Sunbelt XXIV, Portorož, 20041 Pajek Workshop Vladimir Batagelj Andrej Mrvar Wouter de Nooy.

Sunbelt XXIV, Portorož, 2004 1

Pajek Workshop

Vladimir BatageljAndrej Mrvar

Wouter de Nooy


Today’s Program• Introduction to Pajek and social

network analysis

• Analysing large networks with Pajekand fine-tuning layouts

• Discussion and questions


PART 1

Exploratory Network Analysis

with Pajek

(Published at Cambridge University Press, October 2004)

W. de Nooy, A. Mrvar, V. Batagelj

ž


Overview• Network data• Vertex attributes

and properties • Cohesive

subgroups:– in simple networks– in signed networks– in valued networks

• Brokerage:– centrality– structural holes– brokerage roles

• Ranking:– prestige– acyclic networks

• Blockmodeling• Networks and time

– repeated measurement

– diffusion– genealogies, citations

• Network analysis and statistics

• Building your own


Network data• Opening a network in Pajek• Drawing a network in Pajek

– Energizing the layout– Selecting display options– Exporting the sociogram

• Pajek network data– Structure– Store & export from Access

• Example: World trade relations– Imports_manufactures.net


Vertex attributes and structural properties

• Types of data objects– Partitions: discrete properties– Clusters: 1 class from a partition– Vectors: continuous (numeric) properties– Hierarchies: nested classification– Permutations: reordering (renumbering)

• Visualizing partitions and vectors• Menu structure• Pajek project file


Cohesive subgroupsin simple networks

• Connectivity• Example: Attiro.paj• Measures:

– Components: weak and strong– k-cores– Cliques, complete subnetworks

• Analytic strategy


1. Is the netw ork d irected? In fo>N etw ork>G eneral

2b. F ind w eak com ponents

D o c

N et>C om ponents>W eak

om ponents identify subgroups?

2a. F ind com ponents

D o com ponents identify subgroups?

N et>C om ponents>W eak

Fin ish: subgroupsnot found.

3a. F ind -coresk

conta in subgroups?D o k-cores

N et>Partitions>C ore>Input

4a. R em ove vertices of the low est -coresk O perations>Extract from N etw ork>Partition

Fin ish: subgroupsare classes in the

com ponentspartition.

5.5 F ind com ponents (see 2a.)

5a. F ind overlapping com plete subnetw orks

D o subnetw orks identify subgroups?

Select and execute

N ets>Find Fragm ent (1 in 2) >O ptions>Extract Subnetw ork

N ets>Find Fragm ent (1 in 2)>F ind

3b. F ind strong com ponents N et>C om ponents>Strong

D o com ponents identify subgroups?

5b. Sym m etrize the netw orkN et>Transform >Arcs->Edges>A ll

no yes

yes yes

nono

yesno

yes

no

yes

yes

no

4b. F ind overlapping com plete subnetw orks Select

and execute

N ets>Find Fragm ent (1 in 2) >O ptions>Extract Subnetw ork

N ets>Find Fragm ent (1 in 2)>F ind

D o subnetw orks identify subgroups?

no


Cohesive subgroups in signed networks

• Balanced clusters• Example: Sampson.paj• Using line values & signs in layout• Optimization approach

– Set parameters– Search optimal solution– Repeat many times

• Stepping through partitions


Cohesive subgroupsin valued networks

• Cohesion by strong or multiple ties• Example: interlocking directorates

in Scottish banking (circa 1900) Scotland.paj

• Transform 2-mode into 1-mode network

• Measure:– m-core (valued core)

• SVG output


Centrality• Centrality and centralization• Undirected networks (Knoke & Burt,

1983)

• Example: Strike.paj– Degree– Closeness– Betweenness


Brokerage• The flow of information• Example: Strike.paj• Overall network structure:

– Bridges– Cut-vertices or articulation points– Bi-components

• Investigating the ego-network:– Structural holes– Brokerage roles


5 Brokerage roles

v

u w

coord inator

v

u w

itinerant broker

v

u w

lia ison

v

u w

gatekeeper

v

u w

representative


Prestige• Asymmetric choices• Example: SanJuanSur2.paj• Measures:

