searching for k-cliques in unknown graphs
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Searching for k-cliques in unknown graphs
Roni Stern, Meir Kalech, Ariel FelnerDepartment of Information Systems Engineering
Ben Gurion University
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Topics
• Known vs. unknown graphs• Finding k-clique with minimum exploration• Heuristics• MDP and Monte-Carlo approach • Experimental results
– Simulated random graphs– Crawling in Google Scholar
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Known vs. unknown graphs
• Known graph• Unknown graph
– Need exploration actions• World Wide Web
– Dynamic, too large and simply unknown
HTML
HTTP
Parse LinksHTTP
http://www.google.com/search?&q=nice
ExploredKnownUnknown
K-cliques in unknown graphs
D
B C
A
F
E
3-cliques
4-cliques
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K-cliques in unknown graphs• How to find a k-clique in unknown graphs?• Goal: minimize exploration
?
? ?
Which node to explore?
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K-cliques in unknown graphs• How to find a k-clique in unknown graphs?• Goal: minimize exploration
?
? ?
Which node to explore?
ADE
FGHI
C B
ADBGF
AEB
Cost
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4
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Known degree: number of explored neighbors
Heuristic #1: Known Degree
A B CD
FE
12
21
1 1
10
Expand the largest potential k-clique – [Altshuler et. al. ’05]
C & D are a potential 4-clique
Heuristic #2: Clique*
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? ? ??
??
12
21
1 1
D
A
BA,B &C are NOT
a potential 4-clique
CC
1)An m-clique2)K-1-m common neighbors
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Heuristic #2: Clique*
Expand the largest potential k-clique – [Altshuler et. al. ’05]
? ? ??
??
1)An m-clique2)K-1-m common neighbors
?
D
C
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Heuristc #3: RClique*
• Unknown graph but known domain– How can a probabilistic model be used?
• MDP state space is too large
• Monte Carlo sampling approach:– Simulate exploration with domain model– Use average sample results
??
?0.3
0.8
0.1
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Experimental Results
Random and scale free graphs
800 1200 16000
10
20
30
40
50
60
70
80
Random, 100 nodes, 5-Clique
RandomKnown DegreeClique*RClique*Lower bound
Edges
Expl
orati
on C
ost
Heuristics much better than random
Clique* advantage diminishes with density
RClique* is much better
144 5 6
0
20
40
60
80
100
120
RandomKnown DegreeClique*
Desired Clique Size
Expl
orati
on C
ost
Real application, crawling online
• Max. 101 nodes explored
Success rateHeuristic\k 4 5 6
Random 66% 0% 0%Known Degree 58% 28% 20%
Clique* 100% 83% 30%
0% 0%
66%
100%
28%
83%
20%30%
58%
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Summary
• Find k-cliques in unknown graph– Minimize exploration cost
• Heuristics– Known Degree, Clique*, RClique*
• Future work– Incorporate with data mining techniques– Exploring with multiple agents– Generalize to subgraph isomorphism
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Questions?
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