4 Cliques Clusters


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4 Cliques Clusters

  1. 1. Cliques, Clans and Clusters Finding Cohesive Subgroups in Network Data
  2. 2. Social Subgroups <ul><li>Frank & Yasumoto argue that actors seek social capital, defined as the access to resources through social ties </li></ul><ul><li>a) Reciprocity Transactions </li></ul><ul><li> Actors seek to build obligations with others, and thereby </li></ul><ul><li> gain in the ability to extract resources. </li></ul><ul><li>b) Enforceable Trust </li></ul><ul><li> “ Social capital is generated by individual members’ </li></ul><ul><li> disciplined compliance with group expectations.” </li></ul><ul><li>c) Group Cohesion </li></ul>
  3. 3. Goals <ul><li>Find a meaningful way to separate larger networks into groups </li></ul><ul><li>Meaningful = </li></ul><ul><ul><li>Reduce overlap </li></ul></ul><ul><ul><li>Locate cohesive groups </li></ul></ul>
  4. 4. Reciprocity
  5. 5. Reciprocity <ul><li>Ratio of reciprocated pairs of nodes to number of pairs that have at least 1 tie </li></ul><ul><ul><li>In example, reciprocity = 0.5 </li></ul></ul><ul><ul><li>Called “dyad method” </li></ul></ul>
  6. 6. Transitivity <ul><li>Types of triadic relations (in undirected networks): </li></ul><ul><ul><li>Isolation </li></ul></ul><ul><ul><li>Couples only </li></ul></ul><ul><ul><li>Structural holes </li></ul></ul><ul><ul><li>Clusters (also cliques) </li></ul></ul>
  7. 7. In directed networks <ul><li>There are 16 types of triads </li></ul><ul><li>Triad language: </li></ul><ul><ul><li>A-xyz-B form… </li></ul></ul><ul><ul><li>A= 1..16 (number of the triad in the catalogue) </li></ul></ul><ul><ul><li>X = number of pairs of vertices connected by bidirectional arcs </li></ul></ul><ul><ul><li>Y = number of pairs of vertices connected by a single arc; </li></ul></ul><ul><ul><li>z = number of unconnected pairs of vertices. </li></ul></ul>
  8. 9. Triad Catalogue <ul><li>9, 12, 13, 16 are transitive </li></ul><ul><li>6, 7, 8, 10, 11, 14, 15 are intransitive </li></ul><ul><li>1, 2, 3, 4, 5 do not contain arcs to meet the conditions of transitivity (they are vacuously transitive) </li></ul>
  9. 10. Triad #16… <ul><li>… is known as a clique </li></ul><ul><li>Cliques are a particular type of cohesive subgroups </li></ul><ul><li>We can count the number of cliques in the network to estimate overall cohesion or evaluate local properties of nodes </li></ul>
  10. 11. Cliques <ul><li>Definition </li></ul><ul><ul><li>M aximal, complete subgraph </li></ul></ul><ul><li>Properties </li></ul><ul><ul><li>M aximum density (1.0) M inimum distances (all 1) </li></ul></ul><ul><ul><li>o verlapping </li></ul></ul><ul><ul><li>S trict </li></ul></ul>
  11. 13. Relaxation of Strict Cliques <ul><li>Distance (length of paths) </li></ul><ul><ul><li>N-clique, n-clan, n-club </li></ul></ul><ul><li>Density (number of ties) </li></ul><ul><ul><li>K-plex, ls-set, lambda set, k-core, component </li></ul></ul>
  12. 14. N-Cliques <ul><li>Definition </li></ul><ul><ul><li>M aximal subset such that: </li></ul></ul><ul><ul><li>D istance among members less than specified maximum </li></ul></ul><ul><ul><li>W hen n = 1, we have a clique </li></ul></ul><ul><li>Properties </li></ul><ul><ul><li>R elaxes notion of clique </li></ul></ul><ul><ul><li>Avg. distance can </li></ul></ul><ul><ul><li>be greater than 1 </li></ul></ul>
  13. 16. Issues with n-cliques <ul><li>Overlapping </li></ul><ul><ul><li>{ a,b,c,f,e} and {b,c,d,f,e} are both 2-cliques </li></ul></ul><ul><li>Membership criterion satisfiable through non- members </li></ul><ul><li>Even 2-cliques can be fairly non-cohesive </li></ul><ul><ul><li>R ed nodes belong to same 2-clique but none are adjacent </li></ul></ul>
  14. 17. N-Clan <ul><li>Definition </li></ul><ul><ul><li>A n n-clique in which geodesic distance between nodes in the subgraph is no greater then n </li></ul></ul><ul><ul><li>M embers of set within n links of each other without using outsiders </li></ul></ul><ul><li>Properties </li></ul><ul><ul><li>M ore cohesive than n-cliques </li></ul></ul>
  15. 19. N-Club <ul><li>Definition </li></ul><ul><ul><li>A maximal subset S whose diameter is <= n </li></ul></ul><ul><ul><li>N o n-clique requirement </li></ul></ul><ul><li>Properties </li></ul><ul><ul><li>P ainful to compute </li></ul></ul><ul><ul><li>M ore plentiful than n-clans </li></ul></ul><ul><ul><li>O verlapping </li></ul></ul>
  16. 20. K-core: <ul><li>A maximal subgraph such that: </li></ul><ul><li>In English: </li></ul><ul><ul><li>Every node in a subset is connected to at least k other nodes in the same subset </li></ul></ul>
  17. 21. Example
  18. 22. Notes <ul><li>Finds areas within which cohesive subgroups may be found </li></ul><ul><li>Identifies fault lines across which cohesive subgroups do not span </li></ul><ul><li>In large datasets, you can successively examine the 1-cores, the 2-cores, etc. </li></ul><ul><ul><li>Progressively narrowing to core of network </li></ul></ul>
  19. 23. K-plex: <ul><ul><li>Maximal subset such that: </li></ul></ul><ul><ul><li>In English: </li></ul></ul><ul><ul><ul><li>A k-plex is a group of nodes such that every node in the group is connected to every other node except k </li></ul></ul></ul><ul><ul><ul><li>Really a relaxation of a clique </li></ul></ul></ul>
  20. 24. Example
  21. 25. Notes <ul><li>Choosing k is difficult so meaningful results can be found </li></ul><ul><li>One should look at resulting group sizes - they should be larger then k by some margin </li></ul>
  22. 26. Next time… <ul><li>Making sense of triads - structural holes, brokerage and their social effects </li></ul>