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4 Cliques Clusters
 

4 Cliques Clusters

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    4 Cliques Clusters 4 Cliques Clusters Presentation Transcript

    • 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.
    •  
    • 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
        • M aximal, complete subgraph
      • Properties
        • M aximum density (1.0) M inimum distances (all 1)
        • o verlapping
        • S trict
    •  
    • 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
        • 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
    •  
    • 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
    • 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
    •  
    • 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
    • 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
    • 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
    • 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