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Graph-Theoretic Models for Testing the Homogeneity of Data

  • E. Godehardt
  • A. Horsch
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)

Summary

In cluster analysis, the random graph model G n,p and G n,p-based multigraph models have been used for purposes of statistical modelling of data and testing the randomness of outlined clusters. While being appropriate for non-metric data, such models supposing independence of all edges do not take into account the triangle inequality which is valid for metric data. We will introduce graph models I n,dand I t,n,(d1,…,dt) for random intersection graphs in R 1 and multigraphs in R t under which the triangle inequality holds. We derive limit theorems for the distribution of random variables which describe important properties of these random intersection graphs. While being asymptotically equivalent for some properties like the limit distribution of the number of isolated points, the G n,p-model and the I n,d-model differ in numerous aspects.

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References

  1. BARBOUR, A.D., HOLST, L., JANSON, S. (1992): Poisson approximations. Clarendon Press, Oxford.Google Scholar
  2. BOCK, H.H. (1980): Clusteranalyse - Überblick und neuere Entwicklungen. OR Spektrum, 1, 211–232. CrossRefGoogle Scholar
  3. BOLLOBÁS, B. (1985): Random graphs. Academic Press, London - New York Tokyo.Google Scholar
  4. ERDŐS, P., RÉNYI, A. (1960): On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 5, 17–61.Google Scholar
  5. GILBERT, E.N. (1959): Random graphs. Annals of Mathematical Statistics, 30, 114I-1I44.Google Scholar
  6. GODEHARDT, E. (1990): Graphs as structural models: The application of graphs and multigraphs in cluster analysis (Advances in systems analysis, Vol. 4). Friedr. Vieweg & Sohn, Braunschweig - Wiesbaden.Google Scholar
  7. GODEHARDT, E. (1993): Probability models for random multigraphs with applications in cluster analysis. Annals of Discrete Mathematics, 55, 93–108. CrossRefGoogle Scholar
  8. GODEHARDT, E., HORSCH, A. (1994): Testing of data structures with graph- theoretical models, in: Bock, H.H., Lenski, W., Richter, M.M. (eds.): Information systems and data analysis (Proceedings 17th Annual Conference of the Gesellschaft für Klassifikation e.V., Kaiserslautern, March 3–5, 1993). Springer, Berlin - Heidelberg - New York, 226–241Google Scholar
  9. LING, R.F. (1973): A probability theory of cluster analysis. Journal of the American Statistical Association, 68, 159–164-CrossRefGoogle Scholar
  10. LUCZAK, T. (1990): On the equivalence of two basic models of random graphs. In: M. Karoński, J. Jaworski, A. Ruciiiski (eds.): Random Graphs ‘87. John Wiley & Sons, New York - Chichester - Brisbane, 151–157.Google Scholar
  11. KENNEDY, J.W. (1976): Random clumps, graphs, and polymer solutions. In: Y. Alavi, D.R. Lick (eds.): Theory and Applications of Graphs. Springer, Berlin - Heidelberg - New York, 314–329.Google Scholar
  12. ROACH, S.A. (1968): The theory of random clumping. Methuen & Co, London.Google Scholar
  13. ROBERTS, F.S. (1976): Discrete mathematical models. Prentice-Hall, Engle- wood Cliffs.Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1996

Authors and Affiliations

  • E. Godehardt
    • 1
  • A. Horsch
    • 1
  1. 1.AG Biometrie der Klinik für Thorax- und Kardiovaskular-ChirurgieHeinrich Heine-UniversitätDüsseldorfGermany

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