– Popularity: indegree– Input domain: direct and indirect

nominations– Proximity prestige: size of domain

divided by the average distance within the domain

• Structural and social prestige


Ranks: acyclic networks• Discrete ranks or levels• Example: student_government.paj• Local network structure:

– Triadic analysis and the triad census

• Overall network structure:– Strong components and ranks– Symmetric-acyclic decomposition


Balance-theoretic modelsModel Ties within a cluster Ties between ranks Permitted triads

Balance symmetric ties within a

cluster, no ties between

clusters; max 2 clusters

none 102, 300

Clusterability idem

no restriction on the

number of clusters

idem + 003

Ranked

Clusters

idem asymmetric ties from each

vertex to all vertices on

higher ranks

+ 021D, 021U,

030T, 120D, 120U

Transitivity idem null ties may occur

between ranks

+ 012

Hierarchical

Clusters

asymmetric ties within a

cluster allowed provided

that they are acyclic

idem + 120C, 210

no balance-theoretic model (‘forbidden’) 021C, 111D,

111U, 030C, 201


Triad types and models

1 - 003

2 - 012

3 - 102

4 - 021D

5 - 021U

9 - 030T

12 - 120D

13 - 120U

14 - 120C

15 - 210

16 - 300

Balance C lusterability

Transitiv ity H ierarchica lC lusters

R anked C lusters


Blockmodeling• Matrix and permutation for visualization• Blockmodel

– Partition of vertices into classes (positions)– Image matrix of relations among blocks

• Types of blockmodels– Cohesive subgroups– Center-periphery structure– Ranks

• Types of equivalence:– Structural equivalence: hierarchical clustering– Regular equivalence


Cohesive subgroupsD om ingo

C arlosA le jandroEduardo

FrankH al

KarlBobIkeG ill

LannyM ikeJohn

XavierU trecht

N ormR ussQ uint

W endleO zzie

TedSamVernPaul


Image matrix

Class SpanishEnglish –

youngEnglish –

old

Spanish Complete Empty Empty

English – young

Empty Complete Empty

English – old

Empty Empty Complete


Blockmodel types

C ohesion C enter-periphery R anking

Im age m atrix


Regular equivalence and errors

pm inister

m in isters

m inister2

m inister3

m inister4

m inister5

m inister6

m inister7

advisor1

advisor2

advisor3

pm inister advisors

m inister1

X

X

XX

X

X

X

X


Networks and time• Longitudinal network: a network

measured at different time points– Example: Sampson.paj

• Diffusion: vertex property changing over time, e.g., adoption– Example: ModMath.paj

• Descent: a relation spanning time– Genealogies: descent by birth; structural

relinking– Citations: descent of ideas; main path analysis– Example: Gondola_Petrus.ged,

centrality_literature.paj


Genealogies• Data format: GEDCOM 5.5 standard

www.gendex.com/gedcom55/55gcint.htm

• Software:- Genealogical Information Manager www.mind spring.com/~dblaine/gim home.html- Personal Ancestral File www.familysearch.org


Networks and statistics• Statistical relations among properties of

vertices: partitions and vectors• Example: social and structural prestige

(Ch. 9)• In Pajek: discrete (Cramer’s V, Rajski,

rank correlation) and continuous (Pearson correlation, regression)

• Pajek to R: see afternoon session• Pajek to other statistics software: paste

numbers from partition or vector into statistics software datasheet


Building your own• Macro: sequences of commands

performed on selected data objects

• Example: exposure in a diffusion network

• Macro commands:– Record– Add message: add comment– Play


Relations among chapters

C h.1 - Looking for socia l structure

C h.2 - A ttributes and re la tions

C h.3 - C ohesive subgroups

C h.4 - Sentim ents and friendship

C h.5 - A ffilia tions

C h.6 - C enter and periphery

C h.7 - B rokers and bridges

C h.8 - D iffusion

C h.9 - P restige

C h.10 - R anking

C h.11 - G enealogies and cita tions

C h.12 - B lockm odels

Sunbelt XXIV, Portorož, 20041 Pajek Workshop Vladimir Batagelj Andrej Mrvar Wouter de Nooy.

Documents

network dataopening

social network analysis

exploratory network

pajoverall network structure

pajlocal network structure

triadic analysis

citationsnetwork analysis

properties cohesive