Journal of Classification

, Volume 27, Issue 3, pp 279–306 | Cite as

Structural Similarity: Spectral Methods for Relaxed Blockmodeling

  • Ulrik Brandes
  • Jürgen LernerEmail author


In this paper we propose the concept of structural similarity as a relaxation of blockmodeling in social network analysis. Most previous approaches attempt to relax the constraints on partitions, for instance, that of being a structural or regular equivalence to being approximately structural or regular, respectively. In contrast, our approach is to relax the partitions themselves: structural similarities yield similarity values instead of equivalence or non-equivalence of actors, while strictly obeying the requirement made for exact regular equivalences. Structural similarities are based on a vector space interpretation and yield efficient spectral methods that, in a more restrictive manner, have been successfully applied to difficult combinatorial problems such as graph coloring. While traditional blockmodeling approaches have to rely on local search heuristics, our framework yields algorithms that are provably optimal for specific data-generation models. Furthermore, the stability of structural similarities can be well characterized making them suitable for the analysis of noisy or dynamically changing network data.


Social network analysis Blockmodeling Spectral graph partitioning Conflict networks Dynamic network visualization 


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  1. ACHLIOPTAS, D., FIAT, A., KARLIN, A., and MCSHERRY, F. (2001), “Web Search via Hub Synthesis”, in Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science (FOCS’01), pp. 500–509.Google Scholar
  2. ALON, N., and KAHALE, N. (1997), “A Spectral Technique for Coloring Random 3-Colorable Graphs”, SIAM Journal on Computation, 26, 1733–1748.zbMATHMathSciNetCrossRefGoogle Scholar
  3. ARTIN, M. (1991), Algebra, Englewood Cliffs, NJ: Prentice Hall.Google Scholar
  4. AZAR, Y., FIAT, A., KARLIN, A., MCSHERRY, F., and SAIA, J. (2001), “Spectral Analysis of Data”, in Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC), pp. 619–626.Google Scholar
  5. BAI, Z., DEMMEL, J., and MCKENNEY, A. (1993), “On Computing Condition Numbers for the Nonsymmetric Eigenproblem”, ACM Transactions on Mathematical Software, 19, 202–223.zbMATHMathSciNetCrossRefGoogle Scholar
  6. BATAGELJ, V. (1997), “Notes on Blockmodeling”, Social Networks, 19, 143–155.CrossRefGoogle Scholar
  7. BATAGELJ, V., DOREIAN, P., and FERLIGOJ, A. (1992), “An Optimizational Approach to Regular Equivalence”, Social Networks, 14, 121–135.CrossRefGoogle Scholar
  8. BORGATTI, S. P., and EVERETT, M. G. (1992), “Notions of Position in Social Network Analysis”, Sociological Methodology, 22, 1–35.CrossRefGoogle Scholar
  9. BRANDES, U., and ERLEBACH, T. (eds.) (2005), Network Analysis: Methodological Foundations, Berlin, Heidelberg: Springer-Verlag.zbMATHGoogle Scholar
  10. BRANDES, U., FLEISCHER, D., and LERNER, J. (2006), “Summarizing Dynamic Bipolar Conflict Structures”, IEEE Transactions on Visualization and Computer Graphics, special issue on Visual Analytics, 12, 1486–1499.CrossRefGoogle Scholar
  11. BRANDES, U., and LERNER, J. (2004), “Structural Similarity in Graphs”, in Proceedings of the 15th International Symposium on Algorithms and Computation (ISAAC’04), pp. 184–195.Google Scholar
  12. BRANDES, U., and LERNER, J. (2006), “Coloring Random 3-Colorable Graphs with Non-Uniform Edge Probabilities”, in Proceedings of the 31st International Symposium on Mathematical Foundations of Computer Science (MFCS ’06), pp. 202–213.Google Scholar
  13. BRANDES, U., and LERNER, J. (2008), “Visualization of Conflict Networks”, in Building and Using Datasets on Armed Conflicts, ed. Kauffmann, M., IOS Press, vol. 36 of NATO Science for Peace and Security Series E: Human and Societal Dynamics, pp. 169–188.Google Scholar
  14. COJA-OGHLAN, A. (2005), “A Spectral Heuristic for Bisecting Random Graphs”, in Proceeding of the 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 850–859.Google Scholar
  15. CVETKOVIĆ, D. M., DOOB, M., and SACHS, H. (1995), Spectra of Graphs, Heidelberg-Leipzig: Johann Ambrosius Barth.zbMATHGoogle Scholar
  16. DOREIAN, P., BATAGELJ, V., and FERLIGOJ, A. (2005), Generalized Blockmodeling, Cambridge, MA: Cambridge University Press.Google Scholar
