| Peer-Reviewed

Modularity Component Analysis versus Principal Component Analysis

Received: 6 April 2016     Published: 11 April 2016
Views:       Downloads:
Abstract

In this paper the exact linear relation between the leading eigenvectors of the modularity matrix and the singular vectors of an uncentered data matrix is developed. Based on this analysis the concept of a modularity component is defined, and its properties are developed. It is shown that modularity component analysis can be used to cluster data similar to how traditional principal component analysis is used except that modularity component analysis does not require data centering.

Published in American Journal of Applied Mathematics (Volume 4, Issue 2)
DOI 10.11648/j.ajam.20160402.15
Page(s) 99-104
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Data Clustering, Graph Partitioning, Modularity Matrix, Principal Component Analysis

References
[1] V. D. BLONDEL, J.-L. GUILLAUME, R. LAMBIOTTE, AND E. LEFEBVRE, Fast unfolding of communities in large networks, Journal of statistical mechanics: theory and experiment, 2008 (2008), p. P10008.
[2] J. R. BUNCH, C. P. NIELSEN, AND D. C. SORENSEN, Rank-one modification of the symmetric eigenproblem, Numerische Mathematik, 31 (1978), pp. 31–48.
[3] R. CHITTA, R. JIN, AND A. K. JAIN, Efficient kernel clustering using random fourier features, in Data Mining (ICDM), 2012 IEEE 12th International Conference on, IEEE, 2012, pp. 161–170.
[4] B. DASGUPTA AND D. DESAI, On the complexity of newmans community finding approach for biological and social networks, Journal of Computer and System Sciences, 79 (2013), pp. 50–67.
[5] M. A. FORTUNA, D. B. STOUFFER, J. M. OLESEN, P. JORDANO, D. MOUILLOT, B. R. KRASNOV, R. POULIN, AND J. BASCOMPTE, Nestedness versus modularity in ecological networks: two sides of the same coin?, Journal of Animal Ecology, 79 (2010), pp. 811–817.
[6] B. H. GOOD, Y.-A. DE MONTJOYE, AND A. CLAUSET, Performance of modularity maximization in practical contexts, Physical Review E, 81 (2010), p. 046106.
[7] T. HERTZ, A. BAR-HILLEL, AND D. WEINSHALL, Boosting margin based distance functions for clustering, in Proceedings of the twenty-first international conference on Machine learning, ACM, 2004, p. 50.
[8] JOLLIFFE, Principal component analysis, Wiley Online Library, 2002.
[9] A. LANCICHINETTI AND S. FORTUNATO, Limits of modularity maximization in community detection, Physical review E, 84 (2011), p. 066122.
[10] Y. LECUN, L. BOTTOU, Y. BENGIO, AND P. HAFFNER, Gradient-based learning applied to document recognition, Proceedings of the IEEE, 86 (1998), pp. 2278–2324.
[11] R. A. MERCOVICH, A. HARKIN, AND D. MESSINGER, Automatic clustering of multispectral imagery by maximization of the graph modularity, in SPIE Defense, Security, and Sensing, International Society for Optics and Photonics, 2011, pp. 80480Z–80480Z.
[12] D. MEUNIER, R. LAMBIOTTE, A. FORNITO, K. D. ERSCHE, AND E. T. BULLMORE, Hierarchical modularity in human brain functional networks, Hierarchy and dynamics in neural networks, 1 (2010), p. 2.
[13] C. D. MEYER, Matrix analysis and applied linear algebra, Siam, 2000.
[14] M. E. NEWMAN, Modularity and community structure in networks, Proceedings of the National Academy of Sciences, 103 (2006), pp. 8577–8582.
[15] M. E. NEWMAN AND M. GIRVAN, Finding and evaluating community structure in networks, Physical review E, 69 (2004), p. 026113.
[16] S. L. RACE, C. MEYER, AND K. VALAKUZHY, Determining the number of clusters via iterative consensus clustering, in Proceedings of the SIAM Conference on Data Mining (SDM), SIAM, 2013, pp. 94–102.
[17] R. ROTTA AND A. NOACK, Multilevel local search algorithms for modularity clustering, Journal of Experimental Algorithmics (JEA), 16 (2011), pp. 2–3.
[18] U. VON LUXBURG, A tutorial on spectral clustering, Statistics and computing, 17 (2007), pp. 395–416.
[19] J. H. WILKINSON, The algebraic eigenvalue problem, vol. 87, Clarendon Press Oxford, 1965.
[20] R. ZHANG AND A. I. RUDNICKY, A large scale clustering scheme for kernel k-means, in Pattern Recognition, 2002. Proceedings. 16th International Conference on, vol. 4, IEEE, 2002, pp. 289–292.
Cite This Article
  • APA Style

    Hansi Jiang, Carl Meyer. (2016). Modularity Component Analysis versus Principal Component Analysis. American Journal of Applied Mathematics, 4(2), 99-104. https://doi.org/10.11648/j.ajam.20160402.15

    Copy | Download

    ACS Style

    Hansi Jiang; Carl Meyer. Modularity Component Analysis versus Principal Component Analysis. Am. J. Appl. Math. 2016, 4(2), 99-104. doi: 10.11648/j.ajam.20160402.15

    Copy | Download

    AMA Style

    Hansi Jiang, Carl Meyer. Modularity Component Analysis versus Principal Component Analysis. Am J Appl Math. 2016;4(2):99-104. doi: 10.11648/j.ajam.20160402.15

    Copy | Download

  • @article{10.11648/j.ajam.20160402.15,
      author = {Hansi Jiang and Carl Meyer},
      title = {Modularity Component Analysis versus Principal Component Analysis},
      journal = {American Journal of Applied Mathematics},
      volume = {4},
      number = {2},
      pages = {99-104},
      doi = {10.11648/j.ajam.20160402.15},
      url = {https://doi.org/10.11648/j.ajam.20160402.15},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20160402.15},
      abstract = {In this paper the exact linear relation between the leading eigenvectors of the modularity matrix and the singular vectors of an uncentered data matrix is developed. Based on this analysis the concept of a modularity component is defined, and its properties are developed. It is shown that modularity component analysis can be used to cluster data similar to how traditional principal component analysis is used except that modularity component analysis does not require data centering.},
     year = {2016}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Modularity Component Analysis versus Principal Component Analysis
    AU  - Hansi Jiang
    AU  - Carl Meyer
    Y1  - 2016/04/11
    PY  - 2016
    N1  - https://doi.org/10.11648/j.ajam.20160402.15
    DO  - 10.11648/j.ajam.20160402.15
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 99
    EP  - 104
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20160402.15
    AB  - In this paper the exact linear relation between the leading eigenvectors of the modularity matrix and the singular vectors of an uncentered data matrix is developed. Based on this analysis the concept of a modularity component is defined, and its properties are developed. It is shown that modularity component analysis can be used to cluster data similar to how traditional principal component analysis is used except that modularity component analysis does not require data centering.
    VL  - 4
    IS  - 2
    ER  - 

    Copy | Download

Author Information
  • North Carolina State University, Raleigh, NC, USA

  • North Carolina State University, Raleigh, NC, USA

  • Sections