Clique Percolation Method (CPM)
description
Transcript of Clique Percolation Method (CPM)
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Eugene Lim
CLIQUE PERCOLATION
METHOD(CPM)
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What is CPM?
Algorithm
Analysis
Conclusion
CONTENTS
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Method to find overlapping communities
Based on concept:
internal edges of community likely to form cliques
Intercommunity edges unlikely to form cliques
WHAT IS CPM?
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Clique: Complete graph
k-clique: Complete graph with k vertices
CLIQUE
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Clique: Complete graph
k-clique: Complete graph with k vertices
CLIQUE
3-clique
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Clique: Complete graph
k-clique: Complete graph with k vertices
CLIQUE
4-clique
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Clique: Complete graph
k-clique: Complete graph with k vertices
CLIQUE
5-clique
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K-CLIQUE COMMUNITIES
Adjacent k-cliques
Two k-cliques are adjacent when they share k-1 nodes
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K-CLIQUE COMMUNITIES
Adjacent k-cliques
Two k-cliques are adjacent when they share k-1 nodes
k = 3
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K-CLIQUE COMMUNITIES
Adjacent k-cliques
Two k-cliques are adjacent when they share k-1 nodes
k = 3
Clique 1
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Adjacent k-cliques
Two k-cliques are adjacent when they share k-1 nodes
K-CLIQUE COMMUNITIES
k = 3Clique 2
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Adjacent k-cliques
Two k-cliques are adjacent when they share k-1 nodes
K-CLIQUE COMMUNITIES
k = 3Clique 3
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Adjacent k-cliques
Two k-cliques are adjacent when they share k-1 nodes
K-CLIQUE COMMUNITIES
k = 3
Clique 1
Clique 2
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Adjacent k-cliques
Two k-cliques are adjacent when they share k-1 nodes
K-CLIQUE COMMUNITIES
k = 3Clique 3Clique 2
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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
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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
k = 3
Clique 1
Clique 2
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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
k = 3
Community 1
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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
k = 3
Community 1 Clique 3
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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
k = 3
Community 1 Community 2
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Locate maximal cliques
Convert from cliques to k-clique communities
ALGORITHM
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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
LOCATE MAXIMAL CLIQUES
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Finding all cliques: brute-force
1. Set A initially contains vertex v, Set B contains neighbours
of v
2. Transfer one vertex w from B to A
3. Remove vertices that are not neighbours of w from B
4. Repeat until A reaches desired size
5. If fail, step back and try other possibilities
LOCATE MAXIMAL CLIQUES
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Locate maximal cliques
Convert from cliques to k-clique communities
ALGORITHM
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CLIQUES TO K-CLIQUE COMMUNITIES
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CLIQUES TO K-CLIQUE COMMUNITIES
Clique 1: 5-clique
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CLIQUES TO K-CLIQUE COMMUNITIES
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CLIQUES TO K-CLIQUE COMMUNITIES
Clique 2: 4-clique
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CLIQUES TO K-CLIQUE COMMUNITIES
