Introduction to Social Network Analysis theory and its application to CSCL

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Introduction to Social Network Analysis theory
and its application to CSCL

• Christophe Reffay (France)
  Alejandra Martínez-Monés (Spain)

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Outline

• Introduction historical examples
• Milgram, Moreno, Granovetter, Burt, Durkheim,
• Social capital (e.g guanxi), Leaders
  Facilitators
• Graphs Sociograms definition properties
• What is it interesting for?
• Building sociograms in practice

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The world is so little
Fine! But how little exactly?
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Example 1 Milgram, 1967

• Target person
• Name
• Trader
• Lives in Boston

Nebraska
Boston

• Starting senders
• 100 live in Boston
• 96 live in Nebraska
• 100 traders in Nebraska

From 296 documents, only 217 have been sent,
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Only 64 documents reach the target
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So What is the world size?

• Geographically directed intermediaries 6,1
• Professionally directed intermediaries 4,6
• Globally
• For USA 5 intermediaries
• (Kochen,1989) shows stability of this result

gt The 6 degrees of separation
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The New Small-World Experiment (bigger, faster,
and less expensive)

• (Dodds, Muhamad, and Watts, 2002)
• Very similar to Milgrams Experiment, but
  web-based
• Initial results
• 60,000 senders
• 19 targets
• 171 countries
• 380 chains complete (worse attrition than
  Milgram)
• Median chain length ranges from 5 (for same
  country) to 7 (for different countries)

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Conversely how many person do we know?

• Granovetter (1976) defines this relation
   Person that we met, for whom the contact has
  been established 
• Ithiel de Sola Pool (1978)Exp. 100 days gt 3500
  persons every 20 years
• Freeman Thomson (1989)(Phone book 305/112147)
  gt 5520

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Strong vs weak ties (Granovetter 1973)

• Strong ties
• (longevity, emotional intensity, intimacy,
  reciprocity)
• gtTransitivity gt Cliques (sub-groups highly
  connected)
• Strong ties connexions inside the clique
• Weak ties bridges between cliques

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Holism

• An individual acts according to the rules of
  the group he belongs to

Weak Rules are interiorised by socialisation
Ronald Stuart Burt
Strong Structure determines action
Émile Durkheim
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The Strength of weak ties (Granovetter)

• Social Capital (guanxi ??)
• Have a lot of contacts
• To be able to go beyond the clique by activating
  bridges
• Easier to find a good house/job/friend/collaborat
  or
• Nan Lin (1982) shows efficiency of weak ties in
  a Milgram-like experiment (Men/Women x
  Black/White)

bridges
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The Morenos experiments (1943)

• Pupils relation in the classroom
• Pupils of various age range
• Gender study
•  If you could choose freely, which are the (2)
  kids you would like to have as direct
  neighbour? 

• Main results
• At ltagegt gt pupils tend to lt?gt
• 6-8 years old gt mix
• 8-13 years old gt separate
• 13-15 years old gt mix
• 15-17 years old gt separate

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Outline

• Introduction historical examples
• Graphs Sociograms definition properties
• Graphs types, nodes, edges, Relationships,
• Density, Distance, Path, Radius, Diameter
• Centrality, Betweenness, Cohesion
• What is it interesting for?
• Building sociograms in practice

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Graphs Sociograms
See http//en.wikipedia.org/wiki/Social_network
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What SNA deals with?

• Types of the graph
• 1-mode, 2-mode
• Directed/Undirected
• Weighted (valued) graphs
• Measures
• Density, Diameter, Distances, In/Out degree,
• Structural Cohesion, Betweenness Centrality
• Sub graphs
• Cliques
• Clusters

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Graph types examples

• A set of nodes linked by edges/ties
• Examples (one-mode) nodes are homogeneous
• Networks (Cities, roads)
• Communities of pairs (Persons, relationships)
• Examples (two-mode) 2 distinct node types
• Collaborative Activities (Actors-Objects,
  Actions)
• Affiliation network (Actors-Structure,
  Affiliation)
• Directed graphs ties are oriented
• Valued graphs ties have weights

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Sociogram Individuals and relationships

• Nodes are generally individuals (workers,
  actors, learners, teachers, tutors)
• Edges are relationships between
  individuals(communication, resource sharing,
  service exchanges, friendship, family links, )

• Example
• One-mode
• Directed
• Unvalued

Generalisation Nodes can be groups or
corporations
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One-mode or Two-mode networks
Two-mode
One-mode

• All nodes are of the same type

• Administrators
• Societies

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Directed vs Undirected graphs

• Directed

• Undirected

Edges are oriented
Edges are not oriented

• Directed relationship meaning has sent some
  message to
• Undirected relationship meaning has exchanged
  some message with

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Weighted (valued) vs Unvalued graphs

• Weighted/Valued

Edges have values
gt Transform the meaning of relationship!

