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Clique Percolation Method (CPM)

Clique Percolation Method (CPM). Eugene Lim. Contents. What is CPM? Algorithm Analysis Conclusion. What is CPM?. Method to find overlapping communities Based on concept: internal edges of community likely to form cliques Intercommunity edges unlikely to form cliques. Clique.

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Clique Percolation Method (CPM)

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  1. Clique Percolation Method(CPM) Eugene Lim

  2. Contents • What is CPM? • Algorithm • Analysis • Conclusion

  3. What is CPM? • Method to find overlappingcommunities • Based on concept: • internal edges of community likely to form cliques • Intercommunity edges unlikely to form cliques

  4. Clique • Clique: Complete graph • k-clique: Complete graph with k vertices

  5. Clique • Clique: Complete graph • k-clique: Complete graph with k vertices 3-clique

  6. Clique • Clique: Complete graph • k-clique: Complete graph with k vertices 4-clique

  7. Clique • Clique: Complete graph • k-clique: Complete graph with k vertices 5-clique

  8. k-Clique Communities • Adjacent k-cliques Two k-cliques are adjacent when they share k-1nodes

  9. k-Clique Communities • Adjacent k-cliques Two k-cliques are adjacent when they share k-1nodes k = 3

  10. k-Clique Communities • Adjacent k-cliques Two k-cliques are adjacent when they share k-1nodes k = 3 Clique 1

  11. k-Clique Communities • Adjacent k-cliques Two k-cliques are adjacent when they share k-1nodes Clique 2 k = 3

  12. k-Clique Communities • Adjacent k-cliques Two k-cliques are adjacent when they share k-1nodes Clique 3 k = 3

  13. k-Clique Communities • Adjacent k-cliques Two k-cliques are adjacent when they share k-1nodes Clique 2 k = 3 Clique 1

  14. k-Clique Communities • Adjacent k-cliques Two k-cliques are adjacent when they share k-1nodes Clique 2 Clique 3 k = 3

  15. k-Clique Communities • k-clique community Union of all k-cliques that can be reached from each other through a series of adjacent k-cliques

  16. k-Clique Communities • k-clique community Union of all k-cliques that can be reached from each other through a series of adjacent k-cliques Clique 2 k = 3 Clique 1

  17. k-Clique Communities • k-clique community Union of all k-cliques that can be reached from each other through a series of adjacent k-cliques Community 1 k = 3

  18. k-Clique Communities • k-clique community Union of all k-cliques that can be reached from each other through a series of adjacent k-cliques Community 1 Clique 3 k = 3

  19. k-Clique Communities • k-clique community Union of all k-cliques that can be reached from each other through a series of adjacent k-cliques Community 1 Community 2 k = 3

  20. Algorithm • Locate maximal cliques • Convert from cliques to k-clique communities

  21. Locate Maximal Cliques • Largest possible clique size can be determined from degrees of vertices • Starting from this size, find all cliques, then reduce size by 1 and repeat

  22. Locate Maximal Cliques • Finding all cliques: brute-force • Set A initially contains vertex v, Set B contains neighbours of v • Transfer one vertex w from B to A • Remove vertices that are not neighbours of w from B • Repeat until A reaches desired size • If fail, step back and try other possibilities

  23. Algorithm • Locate maximal cliques • Convert from cliques to k-clique communities

  24. Cliques to k-Clique Communities

  25. Cliques to k-Clique Communities Clique 1: 5-clique

  26. Cliques to k-Clique Communities

  27. Cliques to k-Clique Communities Clique 2: 4-clique

  28. Cliques to k-Clique Communities

  29. Cliques to k-Clique Communities Clique 3: 4-clique

  30. Cliques to k-Clique Communities

  31. Cliques to k-Clique Communities Clique 4: 4-clique

  32. Cliques to k-Clique Communities

  33. Cliques to k-Clique Communities Clique 5: 3-clique

  34. Cliques to k-Clique Communities

  35. Cliques to k-Clique Communities Clique 6: 3-clique

  36. Cliques to k-Clique Communities

  37. Cliques to k-Clique Communities

  38. Cliques to k-Clique Communities Clique 1: 5-clique

  39. Cliques to k-Clique Communities Clique 2: 4-clique

  40. Cliques to k-Clique Communities

  41. Cliques to k-Clique Communities k=4

  42. Cliques to k-Clique Communities k=4

  43. Cliques to k-Clique Communities k=4 Delete if less than k

  44. Cliques to k-Clique Communities k=4

  45. Cliques to k-Clique Communities k=4

  46. Cliques to k-Clique Communities k=4 Delete if less than k-1

  47. Cliques to k-Clique Communities k=4

  48. Cliques to k-Clique Communities k=4 Change all non-zeros to 1

  49. Cliques to k-Clique Communities k=4 Clique-clique overlap matrix

  50. Cliques to k-Clique Communities k=4 Community 1

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