4 Cliques Clusters

Cliques, Clans and Clusters

Finding Cohesive Subgroups in

Network Data

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

Goals

• Find a meaningful way to separate larger networks into groups

• Meaningful = • Reduce overlap• Locate cohesive groups

Reciprocity

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”

Transitivity

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

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.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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)

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

Cliques

• Definition • Maximal, complete subgraph

• Properties • Maximum density (1.0)

Minimum distances (all 1) • overlapping • Strict

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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

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

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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

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

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

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

Example

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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

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

Example

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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

Next time…

• Making sense of triads - structural holes, brokerage and their social effects