Consensus Methods for Pyramids and Other Hypergraphs

  • J. Lehel
  • F. R. McMorris
  • R. C. Powers
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)


A classification can most generally be viewed as a hypergraph, which is simply a set of subsets (clusters) of the finite set S of objects being studied. In this paper, we are primarily concerned with consensus functions on tree hypergraphs such as pyramids and totally balanced hypergraphs.


Intersection Rule Consensus Function Consensus Method Unanimity Rule Counting Rule 
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  1. Adams, E.N. III (1986): N-trees as nestings: Complexity, Similarity, and Consensus, Journal of Classification, 3, 2, 299–317.MathSciNetMATHCrossRefGoogle Scholar
  2. Bandelt, H.-J. and Dress, A. (1989): Weak hierarchies associated with similarity measures-an additive clustering technique, Bulletin of Mathematical Biology, 51, 1, 133–166.MathSciNetMATHGoogle Scholar
  3. Bertrand, P. (1995): Structural Properties of Pyramidal Clustering, In: Partitioning Data Sets. Cox, 1. et al. (eds.), DIMACS Series in Discrete Mathematics and Theoretical Cornputer Science, 19, 35–53, AMS, Providence, RI.Google Scholar
  4. Bertrand, P. and Diday, E. (1991): Les pyramides classifiantes: une extension de la structure hiérarchique, C. R. Acad. Sci. Paris, Série I, 693–696.Google Scholar
  5. Duchet, P. (1995): Hypergraphs, In: Handbook of Combinatorics, Graham, R. et al. (eds.), VOL. 1, 381–432, MIT Press, Cambridge, MA.Google Scholar
  6. Gaul, W. and Schadet, M. (1994): Pyramidal classification based on incomplete dissimilarity data, Journal of Classification, 11, 2, 171–193.MathSciNetMATHCrossRefGoogle Scholar
  7. Lehel, J. (1983): Helly-hypergraphs and abstract interval structures, ARS Combinatoria, 16-A, 239–253.Google Scholar
  8. Lehel, J. (1985): A characterization of totally balanced hypergraphs, Discrete Mathematics, 57, 59–65.MathSciNetMATHCrossRefGoogle Scholar
  9. Margush, T. and McMorris, F.R. (1981): Consensus n-trees, Bulletin of Mathematical Biology, 43, 239–344.MathSciNetMATHGoogle Scholar
  10. McMorris, F.R. and Neumann, D.A. (1983): Consensus functions defined on trees, Mathematical Social Sciences, 4, 131–136.MathSciNetMATHCrossRefGoogle Scholar
  11. McMorris, F.R. and Powers, R.C. (1991): Consensus weak hierarchies, Bulletin of Mathematical Biology, 53, 679–684.MATHGoogle Scholar
  12. McMorris, F.R. and Powers, R.C. (1996): Intersection rules for consensus hierarchies, In: Proceedings of the third international conference on ordinal and symbolic data analysis, Di-day, E. et al. (eds.), 301–308, Springer Verlag, Berlin.CrossRefGoogle Scholar
  13. Neumann, D.A. (1983): Faithful consensus methods for n-trees. Mathematical Biosciences, 63, 271–287.MathSciNetMATHCrossRefGoogle Scholar
  14. Powers, R.C. (1995): Intersection rules for consensus n-trees, Applied Mathematics Letters, 8, 4, 51–55.MathSciNetMATHCrossRefGoogle Scholar
  15. Vach, W. (1994): Preserving consensus hierarchies, Journal of Classification, 11, 1, 59–77.MathSciNetMATHCrossRefGoogle Scholar
  16. Van Cutsem, B. (1994): Classification and dissimilarity analysis, Lecture Notes in Statistics, New York.MATHCrossRefGoogle Scholar

Copyright information

© Springer Japan 1998

Authors and Affiliations

  • J. Lehel
    • 1
  • F. R. McMorris
    • 1
  • R. C. Powers
    • 1
  1. 1.Department of MathematicsUniversity of LouisvilleLouisvilleUSA

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