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Geometric Characterizations of Graphs Using Heat Kernel Embeddings

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Mathematics of Surfaces XIII (Mathematics of Surfaces 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5654))

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Abstract

In this paper, we investigate the heat kernel embedding as a route to computing geometric characterisations of graphs. The reason for turning to the heat kernel is that it encapsulates information concerning the distribution of path lengths and hence node affinities on the graph. The heat kernel of the graph is found by exponentiating the Laplacian eigensystem over time. The matrix of embedding co-ordinates for the nodes of the graph is obtained by performing a Young-Householder decomposition on the heat kernel. Once the embedding of its nodes is to hand we proceed to characterise a graph in a geometric manner. To obtain this characterisation, we focus on the edges of the graph under the embedding. Here we use the difference between geodesic and Euclidean distances between nodes to associate a sectional curvature with edges. Once the section curvatures are to hand then the Gauss-Bonnet theorem allows us to compute Gaussian curvatures at nodes on the graph. We explore how the attributes furnished by this analysis can be used to match and cluster graphs.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of York, UK

    H. El-Ghawalby & E. R. Hancock

  2. Faculty of Engineering, Suez Canal University, Egypt

    H. El-Ghawalby

Authors
  1. H. El-Ghawalby
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  2. E. R. Hancock
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Editor information

Editors and Affiliations

  1. Computer Vision and Pattern Recognition Group,Computer Science, University of York Heslington, YO10-5DD, York, United Kingdom

    Edwin R. Hancock

  2. School of Computer Science, Cardiff University, Cardiff, UK

    Ralph R. Martin

  3. Numerical Geometry Ltd, 26 Abbey Lane, Lode, CB5 9EP, Cambridge, UK

    Malcolm A. Sabin

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El-Ghawalby, H., Hancock, E.R. (2009). Geometric Characterizations of Graphs Using Heat Kernel Embeddings. In: Hancock, E.R., Martin, R.R., Sabin, M.A. (eds) Mathematics of Surfaces XIII. Mathematics of Surfaces 2009. Lecture Notes in Computer Science, vol 5654. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03596-8_8

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  • DOI: https://doi.org/10.1007/978-3-642-03596-8_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03595-1

  • Online ISBN: 978-3-642-03596-8

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