Journal of the Brazilian Computer Society

, Volume 17, Issue 1, pp 19–29 | Cite as

A graph clustering algorithm based on a clustering coefficient for weighted graphs

  • Mariá C. V. Nascimento
  • André C. P. L. F. Carvalho
Open Access
Original Paper


Graph clustering is an important issue for several applications associated with data analysis in graphs. However, the discovery of groups of highly connected nodes that can represent clusters is not an easy task. Many assumptions like the number of clusters and if the clusters are or not balanced, may need to be made before the application of a clustering algorithm. Moreover, without previous information regarding data label, there is no guarantee that the partition found by a clustering algorithm automatically extracts the relevant information present in the data. This paper proposes a new graph clustering algorithm that automatically defines the number of clusters based on a clustering tendency connectivity-based validation measure, also proposed in the paper. According to the computational results, the new algorithm is able to efficiently find graph clustering partitions for complete graphs.


Clustering coefficient Graph clustering Combinatorial optimization 


  1. 1.
    Bhattacharjee A, Richards WG, Staunton J, Li C, Monti S, Vasa P, Ladd C, Beheshti J, Bueno R, Gillette M, Loda M, Weber G, Mark EJ, Lander ES, Wong W, Johnson BE, Golub TR, Sugarbaker DJ, Meyerson M (2001) Classification of human lung carcinomas by mRNA expression profiling reveals distinct adenocarcinoma sub-classes. Proc Natl Acad Sci USA 98(24):13790–13795CrossRefGoogle Scholar
  2. 2.
    Boginski V, Butenko S, Pardalos PM (2006) Mining market data: a network approach. Comput Oper Res 33:3171–3184CrossRefMATHGoogle Scholar
  3. 3.
    Clauset A, Newman MEJ, Moore C (2004) Finding community structure in very large networks. Phys Rev E 70(6):066111CrossRefGoogle Scholar
  4. 4.
    Dhillon IS, Guan Y, Kulis B (2007) Weighted graph cuts without eigenvectors a multilevel approach. IEEE Trans Pattern Anal Mach Intell 29(11):1944–1957CrossRefGoogle Scholar
  5. 5.
    Evett IW, Spiehler EJ (1987) Rule induction in forensic science. In: KBS in government, online publications, pp 107–118Google Scholar
  6. 6.
    Feder T, Hell P, Klein S, Motwani R (1999) Complexity of graph partition problems. In: 31ST ANNUAL ACM STOC. Plenum, New York, pp 464–472Google Scholar
  7. 7.
    Fisher RA (1936) The use of multiple measurements in taxonomic problems. Ann Eugen 7:179–188CrossRefGoogle Scholar
  8. 8.
    Golub TR, Slonim DK, Tamayo P, Huard C, Gaasenbeek M, Mesirov JP, Coller H, Loh ML, Downing JR, Caligiuri MA, Bloomfield CD, Lander ES (1999) Molecular classification of cancer: class discovery and class prediction by gene expression monitoring. Science 286(5439):531–537CrossRefGoogle Scholar
  9. 9.
    Hoshida Y, Brunet JP, Tamayo P, Golub TR, Mesiro JP (2007) Subclass mapping: identifying common subtypes in independent disease data sets. PLoS ONE 2(11):e1195CrossRefGoogle Scholar
  10. 10.
    Hubert L, Arabie P (1985) Comparing partitions. J Classif 2:193–218CrossRefMATHGoogle Scholar
  11. 11.
    Huttenhower C, Flamholz AI, Landis JN, Sahi S, Myers CL, Olszewski KL, Hibbs MA, Siemers NO, Troyanskaya OG, Coller HA (2007) Nearest neighbor networks: clustering expression data based on gene neighborhoods. BMC Bioinform 8:250CrossRefGoogle Scholar
  12. 12.
    Karypis G, Kumar V (1996) Parallel multilevel graph partitioning. In: Proceedings of the international parallel processing symposiumGoogle Scholar
  13. 13.
    Karypis G, Kumar V (1998) A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J Sci Comput 20(1):359–392MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lu J, Getz G, Miska EA, Alvarez-Saavedra E, Lamb J, Peck D, Sweet-Cordero A, Ebert BL, Mak RH, Ferrando AA, Downing JR, Jacks T, Horvitz RR, Golub TR (2005) Microrna expression profiles classify human cancers. Nature 435(7043):834–838CrossRefGoogle Scholar
  15. 15.
