Ranking in social
networks
Miha Grčar
Course in Knowledge Management
Lecturer: prof. dr. Nada Lavrac
Ljubljana, January 2007
Outline of the presentation

I. Prestige



Structural prestige, social prestige, correlation
Ways of calculating structural prestige
II. Ranking



Triad census
Acyclic decomposition
Symmetric-acyclic decomposition
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I. Prestige

Prestigious people


Structural prestige measures

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People who receive many positive in-links
Popularity or in-degree
(Restricted) input domain
Proximity prestige
Structural prestige ≠ social prestige (social status)
Correlation between structural and social prestige


Pearson’s correlation coefficient
Spearman’s rank correlation coefficient
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Popularity or in-degree
0
1
0
2
3
0
1
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Miha Grčar
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4
Input domain

Input domain size


How many nodes are path-connected to a particular node?
Overall structure of the network is taken into account
0
3
0
2
7
0
1

0
Problematic in a well-connected network
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Restricted input domain

Resolves the input-domain issue in a
well-connected network
0
3
0
2
5
0
1
0
Maximum distance = 2

Issue: the choice of the maximum distance is
quite arbitrary
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Proximity prestige


Eliminates the maximum-distance parameter
Closer neighbors are weighted higher
0
0.23
0
0.25
0.47
0
0.13
Proximity prestige =
Ljubljana, January 2007
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Input domain size
7/8
Number of nodes
= 0.47
Average
(3+3+2+2+1+1+1)
(3+3+2+2+1+1+1
(3+3+2+2
(3+3
path-distance to /the
7 node
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II. Ranking


We discuss techniques to extract discrete ranks
from social relations
Triad analysis helps us determine if our network is
biased towards…




Unrelated clusters (cluster = clique)
Ranked clusters
Hierarchical clusters
Recipes for determining the hierarchy
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Acyclic decomposition
Symmetric-acyclic decomposition
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Triad analysis
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Triads
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Atomic network structures (local)
16 different types
M-A-N naming convention
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Mutual positive
Asymmetric
Null
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All 16 types of triads
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Triad census

6 balance-theoretic models
Restricted to
Balance
 Clusterability
Unrelated clusters
 Ranked clusters
Ranked clusters
Less and less restricted
 Transitivity
 Hierarchical clusters
Hierarchical clusters
Unrestricted
 (Theoretic model)
Triad census: triads found in the network, arranged
by the balance-theoretic model to which they belong


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Acyclic decomposition


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Cyclic networks (strong components) are
clusters of equals
Acyclic networks perfectly reflect hierarchy
Recipe

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Partition the network into strong components
(i.e. clusters of equals)
Create a new network in which each node
represents one cluster
Compute the maximum depth of each node to
determine the hierarchy
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Acyclic decomposition
An example
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Acyclic decomposition
An example
(1)
(0)
(2)
The maximum depth of a node determines its position in the hierarchy
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Symmetric-acyclic
decomposition

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
Strong components are not strict enough to
detect clusters in the triad-census sense
Symmetric-acyclic decomposition extracts
clusters of vertices that are connected both
ways
After the clusters are identified, we can follow
the same steps as in acyclic decomposition
to determine the hierarchy
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Symmetric-acyclic decomp.
An example
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Reference


Batagelj V., Mrvar A., de Nooy W. (2004):
Exploratory Network Analysis with Pajek.
Cambridge University Press
Some figures used in the presentation are
taken from this book
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