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Uncovering Hidden Block Structure for Clustering

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Machine Learning and Knowledge Discovery in Databases (ECML PKDD 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11906))

Abstract

We present a multistage procedure to cluster directed and undirected weighted graphs by finding the block structure of their adjacency matrices. A central part of the process is to scale the adjacency matrix into a doubly-stochastic form, which permits detection of the whole matrix block structure with minimal spectral information (theoretically a single pair of singular vectors suffices).

We present the different stages of our method, namely the impact of the doubly-stochastic scaling on singular vectors, detection of the block structure by means of these vectors, and details such as cluster refinement and a stopping criterion. Then we test the algorithm’s effectiveness by using it on two unsupervised classification tasks: community detection in networks and shape detection in clouds of points in two dimensions. By comparing results of our approach with those of widely used algorithms designed for specific purposes, we observe that our method is competitive (for community detection) if not superior (for shape detection) in comparison with existing methods.

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References

  1. Bagrow, J.P.: Communities and bottlenecks: trees and treelike networks have high modularity. Phys. Rev. E 85(6), 066118 (2012)

    Article  Google Scholar 

  2. Blondel, V.D., Guillaume, J.L., Lambiotte, R., Lefebvre, E.: Fast unfolding of communities in large networks. J. Stat. Mech: Theory Exp. 2008(10), P10008 (2008)

    Article  MATH  Google Scholar 

  3. Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. Comput. Netw. ISDN Syst. 30(1–7), 107–117 (1998)

    Article  Google Scholar 

  4. Canny, J.: A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 8, 679–698 (1986)

    Article  Google Scholar 

  5. Clauset, A., Newman, M.E.J., Moore, C.: Finding community structure in very large networks. Phys. Rev. E 70(6), 066111 (2004)

    Article  Google Scholar 

  6. Conde-Cespedes, P.: Modélisation et extension du formalisme de l’analyse relationnelle mathématique à la modularisation des grands graphes. Ph.D thesis, Université Pierre et Marie Curie (2013)

    Google Scholar 

  7. Csardi, G., Nepusz, T.: The igraph software package for complex network research. InterJournal Complex Syst. 1695, 1–9 (2006)

    Google Scholar 

  8. Davis, T.A., Hu, Y.: The University of Florida sparse matrix collection. ACM Trans. Math. Softw. 38(1), 1–14 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Dhillon, I.S.: Co-clustering documents and words using bipartite spectral graph partitioning. In: Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2001, pp. 269–274. ACM (2001)

    Google Scholar 

  10. Dhillon, I.S., Mallela, S., Modha, D.S.: Information-theoretic co-clustering. In: Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2003, pp. 89–98. ACM (2003)

    Google Scholar 

  11. Duff, I., Knight, P., Le Gorrec, L., Mouysset, S., Ruiz, D.: Uncovering hidden block structure. Technical report TR/PA/18/90, CERFACS, August 2018

    Google Scholar 

  12. Fortunato, S.: Community detection in graphs. Phys. Rep. 486(3), 75–174 (2010)

    Article  MathSciNet  Google Scholar 

  13. Fortunato, S., Barthélemy, M.: Resolution limit in community detection. Proc. Nat. Acad. Sci. 104(1), 36–41 (2007)

    Article  Google Scholar 

  14. Fortunato, S., Hric, D.: Community detection in networks: a user guide. CoRR abs/1608.00163 (2016)

    Google Scholar 

  15. Fred, A.L.N., Jain, A.K.: Robust data clustering. In: CVPR, no. 2, pp. 128–136. IEEE Computer Society (2003)

    Google Scholar 

  16. Fritzsche, D., Mehrmann, V., Szyld, D.B., Virnik, E.: An SVD approach to identifying metastable states of Markov chains. Electron. Trans. Numer. Anal. 29, 46–69 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Frobenius, G.: Ueber matrizen aus nicht negativen elementen. Sitzungsber. Königl. Preuss. Akad. Wiss, 456–477 (1912)

    Google Scholar 

  18. Knight, P.A., Ruiz, D.: A fast algorithm for matrix balancing. IMA J. Numer. Anal. 33(3), 1029–1047 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lancichinetti, A., Fortunato, S., Radicchi, F.: Benchmark graphs for testing community detection algorithms. Phys. Rev. E 78(4), 046110 (2008)