    Maier M, von Luxburg U, Hein M (2009) Influence of graph construction on graph-based clustering measures. In: Koller D, Schuurmans D, Bengio Y, Bottou L (eds) Advances in neural information processing systems, vol 21, pp 1025–1032. Curran, Red HookGoogle Scholar
  16. 16.
    Monti S, Tamayo P, Mesirov J, Golub T (2003) Consensus clustering: a resampling-based method for class discovery and visualization of gene expression microarray data. Kluwer Academic, Dordrecht. Tech rep, Broad Institute/MITMATHGoogle Scholar
  17. 17.
    Nakai K, Kanehisa M (1991) Expert system for predicting protein localization sites in gram-negative bacteria. Proteins 11:95–110CrossRefGoogle Scholar
  18. 18.
    Nascimento MCV, Toledo FMB, Carvalho ACPLF (2010) Investigation of a new GRASP-based clustering algorithm applied to biological data. Comput Oper Res 37:1381–1388CrossRefMATHGoogle Scholar
  19. 19.
    Newman MEJ, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E 69:026113CrossRefGoogle Scholar
  20. 20.
    Onnela JP, Saramäki J, Kertész J, Kaski K (2005) Intensity and coherence of motifs in weighted complex networks. Phys Rev E 71:065(R), 103(R)CrossRefGoogle Scholar
  21. 21.
    Pons P, Latapy M (2005) Computing communities in large networks using random walks. In: Computer and information sciences—ISCIS 2005, pp 284–293CrossRefGoogle Scholar
  22. 22.
    Ramaswamy S, Tamayo P, Rifkin R, Mukherjee S, Yeang CH, Angelo M, Ladd C, Reich M, Latulippe E, Mesirov JP, Poggio T, Gerald W, Loda M, Lander ES, Golub TR (2001) Multiclass cancer diagnosis using tumor gene expression signatures. Proc Natl Acad Sci USA 98(26):15,149–15,154CrossRefGoogle Scholar
  23. 23.
    Reichardt J, Bornholdt S (2006) Statistical mechanics of community detection. Phys Rev E 74:016 110MathSciNetGoogle Scholar
  24. 24.
    Schaeffer SE (2007) Graph clustering. Comput Sci Rev 1:27–64CrossRefMATHGoogle Scholar
  25. 25.
    Shi J, Malik J (2000) Normalized cuts and image segmentation. IEEE Trans Pattern Anal Mach Intell 22:888–905CrossRefGoogle Scholar
  26. 26.
    Shipp MA, Ross KN, Tamayo P, Weng AP, Kutok JL, Aguiar RCT, Gaasenbeek M, Angelo M, Reich M, Pinkus GS, Ray TS, Koval MA, Last KW, Norton A, Lister TA, Mesirov J (2002) Diffuse large b-cell lymphoma outcome prediction by gene-expression profiling and supervised machine learning. Nat Med 8:68–74CrossRefGoogle Scholar
  27. 27.
    Su AI, Cooke MP, Ching KA, Hakak Y, Walker JR, Wiltshire T, Orth AP, Vega RG, Sapinoso LM, Moqrich A, Patapoutian A, Hampton GM, Schultz PG, Hogenesch JB (2002) Large-scale analysis of the human and mouse transcriptomes. Proc Natl Acad Sci USA 99:4465–4470CrossRefGoogle Scholar
  28. 28.
    van ’t Veer LJ, Dai H, van de Vijver MJ, He YD, Hart AA, Mao M, Peterse HL, van der Kooy K, Marton MJ, Witteveen AT, Schreiber GJ, Kerkhoven RM, Roberts C, Linsley PS, Bernards R, Friend SH (2002) Gene expression profiling predicts clinical outcome of breast cancer. Nature 415(6871):530–536CrossRefGoogle Scholar
  29. 29.
    Venables WN, Smith DM (2010) An introduction to R. R Development Core Team, The R Foundation for Statistical Computing, version 2.11.1Google Scholar
  30. 30.
    Watts D, Strogatz S (1998) Collective dynamics of small-world networks. Nature 393:440CrossRefGoogle Scholar
  31. 31.
    West M, Blanchette C, Dressman H, Huang E, Ishida S, Spang R, Zuzan H, Olson JA, Marks JR, Nevins JR (2001) Predicting the clinical status of human breast cancer by using gene expression profiles. Proc Natl Acad Sci USA 98(20):11462–11467CrossRefGoogle Scholar
  32. 32.
    Yeoh EJ, Ross ME, Shurtleff SA, Williams WK, Patel D, Mahfouz R, Behm F, Raimondi SC, Relling MV, Patel A, Cheng C, Campana D, Wilkins D, Zhou X, Li J, Liu H, Pui CH, Evans WE, Naeve C, Wong L, Downing J (2002) Classification, subtype discovery, and prediction of outcome in pediatric acute lymphoblastic leukemia by gene expression profiling. Cancer Cell 1:133–143CrossRefGoogle Scholar

Copyright information

© The Brazilian Computer Society 2010

Authors and Affiliations

  • Mariá C. V. Nascimento
    • 1
  • André C. P. L. F. Carvalho
    • 1
  1. 1.Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São PauloSão CarlosBrazil

Personalised recommendations