Geometric Characterisation of Graphs

  • Bai Xiao
  • Edwin R. Hancock
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3617)


In this paper, we explore whether the geometric properties of the point distribution obtained by embedding the nodes of a graph on a manifold can be used for the purposes of graph clustering. The embedding is performed using the heat-kernel of the graph, computed by exponentiating the Laplacian eigen-system. By equating the spectral heat kernel and its Gaussian form we are able to approximate the Euclidean distance between nodes on the manifold. The difference between the geodesic and Euclidean distances can be used to compute the sectional curvatures associated with the edges of the graph. To characterise the manifold on which the graph resides, we use the normalised histogram of sectional curvatures. By performing PCA on long-vectors representing the histogram bin-contents, we construct a pattern space for sets of graphs. We apply the technique to images from the COIL database, and demonstrate that it leads to well defined graph clusters.


Sectional Curvature Heat Kernel Spectral Cluster Geodesic Distance Laplacian Matrix 
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.


  1. 1.
    Alexandrov, A.D., Zalgaller, V.A.: Intrinsic geometry of surfaces. Transl. Math. Monographs 15 (1967)Google Scholar
  2. 2.
    Ranicki, A.: Algebraic l-theory and topological manifolds. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  3. 3.
    Wilson, R.C., Luo, B., Hancock, E.R.: Spectral embedding of graphs. Pattern Recognition 36, 2213–2223 (2003)zbMATHCrossRefGoogle Scholar
  4. 4.
    Chung, F.R.K.: Spectral graph theory. American Mathematical Society (1997)Google Scholar
  5. 5.
    Gilkey, P.B.: Invariance theory, the heat equation, and the atiyah-singer index theorem. Publish or Perish Inc. (1984)Google Scholar
  6. 6.
    Harris, C.G., Stephens, M.J.: A combined corner and edge detector. In: Fourth Alvey Vision Conference, pp. 147–151 (1994)Google Scholar
  7. 7.
    Sachs, H., Cvetkovic, D.M., Doob, M.: Spectra of graphs. Academic Press, London (1980)Google Scholar
  8. 8.
    Tenenbaum, J.B., Silva, V.D., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 586–591 (2000)CrossRefGoogle Scholar
  9. 9.
    Luo, B., Wilson, R.C., Hancock, E.R.: Pattern vectors from algebraic graph theory. IEEE Transactions on Pattern Analysis and Machine Intelligence 27 (2005) (to appear)Google Scholar
  10. 10.
    Shokoufandeh, A., Dickinson, S., Siddiqi, K., Zucker, S.: Indexing using a spectral encoding of topological structure. In: CVPR, pp. 491–497 (1999)Google Scholar
  11. 11.
    Rosenberg, S.: The laplacian on a Riemannian manifold. Cambridge University Press, Cambridge (2002)Google Scholar
  12. 12.
    Yau, S.T., Schoen, R.M.: Differential geometry. Science Publication (1988)Google Scholar
  13. 13.
    Bai, X., Hancock, E.R.: Heat kernels, manifolds and graph embedding. Structural, Syntactic, and Statistical Pattern Recognition, 198–206 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Bai Xiao
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceUniversity of YorkYorkUK

Personalised recommendations