zest

bater

bears

bight

cater

dater

dears

eater

eight

ester

fears

fight

gears

hater

hears

hight

later

light

mater

nears

oater

pater

pears

rater

rears

right

Sears

sight

tater

tears

water

wears

wight

years

besterfester

Hester

jester

Lester

nester

pester

rester

tester

westeryester

ad

am

an

at

en

et

he

ho

in

it

on

ow

re

si

so

bad

bag

bam

ban

bar

bat

bay

bee

Ben

bet

bin

bit

bon

bot

bow

cad

cam

can

cap

car

cat

cay

cit

cob

cod

cogcol

comcon

coo

cop

cos

cot

cow

cox

coy

dad

dag

dam

Dan

Dar

day

Dee

den

din

don

dot

Dow

ear

eat

ein

eon

fad

fag

fan

far

fat

fay

fee

fen

fin

fit

gadgag

gam

gap

gar

gat

gay

gee

get

gin

git

gon

got

had

hag

ham

Han

haphat

hay

hee

hen

hit

hob

hoc

hod

hoe

hog

hoi hon

hop

hot

how

hoy

Ianion

jag

jam Jan

jar

jay

jet

Jon

jot

jow

Kay

ken

kin

kit

lad

lag

lam

lap

lay

lee

Len

let

Lin

lit

lot

low

mad

man

map

mar

mat

may

men

met

min

mot

mow

nag

Nan

nap

Nat

nay

nee

net

nit

non

not

now

oar

oat

pad

Pam

pan

pap

par

pat

pay

pee

pen

pet

pin

pit

pon

pot

pow

rad

rag

ram ran

rap rat

ray

red

ree

ret

rin

rit

Ron

rot

row

sad

sag

Sam San

sapsat

say

see

sen

set

sib

sic

sid

sie

sig

sin

sip sir

sis

sitsix

son

sot

sow

tab

tad

tag

tam tantao

tap

tar

tat

tau

taw

tax

tee

ten

tin

tit

ton

tot

tow

van

vat

vee

vet

von

vow

wad

wag

wan

war

way

wee

wen

wet

win

wit

won

wow

yam

yap

yen

yet

yin

yon

yow

zag

Zan

zap

Zen



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Airline connections among 332 US airports(332 vertices, 2116 lines)

Miami Intl



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Paths in a graphIn a directed graph:

• A sequence of vertices (v1, v2, . . . , vk) is called a walk if

(vi, vi+1) ∈ A, i = 1 . . . k − 1

(consequent vertices must be connected with arcs).

• A sequence of vertices (v1, v2, . . . , vk) is called a chain if

(vi, vi+1) ∈ A or (vi+1, vi) ∈ A, i = 1 . . . k − 1

(consequent vertices must be connected, direction of lines is not important).

• A walk is elementary if all its vertices, except maybe initial and terminal, aredifferent. We will call an elementary walk a path.

• A walk is simple if all its lines are different.

• If v1 = vk, the walk is called a closed walk or circuit (the walk starts and endsin the same vertex).

• A cycle is a closed walk of at least three vertices in which all lines are distinct



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and all vertices except the initial and terminal are distinct.

• The chain starting and ending in the same vertex is called a closed chain.

• If v1 = vk and k = 1, the circuit is called a loop (a connection of a vertex toitself).

• The length of a path is k − 1 (the number of lines traveled).

• There can exist several paths between two vertices. The most interesting is theshortest path (all shortest paths).

If a relation means telling news the shortest path will tell the shortest route ofinformation traveling between two persons. – e. g., if all shortest paths frompeople to a selected person are short the selected person is well or quicklyinformed.

In the case of a relation was in contact to, the shortest path tells the mostprobable way of passing an infection. In this case shortest paths are notwelcome.

• The length of the longest shortest path in a graph is called its diameter.



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v1

v2

v3

v4

v5

v6 v7 v8

v9

v10

v1 – v6 – v4 – v8 is not even a chain

v1 – v6 – v7 – v8 is a chain, but not a walk

v8 – v4 – v5 – v2 – v4 – v10 is a simple, not elementary walk

v8 – v4 – v5 – v2 – v3 – v9 is an elementary walk (path)

v4 – v10 – v8 – v4 is a closed chain

v6 – v5 – v7 – v6 is a cycle



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v1 – v2 – v3 – v9 – v10 is a path between v1 and v10, but not the shortest path(length 4).

The shortest path is: v1 – v2 – v4 – v10 (length 3).

