Approximate and Spectral Clustering for Network and Affinity Data

  • Boris Mirkin
Part of the Undergraduate Topics in Computer Science book series (UTICS)


This chapter is devoted to clustering similarity, graph and network data – these are represented by square matrices rather than rectangular ones. This chapter describes methods for finding a cluster or two-cluster split combining three types of approaches from both old and recent developments: (a)combinatorial approach that is oriented at clustering as optimization of some reasonable measure of cluster homogeneity, (b)additive clustering approach that is based on a data recovery model at which the data is decoded from a cluster structure to be found by minimizing the discrepancy between them and observed similarities, and (c)spectral clustering approach exploiting the machinery of matrix eigenvalues and eigenvectors as a relaxation of combinatorial problems for similarity clustering.


Similarity Matrix Spectral Cluster Cluster Similarity Rayleigh Quotient Additive Cluster 
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. Guattery, S., Miller, G.: On the quality of spectral separators. SIAM J. Matrix Anal. Appl. 19(3), 701–719 (1998).MathSciNetMATHCrossRefGoogle Scholar
  2. Johnsonbaugh, R., Schaefer, M.: Algorithms. Pearson Prentice Hall, Upper Saddle River (2004). ISBN 0-13-122853-6.Google Scholar
  3. Klein, C., Randic, M.: Resistance distance. J. Mathematical Chem. 12, 81–95 (1993).MathSciNetCrossRefGoogle Scholar
  4. Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17, 395–416 (2007).MathSciNetCrossRefGoogle Scholar
  5. Mirkin, B.: Additive clustering and qualitative factor analysis methods for similarity matrices. J. Classif. 4, 7–31 (1987); Erratum (1989), 6, 271–272.MathSciNetMATHCrossRefGoogle Scholar
  6. Mirkin, B.: Mathematical Classification and Clustering. Kluwer Academic Press, Boston-Dordrecht (1996).Google Scholar
  7. Mirkin, B., Camargo, R., Fenner, T., Loizou, G., Kellam, P.: Similarity clustering of proteins using substantive knowledge and reconstruction of evolutionary gene histories in herpesvirus. Theor. Chem. Acc.: Theory, Comput. Mod. 125(3–6), 569–582 (2010).CrossRefGoogle Scholar
  8. Newman, M.E.J.: Modularity and community structure in networks. PNAS. 103(23), 8577–8582 (2006).CrossRefGoogle Scholar
  9. Newman, M., Girvan, M.: Finding and evaluating community structure in networks. Phys. Rev. E. 69, 026113 (2004).CrossRefGoogle Scholar
  10. Shepard, R.N., Arabie, P.: Additive clustering: Representation of similarities as combinations of discrete overlapping properties. Psychol. Rev. 86, 87–123 (1979).CrossRefGoogle Scholar
  11. Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Machine Intelligence. 22(8), 888–905 (2000).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  • Boris Mirkin
    • 1
    • 2
  1. 1.Research University – Higher School of Economics, School of Applied Mathematics and InformaticsMoscowRussia
  2. 2.Department of Computer ScienceBirkbeck University of LondonLondonUK

Personalised recommendations