Advertisement

Simplicial Complex Entropy

  • Stefan Dantchev
  • Ioannis IvrissimtzisEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10521)

Abstract

We propose an entropy function for simplicial complices. Its value gives the expected cost of the optimal encoding of sequences of vertices of the complex, when any two vertices belonging to the same simplex are indistinguishable. We focus on the computational properties of the entropy function, showing that it can be computed efficiently. Several examples over complices consisting of hundreds of simplices show that the proposed entropy function can be used in the analysis of large sequences of simplicial complices that often appear in computational topology applications.

Keywords

Entropy Simplicial complices Ambiguous encoding Graph entropy Simplicial complex entropy 

Notes

Acknowledgement

This research was partially supported by the EPRSC Grant EP/K016687/1 “Topology, Geometry and Laplacians of Simplicial Complexes”.

References

  1. 1.
    Attali, D., Lieutier, A., Salinas, D.: Efficient data structure for representing and simplifying simplicial complexes in high dimensions. Int. J. Comput. Geom. Appl. 22(04), 279–303 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ay, N., Olbrich, E., Bertschinger, N., Jost, J.: A geometric approach to complexity. Chaos 21(3), 22 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Csiszár, I., Körner, J., Lovász, L., Marton, K., Simonyi, G.: Entropy splitting for antiblocking corners and perfect graphs. Combinatorica 10(1), 27–40 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dantchev, S., Ivrissimtzis, I.: Efficient construction of the Čech complex. Comput. Graph. 36(6), 708–713 (2012)CrossRefGoogle Scholar
  5. 5.
    de Silva, V., Ghrist, R.: Coverage in sensor networks via persistent homology. Algebraic Geom. Topol. 7, 339–358 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    de Silva, V., Carlsson, G.: Topological estimation using witness complexes. In: Alexa, M., Rusinkiewicz, S. (eds.) Eurographics Symposium on Point-Based Graphics. ETH, Zürich (2004)Google Scholar
  7. 7.
    Edelsbrunner, H.: The union of balls and its dual shape. Discrete Comput. Geom. 13(1), 415–440 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: FOCS 2000, p. 454. IEEE (2000)Google Scholar
  9. 9.
    Guibas, L.J., Oudot, S.Y.: Reconstruction using witness complexes. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, pp. 1076–1085, Philadelphia, PA, USA. SIAM (2007)Google Scholar
  10. 10.
    Korner, J., Marton, K.: New bounds for perfect hashing via information theory. Eur. J. Comb. 9(6), 523–530 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Körner, J.: Coding of an information source having ambiguous alphabet and the entropy of graphs. In: 6th Prague Conference on Information Theory, pp. 411–425 (1973)Google Scholar
  12. 12.
    Simonyi, G.: Graph entropy: a survey. Comb. Optim. 20, 399–441 (1995)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Vejdemo-Johansson, M.: Interleaved computation for persistent homology. CoRR, abs/1105.6305 (2011)Google Scholar
  14. 14.
    Wales, D.J., Ulker, S.: Structure and dynamics of spherical crystals characterized for the Thomson problem. Phys. Rev. B 74(21), 212101 (2006)CrossRefGoogle Scholar
  15. 15.
    Zomorodian, A.: Fast construction of the Vietoris-Rips complex. Comput. Graph. 34, 263–271 (2010)CrossRefGoogle Scholar
  16. 16.
    Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33(2), 249–274 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Duarham UniversityDurhamUK

Personalised recommendations