Skip to main content
SpringerLink
Log in

Community discovery using nonnegative matrix factorization

  • Published:
Data Mining and Knowledge Discovery Aims and scope Submit manuscript

Abstract

Complex networks exist in a wide range of real world systems, such as social networks, technological networks, and biological networks. During the last decades, many researchers have concentrated on exploring some common things contained in those large networks include the small-world property, power-law degree distributions, and network connectivity. In this paper, we will investigate another important issue, community discovery, in network analysis. We choose Nonnegative Matrix Factorization (NMF) as our tool to find the communities because of its powerful interpretability and close relationship between clustering methods. Targeting different types of networks (undirected, directed and compound), we propose three NMF techniques (Symmetric NMF, Asymmetric NMF and Joint NMF). The correctness and convergence properties of those algorithms are also studied. Finally the experiments on real world networks are presented to show the effectiveness of the proposed methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Albert R, Jeong H, Barabasi A-L (1999) The diameter of the world wide web. Nature 401: 130

    Article  Google Scholar 

  • Amaral LAN, Scala A, Bartheélémy M, Stanley HE (2000) Classes of small-world networks. Proc Natl Acad Sci USA 97: 11149–11152

    Article  Google Scholar 

  • Barthelemy M, Amaral LAN (1999) Small-world networks: evidence for a crossover picture. Phys Rev Lett 82: 3180

    Article  Google Scholar 

  • Brunet JP, Tamayo P, Golub TR, Mesirov JP (2004) Metagenes and molecular pattern discovery using matrix factorization. Proc Natl Acad Sci USA 102(12): 4164–4169

    Article  Google Scholar 

  • Chen G, Wang F, Zhang C (2007) Collaborative filtering using orthogonal nonnegative matrix tri-factorization. In: ICDM workshops on high performance computing, pp 303–308

  • Ding C, He X, Simon HD (2005) On the equivalence of nonnegative matrix factorization and spectral clustering. In: Proceedings of SIAM international conference on data mining, pp 606–610

  • Ding C, Li T, Jordan MI (2006a) Convex and semi-nonnegative matrix factorizations. LBNL Tech Report 60428

  • Ding C, Li T, Peng W, Park H (2006b) Orthogonal nonnegative matrix tri-factorizations for clustering. In: SIGKDD, pp 126–135

  • Ding C, Li T, Peng W (2008) On the equivalence between nonnegative matrix factorization and probabilistic latent semantic indexing. Comput Stat Data Anal 52(1): 155–173

    MathSciNet  Google Scholar 

  • Faloutsos M, Faloutsos P, Faloutsos C (1999) On power-law relationships of the internet topology. In: SIGCOMM, pp 251–262

  • Flake GW, Lawrence S, Giles CL (2000) Efficient identification of web communities. In: SIGKDD, pp 150–160

  • Girvan M, Newman ME (2002) Community structure in social and biological networks. Proc Natl Acad Sci USA 99(12): 7821–7826

    Article  MATH  MathSciNet  Google Scholar 

  • Hofmann T, Puzicha J (1999) Latent class models for collaborative filtering. In: IJCAI, pp 688–693

  • Ino H, Kudo M, Nakamura A (2005) Partitioning of web graphs by community topology. In: WWW, pp 661–669

  • Lee DD, Seung HS (1999) Learning the parts of objects by nonnegative matrix factorization. Nature 401: 788–791

    Article  Google Scholar 

  • Lee DD, Seung HS (2000) Algorithms for non-negative matrix factorization, pp 556–562

  • Long B, Zhang Z, Wu X, Yu PS (2007) Relational clustering by symmetric convex coding. In: Proceedings of the 24th international conference on machine learning, pp 569–576

  • Miao G, Song Y, Zhang D, Bai H (2008) Parallel spectral clustering algorithm for large-scale community data mining. In: The 17th WWW workshop on social web search and mining (SWSM)

  • Newman MEJ (2004a) Detecting community structure in networks. Eur Phys J B 38: 321–330

    Article  Google Scholar 

  • Newman MEJ (2004b) Fast algorithm for detecting community structure in very large networks. Phys Rev E 69

  • Paatero P, Tapper U (1994) Positive matrix factorization: a nonnegative factor model with optimal utilization of error estimates of data values. Environmetrics 5: 111–126

