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Structural Similarity in Graphs

A Relaxation Approach for Role Assignment

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Book cover Algorithms and Computation (ISAAC 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3341))

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Abstract

Standard methods for role assignment partition the vertex set of a graph in such a way that vertices in the same class can be considered to have equivalent roles in the graph. Several classes of equivalence relations such as regular equivalence and equitable partitions have been proposed for role assignment, but they all suffer from the strictness of classifying vertices into being either equivalent or not. It is an open problem how to allow for varying degrees of similarity. Proposals include ad-hoc algorithmic approaches and optimization approaches which are computationally hard.

In this paper we introduce the concept of structural similarity by relaxation of equitable partitions, thus providing a theoretical foundation for similarity measures which enjoys desirable properties with respect to existence, structure, and tractability.

We gratefully acknowledge financial support from Deutsche Forschungsgemeinschaft (DFG, grant Br 2158/1-2)

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References

  1. Artin, M.: Algebra. Prentice-Hall, Englewood Cliffs (1991)

    Google Scholar 

  2. Barnes, E.R.: An algorithm for partitioning the nodes of a graph. SIAM Journal on Algebraic and Discrete Methods 3, 541–550 (1982)

    Article  MATH  Google Scholar 

  3. Batagelj, V., Doreian, P., Ferligoj, A.: An optimizational approach to regular equivalence. Social Networks 14, 121–135 (1992)

    Article  Google Scholar 

  4. Bollobás, B.: Modern Graph Theory. Springer, Heidelberg (1998)

    MATH  Google Scholar 

  5. Bonacich, P.: Factoring and weighting approaches to status scores and clique identification. Journal of Mathematical Sociology 2, 113–120 (1972)

    Article  Google Scholar 

  6. Borgatti, S.P., Everett, M.G.: The class of all regular equivalences: Algebraic structure and computation. Social Networks 11, 65–88 (1989)

    Article  MathSciNet  Google Scholar 

  7. Borgatti, S.P., Everett, M.G.: Two algorithms for computing regular equivalence. Social Networks 15, 361–376 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Borgatti, S.P., Everett, M.G.: Models of core/periphery structures. Social Networks 21, 375–395 (1999)

    Article  Google Scholar 

  9. Boyd, J.P., Everett, M.G.: Relations, residuals, regular interiors, and relative regular equivalence. Social Networks 21, 147–165 (1999)

    Article  Google Scholar 

  10. Davis, A., Gardner, B., Gardner, M.: Deep south. The University of Chicago Press (1941)

    Google Scholar 

  11. Diestel, R.: Graph Theory. Springer, New York (2000)

    Google Scholar 

  12. Everett, M.G., Borgatti, S.P.: Regular equivalence: General theory. Journal of Mathematical Sociology 19, 29–52 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fiala, J., Paulusma, D.: The computational complexity of the role assignment problem. In: Proceedings of the ICALP 2003, pp. 817–828. Springer, Heidelberg (2003)

    Google Scholar 

  14. Freeman, L.C.: Finding social groups: A meta-analysis of the southern women data. In: Breiger, R., Carley, K.M., Pattison, P. (eds.) Dynamic Social Network Modeling and Analysis, The National Academies Press, Washington (2003)

    Google Scholar 

  15. Godsil, C., Royle, G.: Algebraic Graph Theory. Springer, Heidelberg (2001)

    MATH  Google Scholar 

  16. Grätzer, G.: General Lattice Theory. Birkhäuser, Basel (1998)

    MATH  Google Scholar 

  17. Kratochvíl, J.: Perfect Codes in General Graphs. Academia Praha (1991)

    Google Scholar 

  18. Lorrain, F., White, H.C.: Structural equivalence of individuals in social networks. Journal of Mathematical Sociology 1, 49–80 (1971)

    Article  Google Scholar 

  19. Milner, R.: A Calculus of Communicating Systems. LNCS, vol. 92. Springer, Berlin (1980)

    MATH  Google Scholar 

  20. Wasserman, S., Faust, K.: Social Network Analysis: Methods and Applications. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  21. White, D.R., Reitz, K.P.: Graph and semigroup homomorphisms on networks of relations. Social Networks 5, 193–234 (1983)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer & Information Science, University of Konstanz,  

    Ulrik Brandes & Jürgen Lerner

Authors
  1. Ulrik Brandes
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  2. Jürgen Lerner
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, Fudan University, 200433, Shanghai, China

    Rudolf Fleischer

  2. Department of Computer Science, The Hong Kong University of Science and Technology, Hong Kong

    Gerhard Trippen

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© 2004 Springer-Verlag Berlin Heidelberg

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Brandes, U., Lerner, J. (2004). Structural Similarity in Graphs. In: Fleischer, R., Trippen, G. (eds) Algorithms and Computation. ISAAC 2004. Lecture Notes in Computer Science, vol 3341. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30551-4_18

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  • DOI: https://doi.org/10.1007/978-3-540-30551-4_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-24131-7

  • Online ISBN: 978-3-540-30551-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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