  17. EVERETT,M. G., and BORGATTI, S. P. (1991), “Role Colouring a Graph”, Mathematical Social Sciences, 21, 183–188.zbMATHMathSciNetCrossRefGoogle Scholar
  18. EVERETT, M. G., and BORGATTI, S. P. (1994), “Regular Equivalence: General Theory”, Journal of Mathematical Sociology, 19, 29–52.zbMATHMathSciNetCrossRefGoogle Scholar
  19. FIALA, J., and PAULUSMA, D. (2003), “The Computational Complexity of the Role Assignment Problem”, in Proceedings of the 30th International Colloquium on Automata, Languages, and Programming (ICALP’03), Berlin Heidelberg: Springer-Verlag, pp. 817–828.CrossRefGoogle Scholar
  20. GAREY, M. R., and JOHNSON, D. S. (1979), Computers and Intractability: A Guide to the Theory of NP-completeness, San Francisco: Freeman and Company.zbMATHGoogle Scholar
  21. GKANTSIDIS, C., MIHAIL, M., and ZEGURA, E. (2003), “Spectral Analysis of Internet Topologies”, in Proceedings of the IEEE INFOCOM’03, pp. 364– 374.Google Scholar
  22. GODSIL, C. (1993), Algebraic Combinatorics, New York: Chapman & Hall.zbMATHGoogle Scholar
  23. GODSIL, C., and ROYLE, G. (2001), Algebraic Graph Theory, New York: Springer.zbMATHGoogle Scholar
  24. GOHBERG, I., LANCASTER, P., and RODMAN, L. (1986), Invariant Subspaces of Matrices with Applications, New York: John Wiley.zbMATHGoogle Scholar
  25. GOLUB, G. H., and VAN LOAN, C. F. (1996), Matrix Computations, Baltimore,MD: John Hopkins University Press.zbMATHGoogle Scholar
  26. KAISER, H. F. (1958), “The Varimax Criterion for Analytic Rotation in Factor Analysis”, Psychometrica, 23, 187–200.zbMATHCrossRefGoogle Scholar
  27. KANNAN, R., VEMPALA, S., and VETTA, A. (2004), “On Clusterings: Good, Bad and Spectral”, Journal of the ACM, 51, 497–515.zbMATHMathSciNetCrossRefGoogle Scholar
  28. KIM, K. H., and ROUSH, F. W. (1984), “Group Relationships and Homomorphisms of Boolean Matrix Semigroups”, Journal of Mathematical Psychology, 28, 448–452.zbMATHMathSciNetCrossRefGoogle Scholar
  29. LORRAIN, F., and WHITE, H. C. (1971), “Structural Equivalence of Individuals in Social Networks”, Journal of Mathematical Sociology, 1, 49–80.CrossRefGoogle Scholar
  30. LUCZKOVICH, J. J., BORGATTI, S. P., JOHNSON, J. C., and EVERETT, M. G. (2003), “Defining andMeasuring Trophic Role Similarity in FoodWebs Using Regular Equivalence”, Journal of Theoretical Biology, 220, 303–321.MathSciNetCrossRefGoogle Scholar
  31. MCSHERRY, F. (2001), “Spectral Partitioning of Random Graphs”, in Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science (FOCS’01), pp. 529–537.Google Scholar
  32. MILNER, R. (1980), A Calculus of Communicating Systems, Berlin: Springer Verlag.zbMATHGoogle Scholar
  33. PAIGE, R., and TARJAN, R. E. (1987), “Three Partition Refinement Algorithms”, SIAM Journal on Computing, 16, 973–983.zbMATHMathSciNetCrossRefGoogle Scholar
  34. PAPADIMITRIOU, C., RAGHAVAN, P., TAMAKI, H., and VEMPALA, S. (2000), “Latent Semantic Indexing: A Probabilistic Analysis”, Journal of Computer and System Sciences, 61, 217–235.zbMATHMathSciNetCrossRefGoogle Scholar
  35. PATTISON, P. (1993), Algebraic Models for Social Networks, Cambridge, MA: Cambridge University Press.CrossRefGoogle Scholar
  36. SAILER, L. D. (1978), “Structural Equivalence: Meaning and Definition, Computation and Application”, Social Networks, 1, 73–90.CrossRefGoogle Scholar
  37. SCHRODT, P. A., DAVIS, S. G., and WEDDLE, J. L. (1994), “Political Science: KEDS-A Program for the Machine Coding of Event Data”, Social Science Computer Review, 12, 561–588.CrossRefGoogle Scholar
  38. STEWART, G. W., and SUN, J.-G. (1990), Matrix Perturbation Theory, New York: Academic Press.zbMATHGoogle Scholar
  39. VAN DE VELDEN,M., and KIERS, H. A. L. (2005), “Rotation in Correspondence Analysis”, Journal of Classification, 22, 251–271.MathSciNetCrossRefGoogle Scholar
  40. VEMPALA, S., and WANG, G. (2002), “A Spectral Algorithm for Learning Mixtures of Distributions”, in Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS’02).Google Scholar
  41. WASSERMAN, S., and FAUST, K. (1994), Social Network Analysis: Methods and Applications, Cambridge MA: Cambridge University Press.Google Scholar
  42. WHITE, D. R. and REITZ, K. P. (1983), “Graph and Semigroup Homomorphisms on Networks of Relations”, Social Networks, 5, 193–234.MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Computer & Information ScienceUniversity of KonstanzKonstanzGermany

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