• Unvalued

Edges have no value
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Conclusion 8 possible network types
Two-mode(Two node types)
One-mode(One node type)
One-Mode Directed Valued One-Mode Undirected Valued
One-Mode Directed Unvalued One-Mode Undirected Unvalued
Two-Mode Directed Valued Two-Mode Undirected Valued
Two-Mode Directed Unvalued Two-Mode Undirected Unvalued
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Network types transformation allowed
Two-Mode
One-Mode
Directed
Undirected
Valued
Unvalued
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Two-Mode
One-Mode
Strategy Decide what shared resource represent
for relationships between (blue) nodes.
Do blue nodes share any orange resource? gt
Unvalued
1
2
2
1
1
1
How many orange resource do blue nodes share ? gt
Valued
NB As made in some CSCL research, we can also
transform the two-mode network to a network of
orange nodes (resource network)
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Directed
Undirected
Strategy Decide if you have/not edges in both
directions.
Are nodes connected (one tie is enough)?
Are nodes connected with reciprocal edges?
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Valued
Unvalued
Strategy Only ties with valuegtThreshold are
considered
Threshold5
Threshold8
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Useful definitions measures on graphs

• Density of the graph,
• Degree, In-degree, Out-degree
• Path, Geodesic distance, Diameter
• Centrality indices (for nodes)
• Degree centrality
• Betweenness centrality,
• Closeness centrality
• Ego-net
• Cliques,

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Density (of edges) for an undirected graph
Eff.0 Poss.10 d0 Eff.2 Poss.10 d0.2 Eff.4 Poss.10 d0.4 Eff.8 Poss.10 d0.8 Eff.10 Poss.10 d1

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Density (of edges) for a directed graph
Reciprocal edges count twice (twice more
possible edges)
Eff.0 Poss.20 d0 Eff.4 Poss.20 d0.2 Eff.8 Poss.20 d0.4 Eff.21 Poss.25 d0.8 Eff.10 Poss.25 d1

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Degree in an undirected graph

• For a node, Degree number of edges

Degree is one of the Centrality measuresgt also
called Degree Centrality
NB These indices have a normalized version where
the of ties is divided by the maximum number of
ties possible
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In- Out- degree in an directed graph

• In-degree number of edges coming in to the node

Out-degree number of edges coming out of the
node
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Path sequence of edges connecting 2 nodes
Example in a directed graph
C
G
A
B
E
F
I
J
H
D

• From A-gtE 2 possible paths
• (A B C E)
• or
• (A B D E)

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Path example in an undirected graph
C
G
A
B
E
F
I
J
H
D

• From A-gtE 2 possible paths
• (D E)
• or
• (D B C E)

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Diameter of the graph

• Diameter longest distance in the graph
  maximal distance between any pair of nodes

What is the diameter of this graph?
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Betweenness centrality

• Number of shortest paths passing through the node

Directed graph
Undirected graph
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Another example for betweenness
Highest score
Lowest score

• Source http//en.wikipedia.org/wiki/Centrality

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Closeness centrality

• Scoring the closeness of one node to all others

Directed graph
Undirected graph
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Eigenvector centrality

• Scoring the importance of a node in the network
• (take into account the importance of connected
  nodes)

Undirected graph
Nb Eigenvector centrality cannot be calculated
for directed graphs
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The structure as a constraint
Net A
2
6
Density DA9/280,321
1
4
5
8
3
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Net B
2
6
Density DB9/280,321
1
4
5
8
3
7
Do nodes 4 and 5 have the same role in nets A
and B?
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Measures for the network B
Betweenness
Degree
Eigenvector
Closeness
2LocalEigenvector
Harmonic Closeness
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Measures for the network A
Betweenness
Degree
Eigenvector
Closeness
2LocalEigenvector
Harmonic Closeness
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Ego-net The network of ego (undirected)

• Ego the selected node
• Alters (neighbours) distance(Ego,Alter) 1
• Ties between ego and alters
• Ties between alters

Ego-net (x34)
Ego-net (x38)
Whole network
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Ego-net (considering edge direction)
Layout / Ego network (new)
Alter-gtEgo (x38)
Ego-gtAlter (x38)
Alterlt-gtEgo (x38)
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Detection/building of sub-regions (NetDraw)

• Components simply connected sub-groups
• K-cores or K-cliques members of core k are
  connected to (at least k-1 other members)
• Choosing the number of sub-regions
• Hiclus of geodesic distance (Hierarchical
  Cluster)
• Factions
• Girvan-Newman Clustering (min max clusters)
• Polished?

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Components
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This results in breaking the component
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K-cores
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Cliques or K-cliques

• Clique maximum subset where all nodes are
  connected
• K-clique Clique with K members

How many cliques?
gt 6 cliques
Which are ?

• One 5-clique
• One 4-clique
• One 3-clique
• Three 2-cliques

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Hierarchical Clusters
6 clusters
8 clusters
2 clusters
4 clusters
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Why are they called Hierarchical clusters
Source (french) http//asi.insa-rouen.fr/enseigne
ment/siteUV/dm/Cours/clustering.pdf
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Outline

• Introduction historical examples
• Graphs Sociograms definition properties
• What is it interesting for?
• Building sociograms in practice
• Conception, transformation, tools format

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General view
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Data representation for graphs

• Graph a set of nodes connected by edges.
• Defining separately
• The set of nodes and
• The set of edges
• Giving the adjacency matrix
• Square matrix defining connections (values)
  between all pairs of nodes

Nodes Gl1, Gl2, Gl4, Gn1, Gn2 Edges (Gn2,Gl4),
(Gl4,Gn1)
Gl1 Gl2 Gl4 Gn1 Gn2
Gl1 0 0 0 0 0
Gl2 0 0 0 0 0
Gl4 0 0 0 1 1
Gn1 0 0 1 0 0
Gn2 0 0 1 0 0
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Format/languages and tools

• Graph languages
• Text based
• Ucinet, Dot, TGF
• XML based
• GraphML, RDF
• GXL, GML, XGMML
• Table based
• DB, CSV, XLS

• Image formats
• Static image
• ps, .gif, .raw, .ppm, .bmp, .jpg, .png,
• Vector graphics/graph
• .svg, .emf, .gexf, .tif
• Interactive scripts
• javascripts

Input
SNA Tools
Output(Visualisation)
Output(transformation)
Output(other)
Other Information Index computation, Sub-graph
detection
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Tools formats interoperability