The diameter is 5: v7 – v6 – v5 – v2 – v3 – v9



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Some interesting resultsNetworks with a very large number of vertices can have quite short shortest pathsamong vertices.

For example:

• The average length of shortest path of the WWW, with over 800 millionvertices, is around 19.

Albert, R., Jeong, H., and Barabasi, A.-L. (1999):

Diameter of the World-Wide Web. Nature, 401, 130-131.

http://www.nd.edu/˜alb/Publication06/062 Diameter of the world wide

web/Diameter of the world wide web.pdf

• Social networks (whom do you know) with over six billion individualsare believed to have a average length of shortest path around six.

Milgram, S. (1967): The small-world problem.Psychol. Today, 2, 60-67.



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ComponentsThree types of components will be defined: strong, weak and biconnected.

A subset of vertices in a network is called a strongly connected component if(taking directions of lines into account) from every vertex of the subset we can reachevery other vertex belonging to the same subset.

If direction of lines is not important (where we consider the network to beundirected), such a subset is called a weakly connected component.

Example: Let the vertices of network correspond to buildings in the city, and linesto streets that connect the buildings. Some streets are ordinary (undirected lines),some are one way only (directed lines).

All buildings that can be reached from one to another using the car, belong to thesame strongly connected component (we must take one way streets into account).

All buildings that can be reached from one to another by walking, belong to thesame weakly connected component (one way streets are no restriction).



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Lets take the relation whom would you invite to the party. For all persons belongingto the same strongly connected component:

• everybody will (at least indirectly) invite everybody else from the same stronglyconnected component

• everybody will be (at least indirectly) invited by everybody else from the samestrongly connected component

If the network is undirected, a strongly connected component os the same as aweakly connected one, so they are simply called connected components.

Connection between walks in a network and components

Between any two vertices from the same strongly connected componentthere always exists a walk. We also say that the two vertices are stronglyconnected.

Between any two vertices from the same weakly connected componentthere always exists a chain. We also say that the two vertices are weaklyconnected.



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A network is called strongly/weakly connected if every pair of vertices isstrongly/weakly connected.

Example

a

b

c

ed f

g

h

i

j

The network is weakly connected.There exist 4 strongly connected components:1. (a), 2. (b, c, e), 3. (d, f), 4. (g, h, i, j)

Vertices that belong to the same strongly connected components are drawnusing the same shape.



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Biconnected components

Lets take the undirected connected network. Vertex a of the network is anarticulation point of the network if there exist two other, different verticesv and w, so that every chain between the two vertices includes also vertexa. Simply saying: vertex a is articulation point, if removing the vertex fromthe network causes the network to become disconnected.

p

rese

cis

ce

A network is called biconnected if for every triple of vertices a, v and w

there exists a chain between v and w which does not include vertex a.



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Example

a

b

c

d

e

f

g

h

i

Te network in the picture (aho1.net) consists of 4 biconnected components:

• (a, b, c)

• (b, d, e)

• (d, f)

• (f, g, h, i)

The articulation points are b, d and f .

Example: Airlines connections networks s y s l s y ss * 6


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CoresA subset of vertices is called a k-core if every vertex from the subset isconnected to at least k vertices from the same subset.

Formally:Let N = (V,L), L ⊆ V × V be a network.

A maximal subgraph H = (W,∩W ×W ) induced by the set W is a k-coreiff ∀v ∈W : degH(v) ≥ k.

Special example:A subset of vertices is called a clique if every vertex from the subset isconnected to every other vertex from the subset.



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0, 1, 2 and 3 core:

Properties of cores:

• Cores are nested (see figure):

i < j =⇒ Hj ⊆ Hi

• They are not necessarily connected subnetworks (see figure).

• Cores can be generalized to networks with values on lines.

Example: Airlines connections network



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Community detection methodsCommunities - dense clusters for which there are more lines inside thanamong clusters (values of lines are taken into account too).

In Pajek two community detection methods are available: Louvain methodand VOS Clustering.

When applying Louvain method we search for partition into clusters withthe highest value of modularity (Q). Modularity is defined in the followingway:

Q =1

2m

∑s

(es − r ∗ K2s

2m)

• m – total number of lines in network,

• s – cluster (community),

• es =∑

ij∈s Aij – 2 times the number of lines in community s



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• Ks =∑

i∈s ki – sum of degrees in community s

• r – resolution parameter, default value 1 means modularity as origi-nally defined

Similar method is VOS Clustering, where VOS quality function is taken intoaccount instead of modularity.

