Pajek Workshop
Vladimir Batagelj
Andrej Mrvar
Wouter de Nooy
Sunbelt XXIV, Portorož, 2004
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Today’s Program
• Introduction to Pajek and social
network analysis
• Analysing large networks with Pajek
and fine-tuning layouts
• Discussion and questions
Sunbelt XXIV, Portorož, 2004
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PART 1
Exploratory Network Analysis
ž
with Pajek
(Published at Cambridge University Press, October 2004)
W. de Nooy, A. Mrvar, V. Batagelj
Sunbelt XXIV, Portorož, 2004
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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
Sunbelt XXIV, Portorož, 2004
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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
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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
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Cohesive subgroups
in simple networks
• Connectivity
• Example: Attiro.paj
• Measures:
– Components: weak and strong
– k-cores
– Cliques, complete subnetworks
• Analytic strategy
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1. Is the network directed?
no
yes
Info>Network>General
2a. Find components
2b. Find weak components
Net>Components>Weak
Net>Components>Weak
Do components identify subgroups?
Do components identify subgroups?
yes
yes
no
no
Finish: subgroups
are classes in the
components
partition.
3b. Find strong components
yes
Net>Components>Strong
Do components identify subgroups?
no
3a. Find k-cores
Net>Partitions>Core>Input
4b. Find overlapping complete
subnetworks
Do k-cores contain subgroups?
no
Select Nets>Find Fragment (1 in 2)
>Options>Extract Subnetwork and execute
Nets>Find Fragment (1 in 2)>Find
yes
Do subnetworks identify subgroups?
4a. Remove vertices of the lowest
k-cores
no
Operations>Extract from Network>Partition
yes
5b. Symmetrize the network
Net>Transform>Arcs->Edges>All
5.5 Find components (see 2a.)
5a. Find overlapping complete
subnetworks
Select Nets>Find Fragment (1 in 2)
>Options>Extract Subnetwork and execute
Nets>Find Fragment (1 in 2)>Find
yes
no
Do subnetworks identify subgroups?
Finish: subgroups
not found.
Sunbelt XXIV, Portorož, 2004
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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
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Cohesive subgroups
in 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
Sunbelt XXIV, Portorož, 2004
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Centrality
• Centrality and centralization
• Undirected networks (Knoke & Burt, 1983)
• Example: Strike.paj
– Degree
– Closeness
– Betweenness
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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
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5 Brokerage roles
v
v
u
w
u
w
v
gatekeeper
itinerant broker
v
u
v
w
coordinator
u
w
representative
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u
w
liaison
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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
Sunbelt XXIV, Portorož, 2004
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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
Sunbelt XXIV, Portorož, 2004
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Balance-theoretic models
Model
Ties within a cluster
Ties between ranks
Permitted triads
Balance
symmetric ties within a
cluster, no ties between
clusters; max 2 clusters
idem
no restriction on the
number of clusters
idem
none
102, 300
idem
+ 003
asymmetric ties from each
vertex to all vertices on
higher ranks
null ties may occur
between ranks
idem
+ 021D, 021U,
030T, 120D, 120U
Clusterability
Ranked
Clusters
Transitivity
Hierarchical
Clusters
idem
asymmetric ties within a
cluster allowed provided
that they are acyclic
no balance-theoretic model (‘forbidden’)
Sunbelt XXIV, Portorož, 2004
+ 012
+ 120C, 210
021C, 111D,
111U, 030C, 201
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Triad types and models
Balance
3 - 102
Clusterability
16 - 300
1 - 003
Ranked Clusters
4 - 021D
5 - 021U
Transitivity
9 - 030T
12 - 120D
13 - 120U
14 - 120C
15 - 210
Hierarchical
Clusters
2 - 012
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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
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Cohesive subgroups
Domingo
Carlos
Alejandro
Eduardo
Frank
Hal
Karl
Bob
Ike
Gill
Lanny
Mike
John
Xavier
Utrecht
Norm
Russ
Quint
Wendle
Ozzie
Ted
Sam
Vern
Paul
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Image matrix
Class
Spanish
English –
young
Spanish
Complete
Empty
Empty
English –
young
Empty
Complete
Empty
English –
old
Empty
Empty
Complete
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English –
old
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Blockmodel types
Image matrix
Cohesion
Center-periphery
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Ranking
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Regular equivalence and errors
pminister
ministers
advisors
X
pminister
minister1
minister2
minister3
X
minister4
minister5
minister6
X
X
minister7
advisor1
X
X
X
advisor2
X
advisor3
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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
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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
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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
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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
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Relations among chapters
Ch.5 - Affiliations
Ch.11 - Genealogies and citations
Ch.9 - Prestige
Ch.10 - Ranking
Ch.2 - Attributes
and relations
Ch.4 - Sentiments and friendship
Ch.12 - Blockmodels
Ch.1 - Looking
for social structure
Ch.3 - Cohesive
subgroups
Ch.6 - Center and periphery
Ch.7 - Brokers and bridges
Ch.8 - Diffusion
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