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

6

4

9

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*

3

? ? ??

??

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?