As Strong as the Weakest Link: Mining Diverse Cliques in Weighted - - PowerPoint PPT Presentation

As Strong as the Weakest Link: Mining Diverse Cliques in Weighted Graphs

Petko Bogdanov (UC Santa Barbara),

with Ben Baumer (Smith College), Prithwish Basu (Raytheon BBN) , Amotz Bar-Noy (CUNY) and Ambuj K. Singh (UC Santa Barbara)

ECML/PKDD, Prague, 2013

Example: Collaboration in sports

Significance of a pair’s success when

• n a team

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Influential groups

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Multiple teams

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Cliques in gene networks

Gene Interaction Networks Complexes - interacting functional units*

* Leemor Joshua-Tor, Structure and Function of Nucleic Acid Regulatory Complexeshttp://www.hhmi.org/research/structure-and- function-nucleic-acid-regulatory-complexes

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Cliques in other domains

• Sets of duplicates and

near-duplicates in similarity networks

○ images ○ video ○ other complex objects with similarity function

• Co-evolving time series

○ stocks of companies related in a supply chain ○ brain regions co-associated in performing a task*

* Hagmann P, Cammoun L, Gigandet X, Meuli R, Honey CJ, Wedeen VJ, Sporns O (2008) Mapping the structural core of human cerebral cortex. PLoS Biology Vol. 6, No. 7, e159

Challenges

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• Enumeration of cliques

○ MAX CLIQUE is NP-hard

• Ensuring diversity in the result set

○ Managing overlap “adds” complexity

• Size and density of real-world networks
• How to find the best diverse cliques

efficiently while maintaining good quality

• f the solution

Outline

• Motivation and examples
• Problem statement and properties
• Proposed solutions
• Experiments
• Conclusion

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Basic notions

• A graph G(V,E,w) represents a network of

entities V with edges E among them

• w defines weights on edges

○ higher weight means stronger association

• A clique is a complete subgraph, i.e. all

edges among the selected entities exist 9

Clique strength

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• Strong ties of all

pairwise edges

• A clique is as strong

as its weakest link

• “Flat” teams in which

all connections are important

• Bigger cliques

featuring all strong edges are better

• Linear

combination

• f score and

diversity via α

• Higher

number of distinct nodes in solution means higher diversity Score Diversity

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Diversity

Example: Top-2 cliques

Too much

• verlap

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Slightly lower score but less

• verlap

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Example: Top-2 cliques

Complexity

• m-Diverse k-Structures

(mDkS) is NP-hard

○ reduction from SET COVER

• Even if we are interested

in sets of arbitrary structure, maximizing diversity is NP-hard 14

Included in solution

Approximation

• Good news

○ monotonic ○ submodular ○ Allows a (1-1/e)-APX

• Challenges

○ Requires greedily finding the next best clique ○ MAX CLIQUE NP-hard to approximate to a constant

• Questions

○ Can we develop a solution with APX guarantees that is fast? Limitations? ○ Can we develop a very fast solution of good quality?

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Candidate to add to solution Diminishing return

Outline

• Motivation and examples
• Problem statement and properties
• Proposed solutions
• Experiments
• Conclusion

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Intuition

• How to obtain good cliques while reducing

the cost of enumeration?

• Exploit the distribution of edge

weights in a real network.

• Consider good edges first.
• Include good cliques in

solution before considering all edges based on bounding the contribution of partial cliques 17

Optimistic completion

C

Current lowest weight will be the lowest in the whole clique The rest of the nodes will not overlap

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Upper bound for an incomplete clique contribution

DiCliQ - threshold and prune

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DiCliQ - threshold and prune

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• 1. Enumerate

cliques in a thresholded graph

• 2. Upper bound
• 3. If there is a

candidate with a better score contribution than the best UB, add it to the solution

DiCliQ - threshold and prune

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• 1. Enumerate

cliques in a thresholded graph

• 2. Upper bound
• 3. If there is a

candidate with a better score contribution than the best UB, add it to the solution

• 4. Lower threshold

and repeat

DiCliQ - threshold and prune

• Implements a GREEDY and hence has a (1-

1/e)-approximation factor

• Exhaustive enumeration of all cliques might

incur high cost in very large/dense instances

• How to scale up the discovery of diverse

cliques without compromising the quality much? 22

BUDiC - Bottom-up greedy heuristic

• Greedy expansion

around a node based

• n the UB contribution
• Incorporates diversity

C

Already in the solution A

UB? UB?

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BUDiC - Bottom-up greedy heuristic C

• Greedy expansion

around a node based

• n the UB contribution
• Incorporates diversity

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Already in the solution A

Grow away from included nodes based on UB

BUDiC - Bottom-up greedy heuristic C

Already in the solution A

• Greedy expansion

around a node based

• n the UB contribution
• Incorporates diversity
• Repeat for all nodes
• Scales much better: O

(m*k*|E|)

• No APX guarantee
• Good quality on real

datasets 25

Grow away from included nodes based on UB

Outline

• Motivation and examples
• Problem statement and properties
• Proposed solutions
• Experiments
• Conclusion

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Data

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Scalability

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• Compare running time to a Baseline (No

thresholding) and relative quality to iMDV*

• α = 0.5, m = 10, k = 5

* S. Bandyopadhyay and M. Bhattacharyya. Mining the largest dense vertexlet in a weighted scale-free graph. Fundam. Inform., 96(1-2):1–25, 2009

• Apx. guarantee

Scalable, High Quality

Scalability on YeastNet

27 α=0.5, k=5 α=0.5, m=5

Quality

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Discovering gene complexes

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Conclusion

• General results for diverse clique mining

○ application to discovery of effective groups in collaboration ○ complexes in gene networks ○ similarity/correlation graphs

• Two scalable algorithms, one with constant

factor approximation

• More than 3 orders of magnitude running time

improvement while preserving good quality 30

Thank You

Q&A

The research was supported by the Army Research Laboratory under cooperative agreement W911NF-09-2-0053 (NS-CTA).

Effect of diversity parameter α

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Groups in the other datasets

• The Harry Potter cast in the movies data set
• NBA: Nowitzki-Chandler-Stevenson of the

defending champion Dallas Mavericks (addition of Chandler positive)

• MLB: Ramirez-Blake-Kuo of the LA Dodgers

(13/14 with an otherwise unremarkable lineup reached the playoffs in 2008) 33

Related work

• Quasi-cliques

○ frequency of clique occurrence (not score) ○ non-unique labels

• Weighted cliques

○ Bandyopadhyay et al. 2009: no APX guarantees, single clique, extended version does not have as good quality

• Other subgraph types

○ Steiner trees ○ Clique percolation (CFinder) ○ Edge weights are constraints and not part of score

• Diversity of nodes labels within a clique

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