1.
Cliques, Clans and Clusters
Finding Cohesive Subgroups in
Network Data

2.
Social Subgroups
Frank & Yasumoto argue that actors seek social capital, defined as the
access to resources through social ties
a) Reciprocity Transactions
Actors seek to build obligations with others, and thereby
gain in the ability to extract resources.
b) Enforceable Trust
“Social capital is generated by individual members’
disciplined compliance with group expectations.”
c) Group Cohesion

3.
Goals
• Find a meaningful way to separate larger networks into
groups
• Meaningful =
• Reduce overlap
• Locate cohesive groups

4.
Reciprocity

5.
Reciprocity
• Ratio of reciprocated pairs of nodes to
number of pairs that have at least 1 tie
• In example, reciprocity = 0.5
• Called “dyad method”

6.
Transitivity
• Types of triadic relations (in undirected
networks):
• Isolation
• Couples only
• Structural holes
• Clusters (also cliques)

7.
In directed networks
• There are 16 types of triads
• Triad language:
• A-xyz-B form…
• A= 1..16 (number of the triad in the catalogue)
• X = number of pairs of vertices connected by
bidirectional arcs
• Y = number of pairs of vertices connected by a
single arc;
• z = number of unconnected pairs of vertices.

8.
QuickTimeª and a
TIFF (LZW) decompressor
are needed to see this picture.

9.
Triad Catalogue
• 9, 12, 13, 16 are transitive
• 6, 7, 8, 10, 11, 14, 15 are intransitive
• 1, 2, 3, 4, 5 do not contain arcs to meet
the conditions of transitivity (they are
vacuously transitive)

10.
Triad #16…
• …is known as a clique
• Cliques are a particular type of
cohesive subgroups
• We can count the number of cliques in
the network to estimate overall
cohesion or evaluate local properties of
nodes

11.
Cliques
• Definition
• Maximal, complete subgraph
• Properties
• Maximum density (1.0)
Minimum distances (all 1)
• overlapping
• Strict

12.
QuickTimeª and a
TIFF (LZW) decompressor
are needed to see this picture.

13.
Relaxation of Strict Cliques
• Distance (length of paths)
• N-clique, n-clan, n-club
• Density (number of ties)
• K-plex, ls-set, lambda set, k-core, component

14.
N-Cliques
• Definition
• Maximal subset such that:
• Distance among members less than specified
maximum
• When n = 1, we have a clique
• Properties
• Relaxes notion of clique
• Avg. distance can
• be greater than 1

15.
QuickTimeª and a
TIFF (LZW) decompressor
are needed to see this picture.

16.
Issues with n-cliques
• Overlapping
• {a,b,c,f,e} and {b,c,d,f,e} are
both 2-cliques
• Membership criterion
satisfiable through non-
members
• Even 2-cliques can be
fairly non-cohesive
• Red nodes belong to same
2-clique but none are
adjacent

17.
N-Clan
• Definition
• An n-clique in which geodesic distance
between nodes in the subgraph is no
greater then n
• Members of set within n links of each other
without using outsiders
• Properties
• More cohesive
than n-cliques

18.
N-Club
• Definition
• A maximal subset S whose
diameter is <= n
• No n-clique requirement
• Properties
• Painful to compute
• More plentiful than n-clans
• Overlapping

19.
K-core:
• A maximal subgraph such that:
• In English:
• Every node in a subset is connected to at
least k other nodes in the same subset

20.
Example
QuickTimeª and a
TIFF (LZW) decompressor
are needed to see this picture.

21.
Notes
• Finds areas within which cohesive subgroups
may be found
• Identifies fault lines across which cohesive
subgroups do not span
• In large datasets, you can successively
examine the 1-cores, the 2-cores, etc.
• Progressively narrowing to core of network

22.
K-plex:
• Maximal subset such that:
• In English:
• A k-plex is a group of nodes such that every
node in the group is connected to every other
node except k
• Really a relaxation of a clique

23.
Example
QuickTimeª and a
TIFF (LZW) decompressor
are needed to see this picture.

24.
Notes
• Choosing k is difficult so meaningful
results can be found
• One should look at resulting group
sizes - they should be larger then k by
some margin

25.
Next time…
• Making sense of triads - structural
holes, brokerage and their social effects