Journal
AXIOMS
Volume 4, Issue 3, Pages 275-293Publisher
MDPI
DOI: 10.3390/axioms4030275
Keywords
graph spectra; kernel-based methods; graph embedding; graph clustering; differential geometry
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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.
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