Advertisement

Cluster Identification in Nearest-Neighbor Graphs

  • Markus Maier
  • Matthias Hein
  • Ulrike von Luxburg
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4754)

Abstract

Assume we are given a sample of points from some underlying distribution which contains several distinct clusters. Our goal is to construct a neighborhood graph on the sample points such that clusters are “identified”: that is, the subgraph induced by points from the same cluster is connected, while subgraphs corresponding to different clusters are not connected to each other. We derive bounds on the probability that cluster identification is successful, and use them to predict “optimal” values of k for the mutual and symmetric k-nearest-neighbor graphs. We point out different properties of the mutual and symmetric nearest-neighbor graphs related to the cluster identification problem.

Keywords

Sample Point Random Graph Neighborhood Graph Cluster Identification High Density Region 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bettstetter, C.: On the minimum node degree and connectivity of a wireless multihop network. In: Proceedings of MobiHoc, pp. 80–91 (2002)Google Scholar
  2. Biau, G., Cadre, B., Pelletier, B.: A graph-based estimator of the number of clusters. ESIAM: Prob. and Stat. 11, 272–280 (2007)MathSciNetzbMATHGoogle Scholar
  3. Bollobas, B.: Random Graphs. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  4. Bollobas, B., Riordan, O.: Percolation. Cambridge Universiy Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  5. Brito, M., Chavez, E., Quiroz, A., Yukich, J.: Connectivity of the mutual k-nearest-neighbor graph in clustering and outlier detection. Stat. Probabil. Lett. 35, 33–42 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, 13–30 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Kunniyur, S.S., Venkatesh, S.S.: Threshold functions, node isolation, and emergent lacunae in sensor networks. IEEE Trans. Inf. Th. 52(12), 5352–5372 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Maier, M., Hein, M., von Luxburg, U.: Cluster identification in nearest-neighbor graphs. Technical Report 163, MPI for Biological Cybernetics, Tübingen (2007)Google Scholar
  9. Penrose, M.: Random Geometric Graphs. Oxford University Press, Oxford (2003)CrossRefzbMATHGoogle Scholar
  10. Santi, P., Blough, D.: The critical transmitting range for connectivity in sparse wireless ad hoc networks. IEEE Trans. Mobile Computing 02(1), 25–39 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Markus Maier
    • 1
  • Matthias Hein
    • 1
  • Ulrike von Luxburg
    • 1
  1. 1.Max Planck Institute for Biological Cybernetics, TübingenGermany

Personalised recommendations