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# 4 Cliques Clusters

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• 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
• 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
• 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.
• 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)
• … 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
• M aximal, complete subgraph
• Properties
• M aximum density (1.0) M inimum distances (all 1)
• o verlapping
• S trict
• 12.
• 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
• M aximal subset such that:
• D istance among members less than specified maximum
• W hen n = 1, we have a clique
• Properties
• R elaxes notion of clique
• Avg. distance can
• be greater than 1
• 15.
• 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
• R ed nodes belong to same 2-clique but none are adjacent
• 17. N-Clan
• Definition
• A n n-clique in which geodesic distance between nodes in the subgraph is no greater then n
• M embers of set within n links of each other without using outsiders
• Properties
• M ore cohesive than n-cliques
• 18.
• 19. N-Club
• Definition
• A maximal subset S whose diameter is <= n
• N o n-clique requirement
• Properties
• P ainful to compute
• M ore plentiful than n-clans
• O verlapping
• 20. 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
• 21. Example
• 22. 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
• 23. 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
• 24. Example
• 25. 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
• 26. Next time…
• Making sense of triads - structural holes, brokerage and their social effects