Skip to main content
SpringerLink
Log in

Topological graph clustering with thin position

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

A clustering algorithm partitions a set of data points into smaller sets (clusters) such that each subset is more tightly packed than the whole. Many approaches to clustering translate the vector data into a graph with edges reflecting a distance or similarity metric on the points, then look for highly connected subgraphs. We introduce such an algorithm based on ideas borrowed from the topological notion of thin position for knots and 3-dimensional manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Carlsson, G., Mémoli, F.: Multiparameter hierarchical clustering methods. Classification as a tool for research, Stud. Classification Data Anal. Knowledge Organ., pp. 63–70. Springer, Berlin (2010)

  2. Defays, D.: An efficient algorithm for a complete link method. Comput. J. 20(4), 364–366 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  3. Donath, W.E., Hoffman, A.J.: Lower bounds for the partitioning of graphs. IBM J. Res. Dev. 17, 420–425 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  4. Everitt, B.S., Landau, S., Leese, M., Stahl, D.: Cluster Analysis, 5th edn. Wiley, New York (2011)

    Book  MATH  Google Scholar 

  5. David, G.: Foliations and the topology of 3-manifolds. III. J. Differ. Geom. 26(3), 479–536 (1987)

    MATH  Google Scholar 

  6. Hartigan, J.A., Wong, M.A.: A k-means clustering algorithm. J. R. Stat. Soc. Ser. C. (Applied Statistics) 28(1), 100–108 (1979)

    MATH  Google Scholar 

  7. Marc, L.: Heegaard splittings, the virtually Haken conjecture and property \((\tau )\). Invent. Math. 164(2), 317–359 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Minoux, M., Pinson, E.: Lower bounds to the graph partitioning problem through generalized linear programming and network flows. RAIRO Rech. Opér. 21(4), 349–364 (1987)

    MATH  MathSciNet  Google Scholar 

  9. Pitts, J.T., Rubinstein, J.H.: Existence of minimal surfaces of bounded topological type in three-manifolds. Miniconference on geometry and partial differential equations (Canberra, 1985), Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 10, Austral. Nat. Univ., Canberra, pp. 163–176 (1986)

  10. Scharlemann, M., Thompson, A.: Thin position for 3-manifolds. Geometric topology (Haifa, 1992), Contemp. Math., vol. 164, Am. Math. Soc., Providence, RI, pp. 231–238 (1994)

  11. Sibson, R.: SLINK: an optimally efficient algorithm for the single-link cluster method. Comput. J. 16, 30–34 (1973)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Oklahoma State University, Stillwater, OK, 74078, USA

    Jesse Johnson

Authors
  1. Jesse Johnson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jesse Johnson.

Additional information

This project was supported by NSF Grant DMS-1006369 and was inspired by the workshop The Geometry of Large Networks at the American Institute of Mathematics in November, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Johnson, J. Topological graph clustering with thin position. Geom Dedicata 169, 165–173 (2014). https://doi.org/10.1007/s10711-013-9848-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-013-9848-z

Keywords

Mathematics Subject Classification

Search

Navigation