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

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

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

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

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

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

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

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

Buriol, Frahling, Leonardi, Marchetti-Spaccamela, Sohler k 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

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

Our Sampling Approach HEADS! (i,j) “survives” G(V,E) 1/p i j DOULION, KDD 09

Our Sampling Approach G(V,E) k m TAILS! (k,m) “dies” DOULION, KDD 09

Sampling approach DOULION, KDD 09

Our Sampling Approach on Kn Gn,0.5 In Expectation Initially Weighted * DOULION, KDD 09

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

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

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

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

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

Some results (II) ~3.1M, ~37M ~3.6M, ~42M DOULION, KDD 09

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

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

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

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

Approximate Triangle Counting Approximate Triangle Counting Arxiv preprint http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.3761v1.pdf C.E.T M.N. Kolountzakis G.L. Miller DOULION, KDD 09

Theorem C.E.T, Kolountzakis, Miller 2009 How to choose p? Mildness, pick p=1 Concentration DOULION, KDD 09

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

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

Thanks! http://www.cs.cmu.edu/~ctsourak/projects.html Code and datasets available graphminingtoolbox@gmail.com (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] DOULION, KDD 09

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] DOULION, KDD 09

References Curvature of co-links uncovers hidden thematic layers in the World Wide Web Eckmann, Moses [EM02] DOULION, KDD 09

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

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

Doulion DOULION, KDD 09