By changing resolution parameter - r we can get larger or smaller com-munities. By default resolution parameter is set to 1. Setting r larger than1 means searching for larger number of smaller communities. Setting r

smaller than 1 means searching for smaller number of larger communities.

Examples: football.net, import.net.

Some more examples:http://mrvar.fdv.uni-lj.si/pajek/community/LouvainVOS.htm



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Measures of centrality and prestige

One of the most important uses of network analysis is identification of’most central’ units in a network.

Measures of centrality and prestige can be defined in two different ways:

• for each unit respectively – unit centrality (one number for each unit)

• for the whole network – network centralisation (only one number forthe whole network).



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Unit centrality and prestigeWhen we talk about centrality we must distinguish between undirected anddirected networks:

• the term centrality measures is used for undirected networks;Example: A city is central, if a lot of roads are passing through it.

• the term prestige is used for directed networks. In this case we candefine two different types of prestige:

– one for outgoing arcs (measures of influence),

– one for incoming arcs (measures of support).

Examples:An actor has high influence, if he gives commands to several otheractors.An actor has high support, if a lot of people vote for him.



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Unit centrality measuresSelected unit is central

• if it has high degree,

• if it is easily accessible (close to) all other units,

• if it lies on several geodesics (shortest paths) between other units.

Star and cycle

According to all criteria mentioned, unit 7 in the star is the most central,while all units in the cycle are equally central.



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Degree CentralityThe simplest measure – unit is central in a network, if it is active enough inthe sense that is has a lot of links to other units (unit 7 in the star). In thecase of cycle all units are equally central.

Degree centrality is defined by a degree of unit x

cD(x) = degree of unit x

Such measures are called absolute measures of centrality. Absolutemeasures cannot be used to compare centralities of networks with differentnumber of units. Therefore such measures are normalised to get measurein interval from 0 to 1, where 0 means the smallest possible and value 1the highest possible centrality. Measures normalised in this way are calledrelative measures of centrality.



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Relative degree centrality is

CD(x) =cD(x)

highest degree=

cD(x)

n− 1

if n is number of units in a network, the highest possible degree (in anetwork without loops) is n− 1.

The same centrality measure can be used as a measure of prestige fordirected networks. In this case there are two possibilities

• to choose (influence – out degree: number of arcs going out)

• to be chosen (support – in degree: number of arcs coming into).



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Closeness CentralitySabidussi (1966) suggested the measure of centrality according to closenessof unit x:

cC(x) =1∑

y∈U d(x, y)

where d(x, y) is the graph theoretic distance (length of shortest path)between units x and y, U is set of all units.

If network is not strongly connected, we take only reachable vertices intoaccount, but we weight the result with number of reachable vertices.

Ce omrezje ni krepko povezano, upostevamo samo dosegljive tocke, s temda mero utezimo z mocjo mnozice dosegljivih tock.

The most central units according to closeness centrality can quickly interactto all others because they are close to all others.

This measure is preferable to degree centrality, because it does not take intoaccount only direct connections among units but also indirect connections.



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Betweenness CentralityIn the case of communication networks the distance from other units is notthe only important property of a unit. More important is which units lie onthe shortest paths among pairs of other units. Such units have control overthe flow of information in the network.

Idea of betweenness centrality measures: unit is central, if it lies on severalshortest paths among other pairs of units.

Freeman (1977) defined the centrality measure of unit x according tobetweenness in the following way:

cB(x) =∑y<z

# of shortest paths between y and z through unit x# of shortest paths between y and z

Suppose that communication in a network always passes through shortestavailable paths: Betweenness centrality of unit x is the sum of probabilitiesacross all possible pairs of units, that the shortest path between y and z willpass through unit x.



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Choosing suitable centrality measureDifferent centrality measures can give quite different results. Thereforewe must be very careful which centrality measure to choose for a givennetwork:

It can happen, that some units have low degrees, but high betweennesscentrality.

Example: units 33, 53, 32 in 48 are the most central according to between-ness centrality, although several other vertices with higher degree centralityexist (unit 61 has degree 6, some units have degree 5). The four unitsmentioned have degree only 3 or 4.



