1. DOULION: Counting Triangles in Massive Graphs with a Coin
Charalampos (Babis) Tsourakakis
Carnegie Mellon University KDD ‘09 Paris
Joint work with: U Kang, Gary L. Miller, Christos Faloutsos
DOULION, KDD 09

2. Outline Motivation Related Work Proposed Method Results Conclusion
Extra
DOULION, KDD 09

3. Why is Triangle Counting important?
Clustering coefficient
Transitivity ratio
Social Network Analysis fact: “Friends of friends are friends” A C B
[WF94)]
Hidden Thematic Structure of the Web (Eckmann et al. PNAS [EM02])
Motif Detection, (e.g., [YPSB05] )
Web Spam Detection (Becchetti et.al. KDD ’08 [BBCG08])
DOULION, KDD 09

4. Personal Motivation [CET08] eigenvalues of adjacency matrix
Political Blogs
eigenvalues of adjacency matrix
Keep only 3! 3
i-th eigenvector
DOULION, KDD 09

5. Outline Motivation Related Work Proposed Method Results Conclusion
Extra
DOULION, KDD 09

6. Counting methods Dense graphs Sparse graphs Fast Low space
Time complexity
O(n2.37)
O(n3)
Space complexity
O(n2)
O(m)
Sparse graphs
Fast
Low space
Time complexity
O(m0.7n1.2+n2+o(1))
e.g. O( n )
Space complexity
Θ(n2) (eventually)
Θ(m)
Matrix Multiplication not practical
M. Latapy,
Theory and Experiments
DOULION, KDD 09

7. Naive Sampling X=1 T3 X=0 T0 T1 T2
r independent samples of three distinct vertices
X=1 T3
X=0 T0 T1 T2
DOULION, KDD 09

8. Naive Sampling
r independent samples of three distinct vertices
Then the following holds:
with probability at least 1-δ
Works
Prohibitive for graphs with T3=o(n2). e.g., T3 n2logn
DOULION, KDD 09

9. Buriol, Frahling, Leonardi, Marchetti-Spaccamela, Sohlerk
Sample uniformly
at random an edge
(i,j) and a node k in V-{i,j} ? ? i j
Check if edges (i,k) and (j,k) exist in E(G)
samples
DOULION, KDD 09

10. Outline Motivation Related Work Proposed Method Results Conclusion
Extra
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11. Our Sampling Approach HEADS! (i,j) “survives” G(V,E) 1/p i j
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12. Our Sampling Approach
G(V,E) k m
TAILS!
(k,m) “dies”
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13. Sampling approach
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14. Our Sampling Approach on Kn
Gn,0.5
In Expectation
Initially
Weighted *
DOULION, KDD 09

15. E[Χ]=Δ Mean and Variance Δ=#triangles=k+(Δ-k)
k non-edge-disjoint triangles
X r.v, our estimate
E[Χ]=Δ
DOULION, KDD 09

16. Outline Motivation Related Work Proposed Method Results Conclusion
Extra
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17. Doulion and NodeIterator
Sparsify first and then use Node Iterator to count triangles.
Node Iterator: Consider each node and count how many edges among its neighbors
DOULION, KDD 09

18. Expected Speedup Expected Speedup: 1/p2 Proof
Let R be the running time of Node Iterator after the
sparsification:
Therefore, expected speedup:
DOULION, KDD 09

19. Some results (I)
~3M, ~35M
~400K, ~2.1M
DOULION, KDD 09

20. Some results (II)
~3.1M, ~37M
~3.6M, ~42M
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21. Outline Motivation Related Work Proposed Method Results Conclusion
Extra
DOULION, KDD 09

22. Conclusions New Sampling approach that counts triangles approximately.
Basic analysis of the estimate (expectation, variance, expected speedup)
Experimentation on many real world datasets where we showed that for p=constant we get high quality estimates and 1/p2 constant speedups.
DOULION, KDD 09

23. Question
Can p be smaller than constant? How small can we afford p to be and at the same time guarantee concentration?
Could e.g., p be as small as 1/ ???
Motivation: p
Speedup
0.001
106
0.005
4*104
0.01
104
DOULION, KDD 09

24. Outline Motivation Related Work Proposed Method Results Conclusion
Extra
DOULION, KDD 09

25. Approximate Triangle Counting
Approximate Triangle Counting Arxiv preprint
C.E.T M.N. Kolountzakis G.L. Miller
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26. Theorem C.E.T, Kolountzakis, Miller 2009
How to choose p?
Mildness, pick p=1
Concentration
DOULION, KDD 09

27. Practitioner’s Guide Wikipedia 2005 1,6M nodes 18,5M edges
Pick p=1/
Keep doubling until
concentration
Concentration appears
Concentration becomes
stronger
DOULION, KDD 09

28. “Bad” Instances Remove edge (1,2)
Remove any weighted edge w sufficiently large
DOULION, KDD 09

29. Thanks! http://www.cs.cmu.edu/~ctsourak/projects.html
Code and datasets available
(HADOOP, MATLAB, JAVA implementations along with
small real-world graphs, all datasets used are on the
web)
An article about computational science in a scientific publication is not the scholarship
itself, it is merely advertising of the scholarship. The actual scholarship is the complete
software environment and the complete set of instructions which generated the figures.
Buckheit and Donoho[BD95]
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30. References
Efficient semi-streaming algorithms for local triangle counting in massive graphs
Becchetti, Boldi, Castillio, Gionis [BBCG08]
Commensurate distances and similar motifs in genetic congruence and protein interaction networks in yeast
Ye, Peyser, Spencer, Bader [YPSB05]
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31. References
Curvature of co-links uncovers hidden thematic layers in the World Wide Web
Eckmann, Moses [EM02]
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32. References
Fast Counting of Triangles in Large Real-World Networks: Algorithms and Laws C. Tsourakakis
[BD95] Wavelab and reproducible research Buckheit, Donoho
DOULION, KDD 09

33. References Social Network Analysis: Methods and Applications
Wasserman, Faust [WF94]
Counting triangles in data streams
Buriol, Frahling, Leonardi, Spaccamela, Sohler [BFLSS06]
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34. Doulion
DOULION, KDD 09