    Article  Google Scholar 

  20. Lei, J., Rinaldo, A., et al.: Consistency of spectral clustering in stochastic block models. Ann. Stat. 43(1), 215–237 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mouysset, S., Noailles, J., Ruiz, D.: Using a global parameter for gaussian affinity matrices in spectral clustering. In: Palma, J.M.L.M., Amestoy, P.R., Daydé, M., Mattoso, M., Lopes, J.C. (eds.) VECPAR 2008. LNCS, vol. 5336, pp. 378–390. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-92859-1_34

    Chapter  Google Scholar 

  22. Newman, M.E.J.: Finding community structure in networks using the eigenvectors of matrices. Phys. Rev. E 74(3), 036104 (2006)

    Article  MathSciNet  Google Scholar 

  23. Newman, M.: Analysis of weighted networks. Phys. Rev. E 70, 056131 (2004)

    Article  Google Scholar 

  24. Ng, A.Y., Jordan, M.I., Weiss, Y.: On spectral clustering: analysis and an algorithm. In: Proceedings of the 14th International Conference on Neural Information Processing Systems: Natural and Synthetic, NIPS 2001, pp. 849–856. MIT Press (2001)

    Google Scholar 

  25. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., Duchesnay, E.: Scikit-learn: machine learning in Python. J. Mach. Learn. Res. 12, 2825–2830 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Perron, O.: Zur theorie der matrices. Mathematische Annalen 64(2), 248–263 (1907)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pons, P., Latapy, M.: Computing communities in large networks using random walks (long version). arXiv Physics e-prints, December 2005

    Google Scholar 

  28. Schaeffer, S.E.: Graph clustering. Comput. Sci. Rev. 1(1), 27–64 (2007)

    Article  MATH  Google Scholar 

  29. Sinkhorn, R., Knopp, P.: Concerning nonnegative matrices and doubly stochastic matrices. Pacific J. Math. 21, 343–348 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  30. Von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)

    Article  MathSciNet  Google Scholar 

  31. Yang, Z., Algesheimer, R., Tessone, C.J.: A comparative analysis of community detection algorithms on artificial networks. Sci. Rep. 6, 30750 (2016)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Université de Toulouse - IRIT, 2 rue Camichel, Toulouse, France

    Luce le Gorrec, Sandrine Mouysset & Daniel Ruiz

  2. R7l, STFC Rutherford Appleton Laboratory, Didcot, OX, OX11 0QX, UK

    Iain S. Duff

  3. CERFACS, Toulouse, France

    Iain S. Duff

  4. University of Strathclyde, Richmond Street, Glasgow, UK

    Philip A. Knight

Authors
  1. Luce le Gorrec
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  2. Sandrine Mouysset
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  3. Iain S. Duff
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  4. Philip A. Knight
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  5. Daniel Ruiz
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Corresponding author

Correspondence to Luce le Gorrec .

Editor information

Editors and Affiliations

  1. Leuphana University, Lüneburg, Germany

    Ulf Brefeld

  2. IRISA/Inria, Rennes, France

    Elisa Fromont

  3. University of Würzburg, Würzburg, Germany

    Andreas Hotho

  4. Leiden University, Leiden, The Netherlands

    Arno Knobbe

  5. ETH Zurich, Zurich, Switzerland

    Marloes Maathuis

  6. Institut National des Sciences Appliquées, Villeurbanne, France

    Céline Robardet

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le Gorrec, L., Mouysset, S., Duff, I.S., Knight, P.A., Ruiz, D. (2020). Uncovering Hidden Block Structure for Clustering. In: Brefeld, U., Fromont, E., Hotho, A., Knobbe, A., Maathuis, M., Robardet, C. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2019. Lecture Notes in Computer Science(), vol 11906. Springer, Cham. https://doi.org/10.1007/978-3-030-46150-8_9

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  • DOI: https://doi.org/10.1007/978-3-030-46150-8_9

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-46149-2

  • Online ISBN: 978-3-030-46150-8

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