    Article  Google Scholar 

  • Palla G, Derényi I, Farkas I, Vicsek T. Uncovering the overlapping community structure of complex networks in nature and society. Nature 435(7043):814–818

  • Pauca VP, Shahnaz F, Berry MW, Plemmons RJ (2004) Positive matrix factorization: a nonnegative factor model with optimal utilization of error estimates of data values. In: The 4th SIAM international conference on data mining, pp 452–456

  • Pennock DM, Horvitz E, Lawrence S, Giles CL (2000) Collaborative filtering by personality diagnosis: a hybrid memory- and model-based approach. In: UAI

  • Priebe CE, Conroy JM, Marchette DJ, Park Y (2005) Scan statistics on Enron graphs. In: Proceedings of SIAM international conference on data mining

  • Resnick P., Iacovou N, Suchak M, Bergstrom P, Riedl J (1994) Grouplens: an open architecture for collaborative filtering of netnews, pp 175–186

  • Ruan J, Zhang W (2007) An efficient spectral algorithm for network community discovery and its applications to biological and social networks. In: ICDM

  • Scott J (2000) Social network analysis: a handbook, 2nd edn. Sage Publications, London

    Google Scholar 

  • Sharan R et al (2005) From the cover: conserved patterns of protein interaction in multiple species. Proc Natl Acad Sci USA 102(6): 1974–1979

    Article  Google Scholar 

  • Strehl A, Ghosh J, Cardie C (2002) Cluster ensembles: a knowledge reuse framework for combining multiple partitions. J Mach Learn Res 3: 583–617

    Article  Google Scholar 

  • von Luxburg U (2007) A tutorial on spectral clustering. Stat Comput 17(4): 395–416

    Article  MathSciNet  Google Scholar 

  • Wang F, Ma S, Yang L, Li T (2006) Recommendation on item graphs. In: ICDM, pp 1119–1123

  • Wang D, Li T, Zhu S, Ding CHQ (2008a) Multi-document summarization via sentence-level semantic analysis and symmetric matrix factorization. In: Proceedings of the 31st annual international ACM SIGIR conference on research and development in information retrieval, pp 307–314

  • Wang F, Li T, Zhang C (2008b) Semi-supervised clustering via matrix factorization. In: The 8th SIAM international conference on data mining

  • Wasserman S, Faust K (1994) Social network analysis. Cambridge University Press, Cambridge

    Google Scholar 

  • Watts DJ, Strogatz SH (1998) Collective dynamics of ’small-world’ networks. Nature 393: 440–442

    Article  Google Scholar 

  • Weiss Y (1999) Segmentation using eigenvectors: a unifying view. In: ICCV, pp 975–982

  • Xie YL, Hopke PK, Paatero P (1999) Positive matrix factorization applied to a curve resolution problem. J Chemometr 12(6): 357–364

    Article  Google Scholar 

  • Yu K, Tresp V (2005) Learning to learn and collaborative filtering. In: NIPS workshop on inductive transfer: 10 years later

  • Zhou D, Huang J, Schölkopf B (2005) Learning from labeled and unlabeled data on a directed graph. In: Proceedings of the 22nd international conference on machine learning, pp 1036–1043

  • Zhang H, Giles CL, Foley HC, Yen J (2007) Probabilistic community discovery using hierarchical latent gaussian mixture model. In: AAAI, pp 663–668

Download references

Author information

Author notes

  1. Fei Wang

    Present address: IBM T.J. Watson Research Lab, Hawthorne, NY, USA

Authors and Affiliations

  1. School of Computing and Information Sciences, Florida International University, Miami, FL, USA

    Fei Wang, Tao Li & Xin Wang

  2. NEC Research Lab America at Cupertino, Cupertino, CA, USA

    Shenghuo Zhu

  3. Department of Computer Science and Engineering, University of Texas at Arlington, Arlington, TX, USA

    Chris Ding

Authors
  1. Fei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shenghuo Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Chris Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fei Wang.

Additional information

Responsible editor: Eamonn Keogh.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, F., Li, T., Wang, X. et al. Community discovery using nonnegative matrix factorization. Data Min Knowl Disc 22, 493–521 (2011). https://doi.org/10.1007/s10618-010-0181-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10618-010-0181-y

Keywords

Search

Navigation