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CLIQUES TO K-CLIQUE COMMUNITIES
Clique 3: 4-clique
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CLIQUES TO K-CLIQUE COMMUNITIES
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CLIQUES TO K-CLIQUE COMMUNITIES
Clique 4: 4-clique
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CLIQUES TO K-CLIQUE COMMUNITIES
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CLIQUES TO K-CLIQUE COMMUNITIES
Clique 5: 3-clique
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CLIQUES TO K-CLIQUE COMMUNITIES
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CLIQUES TO K-CLIQUE COMMUNITIES
Clique 6: 3-clique
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1 2 3 4 5 6
1 5
2 4
3 4
4 4
5 3
6 3
CLIQUES TO K-CLIQUE COMMUNITIES
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1 2 3 4 5 6
1 5 3 1 3 1 2
2 3 4 1 1 1 2
3 1 1 4 2 1 2
4 3 1 2 4 0 1
5 1 1 1 0 3 2
6 2 2 2 1 2 3
CLIQUES TO K-CLIQUE COMMUNITIES
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CLIQUES TO K-CLIQUE COMMUNITIES
Clique 1: 5-clique
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CLIQUES TO K-CLIQUE COMMUNITIES
Clique 2: 4-clique
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1 2 3 4 5 6
1 5 3 1 3 1 2
2 3 4 1 1 1 2
3 1 1 4 2 1 2
4 3 1 2 4 0 1
5 1 1 1 0 3 2
6 2 2 2 1 2 3
CLIQUES TO K-CLIQUE COMMUNITIES
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1 2 3 4 5 6
1 5 3 1 3 1 2
2 3 4 1 1 1 2
3 1 1 4 2 1 2
4 3 1 2 4 0 1
5 1 1 1 0 3 2
6 2 2 2 1 2 3
CLIQUES TO K-CLIQUE COMMUNITIES
k=4
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1 2 3 4 5 6
1 5 3 1 3 1 2
2 3 4 1 1 1 2
3 1 1 4 2 1 2
4 3 1 2 4 0 1
5 1 1 1 0 3 2
6 2 2 2 1 2 3
CLIQUES TO K-CLIQUE COMMUNITIES
k=4
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1 2 3 4 5 6
1 5 3 1 3 1 2
2 3 4 1 1 1 2
3 1 1 4 2 1 2
4 3 1 2 4 0 1
5 1 1 1 0 0 2
6 2 2 2 1 2 0
CLIQUES TO K-CLIQUE COMMUNITIES
k=4
Delete if less than k
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1 2 3 4 5 6
1 5 3 1 3 1 2
2 3 4 1 1 1 2
3 1 1 4 2 1 2
4 3 1 2 4 0 1
5 1 1 1 0 0 2
6 2 2 2 1 2 0
CLIQUES TO K-CLIQUE COMMUNITIES
k=4
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1 2 3 4 5 6
1 5 3 1 3 1 2
2 3 4 1 1 1 2
3 1 1 4 2 1 2
4 3 1 2 4 0 1
5 1 1 1 0 0 2
6 2 2 2 1 2 0
CLIQUES TO K-CLIQUE COMMUNITIES
k=4
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1 2 3 4 5 6
1 5 3 0 3 0 0
2 3 4 0 0 0 0
3 0 0 4 0 0 0
4 3 0 0 4 0 0
5 0 0 0 0 0 0
6 0 0 0 0 0 0
CLIQUES TO K-CLIQUE COMMUNITIES
k=4
Delete i f less than k-1
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1 2 3 4 5 6
1 5 3 0 3 0 0
2 3 4 0 0 0 0
3 0 0 4 0 0 0
4 3 0 0 4 0 0
5 0 0 0 0 0 0
6 0 0 0 0 0 0
CLIQUES TO K-CLIQUE COMMUNITIES
k=4
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1 2 3 4 5 6
1 1 1 0 1 0 0
2 1 1 0 0 0 0
3 0 0 1 0 0 0
4 1 0 0 1 0 0
5 0 0 0 0 0 0
6 0 0 0 0 0 0
CLIQUES TO K-CLIQUE COMMUNITIES
k=4
Change all non-zeros to 1
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1 2 3 4 5 6
1 1 1 0 1 0 0
2 1 1 0 0 0 0
3 0 0 1 0 0 0
4 1 0 0 1 0 0
5 0 0 0 0 0 0
6 0 0 0 0 0 0
CLIQUES TO K-CLIQUE COMMUNITIES
k=4
Clique-clique overlap matrix
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CLIQUES TO K-CLIQUE COMMUNITIES
k=4
Community 1
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CLIQUES TO K-CLIQUE COMMUNITIES
k=4
Community 2
![Page 52: Clique Percolation Method (CPM)](https://reader031.fdocuments.net/reader031/viewer/2022020919/568165c4550346895dd8cff5/html5/thumbnails/52.jpg)
Believed to be non-polynomial
No closed formula can be given
However, claimed to be efficient on real systems
ANALYSIS
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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?
CONCLUSION
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Palla et al. – Uncovering the overlapping community
structure of complex networks in nature and society
Santo Fortunato - Community detection in graphs
REFERENCES
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Thank you!