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Axioms 2015, 4(3), 275-293; doi:10.3390/axioms4030275

Heat Kernel Embeddings, Differential Geometry and Graph Structure

1
Faculty of Engineering, Port-Said University, Port Said 42526, Egypt
2
Department of Computer Science, University of York, York YO10 5GH, UK
*
Author to whom correspondence should be addressed.
Academic Editor: Angel Garrido
Received: 24 April 2015 / Revised: 25 June 2015 / Accepted: 2 July 2015 / Published: 21 July 2015
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Abstract

In this paper, we investigate the heat kernel embedding as a route to graph representation. The heat kernel of the graph encapsulates information concerning the distribution of path lengths and, hence, node affinities on the graph; and is found by exponentiating the Laplacian eigen-system over time. A Young–Householder decomposition is performed on the heat kernel to obtain the matrix of the embedded coordinates for the nodes of the graph. With the embeddings at hand, we establish a graph characterization based on differential geometry by computing sets of curvatures associated with the graph edges and triangular faces. A sectional curvature computed from the difference between geodesic and Euclidean distances between nodes is associated with the edges of the graph. Furthermore, we use the Gauss–Bonnet theorem to compute the Gaussian curvatures associated with triangular faces of the graph. View Full-Text
Keywords: graph spectra; kernel-based methods; graph embedding; graph clustering; differential geometry graph spectra; kernel-based methods; graph embedding; graph clustering; differential geometry
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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ElGhawalby, H.; Hancock, E.R. Heat Kernel Embeddings, Differential Geometry and Graph Structure. Axioms 2015, 4, 275-293.

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