1

Searching for k-cliques in unknown graphs

Roni Stern, Meir Kalech, Ariel FelnerDepartment of Information Systems Engineering

Ben Gurion University

2

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

5

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

7

K-cliques in unknown graphs• How to find a k-clique in unknown graphs?• Goal: minimize exploration

?

? ?

Which node to explore?

8

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

11

Heuristic #2: Clique*

Expand the largest potential k-clique – [Altshuler et. al. ’05]

? ? ??

??

1)An m-clique2)K-1-m common neighbors

?

D

C

12

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

13

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%

15

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

16

Questions?