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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
      • 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.  
  • 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
      • 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