Social Network Analysis and Mining

, Volume 1, Issue 2, pp 75–81 | Cite as

Spectral counting of triangles via element-wise sparsification and triangle-based link recommendation

  • Charalampos E. TsourakakisEmail author
  • Petros Drineas
  • Eirinaios Michelakis
  • Ioannis Koutis
  • Christos Faloutsos
Original Article


Triangle counting is an important problem in graph mining. The clustering coefficient and the transitivity ratio, two commonly used measures effectively quantify the triangle density in order to quantify the fact that friends of friends tend to be friends themselves. Furthermore, several successful graph-mining applications rely on the number of triangles in the graph. In this paper, we study the problem of counting triangles in large, power-law networks. Our algorithm, SparsifyingEigenTriangle, relies on the spectral properties of power-law networks and the Achlioptas–McSherry sparsification process. SparsifyingEigenTriangle is easy to parallelize, fast, and accurate. We verify the validity of our approach with several experiments in real-world graphs, where we achieve at the same time high accuracy and considerable speedup versus a straight-forward exact counting competitor. Finally, our contributions include a simple method for making link recommendations in online social networks based on the number of triangles as well as a procedure for estimating triangles using sketches.


Adjacency Matrix Cluster Coefficient Online Social Network Lanczos Method Exact Counting 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Achlioptas D, McSherry F (2001) Fast computation of low rank matrix approximation. In: Proceedings of 33rd STOC. ACM Press, New YorkGoogle Scholar
  2. Alon N, Matias Y, Szegedy M (1996) The space complexity of approximating the frequency moments. In: STOC ’96: proceedings of the twenty-eighth annual ACM symposium on theory of computing, New York, NY, USA. ACM, New York, pp 20–29Google Scholar
  3. Alon N, Yuster R, Zwick U (1997) Finding and counting given length cycles. Algorithmica 17(3):209–223Google Scholar
  4. Alon N, Gibbons PB, Matias Y, Szegedy M (2002) Tracking join and self-join sizes in limited storage, pp 10–20Google Scholar
  5. Bar-Yosseff Z, Kumar R, Sivakumar D (2002) Reductions in streaming algorithms, with an application to counting triangles in graphs. In: SODA ’02: proceedings of the thirteenth annual ACM-SIAM symposium on discrete algorithms, Philadelphia, PA, USA. Society for Industrial and Applied Mathematics, pp 623–632Google Scholar
  6. Becchetti L, Boldi P, Castillo C, Gionis A (2008) Efficient semi-streaming algorithms for local triangle counting in massive graphs. In: Proceedings of ACM KDD, Las Vegas, NV, USAGoogle Scholar
  7. Buriol LS, Frahling G, Leonardi S, Marchetti-Spaccamela A, Sohler C (2006) Counting triangles in data streams. In: PODS ’06: proceedings of the twenty-fifth ACM SIGMOD-SIGACT-SIGART symposium on principles of database systems, New York, NY, USA. ACM, pp 253–262Google Scholar
  8. Chung F, Lu L, Vu V (2003) Eigenvalues of random power law graphs. Ann Comb 7(1):21–33zbMATHMathSciNetGoogle Scholar
  9. Coppersmith D, Winograd S (1987) Matrix multiplication via arithmetic progressions. In: STOC ’87: proceedings of the nineteenth annual ACM conference on theory of computing, New York, NY, USA. ACM, pp 1–6Google Scholar
  10. Deerwester S, Dumais ST, Furnas GW, Landauer TK, Harshman R (1990) Indexing by latent semantic analysis. J Am Soc Inf Sci 41:391–407CrossRefGoogle Scholar
  11. Demmel J (1997) Applied numerical linear algebra. SIAM, PhiladelphiazbMATHGoogle Scholar
  12. Drineas P, Kannan R (2003) Pass efficient algorithms for approximating large matrices. In: SODA ’03: proceedings of the fourteenth annual ACM-SIAM symposium on discrete algorithms, Philadelphia, PA, USA. Society for Industrial and Applied Mathematics, pp 223–232Google Scholar
  13. Eckmann J-P, Moses E (2002) Curvature of co-links uncovers hidden thematic layers in the world wide web. PNAS 99(9):5825–5829CrossRefMathSciNetGoogle Scholar
  14. Faloutsos M, Faloutsos P, Faloutsos C (1999) On power-law relationships of the internet topology. In: SIGCOMM, pp 251–262Google Scholar
  15. Farkas IJ, Derenyi I, Barabasi A-L, Vicsek T (2001) Spectra of “real-world” graphs: beyond the semi-circle law. Phys Rev E 64:1CrossRefGoogle Scholar
  16. Frieze A, Kannan R, Vempala S (1998) Fast Monte-Carlo algorithms for finding low-rank approximations. In: Proceedings of the 39th annual IEEE symposium on foundations of computer science, pp 370–378Google Scholar
  17. Gilbert AC, Kotidis Y, Muthukrishnan S, Strauss MJ (2003) One-pass wavelet decompositions of data streams. IEEE TKDE 15:2003Google Scholar
  18. Golub G, Van Loan C (1989) Matrix computations, 2nd edn. Johns Hopkins Press, BaltimoreGoogle Scholar
  19. Latapy M (2008) Main-memory triangle computations for very large (sparse (power-law)) graphs. Theor Comput Sci 407(1–3):458–473CrossRefzbMATHMathSciNetGoogle Scholar
  20. Liben-Nowell D (2005) An algorithmic approach to social networks. PhD thesis, Massachusetts Institute of Technology, Electrical Engineering and Computer Science Department, June 2005Google Scholar
  21. Mihail M, Papadimitriou C (2002) On the eigenvalue power law. In: Lecture notes on computer sciences, vol 2483. Springer, BerlinGoogle Scholar
  22. Newman MEJ (2001) Clustering and preferential attachment in growing networks. Phys Rev E 64:016131Google Scholar
  23. Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45:167–256CrossRefzbMATHMathSciNetGoogle Scholar
  24. Schank T, Wagner D (2004) DELIS-TR-0043—finding, counting and listing all triangles in large graphs, an experimental study. Techreport 0043Google Scholar
  25. Tsourakakis C (2008) Fast counting of triangles in large real networks, without counting: algorithms and laws. In: ICDMGoogle Scholar
  26. Tsourakakis CE, Kang U, Miller GL, Faloutsos C (2009a) Doulion: counting triangles in massive graphs with a coin. In: Elder JF, Fogelman-Souli F, Flach PA, Zaki M (eds) Proceedings of KDD. ACM, pp 837–846Google Scholar
  27. Tsourakakis CE, Kolountzakis MN, Miller GL (2009b) Approximate triangle counting. CoRR, abs/0904.3761Google Scholar
  28. Turk M, Pentland A (1991) Eigenfaces for recognition. J Cogn Neurosci 3(1):71–86CrossRefGoogle Scholar
  29. Wasserman S, Faust K (1994) Social network analysis. Cambridge University Press, CambridgeGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Charalampos E. Tsourakakis
    • 1
    Email author
  • Petros Drineas
    • 2
  • Eirinaios Michelakis
    • 3
  • Ioannis Koutis
    • 1
  • Christos Faloutsos
    • 1
  1. 1.School of Computer ScienceCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Computer ScienceRensselaer Polytechnic InstituteTroyUSA
  3. 3.EECS University of California, BerkeleyBerkeleyUSA

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