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1

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5152

53

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0



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Example: Padgett’s Florentine Families

ACCIAIUOL

ALBIZZI

BARBADORI

BISCHERI

CASTELLAN

GINORI

GUADAGNI

LAMBERTES

MEDICI PAZZI

PERUZZI RIDOLFI

SALVIATI

STROZZI TORNABUON



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Relative centrality measuresNo. Family CD CC CB

1. Acciaiuoli 0.071 0.368 0.0002. Albizzi 0.214 0.483 0.2123. Barbadori 0.143 0.438 0.0934. Bischeri 0.214 0.400 0.1045. Castellani 0.214 0.389 0.0556. Ginori 0.071 0.333 0.0007. Guadagni 0.286 0.467 0.2558. Lamberteschi 0.071 0.326 0.0009. Medici 0.429 0.560 0.522

10. Pazzi 0.071 0.286 0.00011. Peruzzi 0.214 0.368 0.02212. Ridolfi 0.214 0.500 0.11413. Salviati 0.143 0.389 0.14314. Strozzi 0.286 0.438 0.10315. Tornabuoni 0.214 0.483 0.092



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Brokerage rolesIn a network in which groups are defined (e.g. men and women) each unitcan be involved in the following brokerage roles:

• coordinator,

• itinerant broker,

• representative,

• gatekeeper,

• liaison.

j

i k

coordinator

j

i k

itinerant broker

j

i k

liaison

j

i k

gatekeeper

j

i k

representative



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Genealogies as large networksGEDCOM is standard for storing genealogical data, which is used to interchange

and combine data from different programs. The following lines are extracted from

the GEDCOM file of European Royal families.

0 HEAD 0 @I115@ INDI1 FILE ROYALS.GED 1 NAME William Arthur Philip/Windsor/... 1 TITL Prince0 @I58@ INDI 1 SEX M1 NAME Charles Philip Arthur/Windsor/ 1 BIRT1 TITL Prince 2 DATE 21 JUN 19821 SEX M 2 PLAC St.Mary’s Hospital, Paddington1 BIRT 1 CHR2 DATE 14 NOV 1948 2 DATE 4 AUG 19822 PLAC Buckingham Palace, London 2 PLAC Music Room, Buckingham Palace1 CHR 1 FAMC @F16@2 DATE 15 DEC 1948 ...2 PLAC Buckingham Palace, Music Room 0 @I116@ INDI1 FAMS @F16@ 1 NAME Henry Charles Albert/Windsor/1 FAMC @F14@ 1 TITL Prince... 1 SEX M... 1 BIRT0 @I65@ INDI 2 DATE 15 SEP 19841 NAME Diana Frances /Spencer/ 2 PLAC St.Mary’s Hosp., Paddington1 TITL Lady 1 FAMC @F16@1 SEX F ...1 BIRT 0 @F16@ FAM2 DATE 1 JUL 1961 1 HUSB @I58@2 PLAC Park House, Sandringham 1 WIFE @I65@1 CHR 1 CHIL @I115@2 PLAC Sandringham, Church 1 CHIL @I116@1 FAMS @F16@ 1 DIV N1 FAMC @F78@ 1 MARR... 2 DATE 29 JUL 1981... 2 PLAC St.Paul’s Cathedral, London



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Representation of genealogies using networks

Genealogies can be represented as networks in different ways:

• as Ore-graph,

• as p-graph,

• as bipartite p-graph.



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Ore-graph: In Ore-graph every person is represented by a vertex, marriages arerepresented with edges and relation is a parent of as arcs pointing from each ofthe parents to their children.

grandfather-f grandmother-f

father mother

ME wife

sondaughter-in-law daughter son-in-law

stepmother

grandfather-m grandmother-m

sisterbrothersister-in-law



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p-graph: In p-graph vertices represent individuals or couples. In the case thatperson is not married yet (s)he is represented by a vertex, otherwise personis represented with the partner in a common vertex. There are only arcs inp-graphs – they point from children to their parents.

son-in-law & daughter

ME & wife

father & mother

grandfather-f & grandmother-f grandfather-m & grandmother-m

son & daughter-in-law

brother & sister-in-law sister

father & stepmother



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Bipartite p-graph: has two types of vertices – vertices representing couples (rect-angles) and vertices representing individuals (circles for women and trianglesfor men). Arcs again point from children to their parents.

son-in-law & daughter

ME & wife

father & mother

grandfather-f & grandmother-f grandfather-m & grandmother-m

son & daughter-in-law

brother & sister-in-law

father & stepmother

son daughter

brother sister

fathermother

ME

daughter-in-law son-in-law

sister-in-law wife

stepmother

grandfather-fgrandmother-f

grandfather-mgrandmother-m



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Largest component of genealogy of American presidents

George H.W. Bush

George W. Bush

Franklin D. Roosevelt



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Relinking marriages (p-graphs with 2 up to 6 vertices)

A3A2 A4.1B4

A4.2

B6.2

A5.1 A5.2

B5

A6.1 A6.2A6.3

C6

B6.1B6.3

B6.4



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Relinking marrages in genealogy of European noble families

Genealogies of noble families are much more relinked than the ’usual’ one.In 2016 we celebrated 100 years since Franc Jozef died. He was married tohis cousin ’Sisi’.



