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 overlapping communities 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

6 Clique Clique: Complete graph k-clique: Complete graph with k vertices

7 Clique Clique: Complete graph k-clique: Complete graph with k vertices

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

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

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

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

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

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

14 k-Clique Communities Adjacent k-cliques
Two k-cliques are adjacent when they share k-1 nodes 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
1 2 3 4 5 6

37 Cliques to k-Clique Communities
1 2 3 4 5 6

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
1 2 3 4 5 6

41 Cliques to k-Clique Communities
1 2 3 4 5 6

42 Cliques to k-Clique Communities
1 2 3 4 5 6

43 Cliques to k-Clique Communities
1 2 3 4 5 6 Delete if less than k

44 Cliques to k-Clique Communities
1 2 3 4 5 6

45 Cliques to k-Clique Communities
1 2 3 4 5 6

46 Cliques to k-Clique Communities
1 2 3 4 5 6 Delete if less than k-1

47 Cliques to k-Clique Communities
1 2 3 4 5 6

48 Cliques to k-Clique Communities
1 2 3 4 5 6 Change all non-zeros to 1

49 Cliques to k-Clique Communities
1 2 3 4 5 6 Clique-clique overlap matrix

50 Cliques to k-Clique Communities
Community 1

51 Cliques to k-Clique Communities
Community 2

52 ANALYSIS Believed to be non-polynomial No closed formula can be given
However, claimed to be efficient on real systems

53 CONCLUSION Widely used algorithm for detecting overlapping communities
However: Fail to give meaningful covers for graph with few cliques With too many cliques, might give a trivial community structure Left out vertices? Subgraphs containing many cliques == community? What value of k to choose to give a meaningful structure?

54 References Palla et al. – Uncovering the overlapping community structure of complex networks in nature and society Santo Fortunato - Community detection in graphs

55 Thank you!