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...More relinking marraiges



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...More relinking marraiges



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Blood marriage grandson-granddaughter in Genealogy of Ragusan noble families

Benedictus (Benko)/Gondola/ & Rade/Goce/

Petrus/Gondola/ & Gondola/Gondola/ Symoneto/Bona/ & Madussa/Gondola/

Benedictus/Gondola/ & Anucla/Bona/



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Non-blood relinking marriages

Damianus/Georgio/ & Legnussa/Babalio/

Marin/Gondola/ & Magdalena/Grede/

Nicolinus/Gondola/ & Franussa/Bona/

Marinus/Bona/ & Phylippa/Mence/

Sarachin/Bona/ & Nicoletta/Gondola/

Marinus/Zrieva/ & Maria/Ragnina/

Lorenzo/Ragnina/ & Slavussa/Mence/Junius/Zrieva/ & Margarita/Bona/

Junius/Georgio/ & Anucla/Zrieva/ Michael/Zrieva/ & Francischa/Georgio/ Nicola/Ragnina/ & Nicoleta/Zrieva/



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BlockmodelingThe goal of blockmodeling is to reduce a large, potentially incoherentnetwork to a smaller comprehensible structure that can be interpreted morereadily. Blockmodeling, as an empirical procedure, is based on the idea thatunits in a network can be grouped according to the extent to which they areequivalent, according to some meaningful definition of equivalence.



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Structural equivalence

Units X and Y are structurally equivalent if they are connected to the restof the network in identical ways (Lorrain and White, 1971). If X and Y arestructurally equivalent they are interchangeable.

X and Y are structurally equivalent iff:

s1. XRY ⇔ Y RX

s2. XRX ⇔ Y RY

s3. ∀Z ∈ E \ {X,Y } : (XRZ ⇔ Y RZ)

s4. ∀Z ∈ E \ {X,Y } : (ZRX ⇔ ZRY )



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Ideal blocks compatible with structural equivalence are:



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Regular equivalence

X and Y are regularly equivalent if they link in equivalent ways to otherunits that are also equivalent.

X and Y are regularly equivalent iff:

R1. XRZ ⇒ ∃W ∈ E : (Y RW ∧W ≈ Z)

R2. ZRX ⇒ ∃W ∈ E : (WRY ∧W ≈ Z)



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Ideal blocks compatible with regular equivalence are:

• null block

• blocks that have the property that there is at least one 1 in each of theirrows and in each of its columns in regularni bloki

Structural euqivalence is also regular equivalence.



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Example

The analyzed network consists of social support exchange relation amongfifteen students of the Social Science Informatics fourth year class(2002/2003) at the Faculty of Social Sciences, University of Ljubljana.Interviews were conducted in October 2002.

Support relation among students was identified by the following question:Introduction: You have done several exams since you are in the secondclass now. Students usually borrow studying material from their colleagues.Enumerate (list) the names of your colleagues that you have most oftenborrowed studying material from. (The number of listed persons is notlimited.)



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m02

m03

w07

w09

w10

w12

w22 w24

w28

w42

m51

w63

m85

m89

m96



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Indirect approach - using dissimilarity matrix

m51

m89

m02

m96

m03

m85

w10

w24

w09

w63

w12

w07

w28

w22

w42

m02

m03

w07

w09

w10

w12

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Reoredered matrix

m02

m03

w10

w24

m51

m85

m89

m96

w07

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w42

w09

w12

w63

m02

m03

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m51

m85

m89

m96

w07

w22

w28

w42

w09

w12

w63



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Direct approach - optimizationDoes center-periphery model fit our data?

C1 C2

C1 com, reg -

C2 com, reg -

m02

m03

w07

w09

w10

w12

w22 w24

w28

w42

m51

w63

m85

m89

m96

w07

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w24

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m85

m02

m03

w09

m51

m89

m96

w07

w10

w12

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w24

w28

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m85

m02

m03

w09

m51

m89

m96


Documents

Introduction to Social Network Analysis s with Pajekmrvar.fdv.uni-lj.si/sola/dr/slideseng.pdf · 2017. 3. 22. · A. Mrvar: Social Network Analysis 5 ’ & $ % X iRX j)there exists