Heat kernels, manifolds and graph embedding

In this paper, we investigate the use of heat kernels as a means of embedding graphs in a pattern space. We commence by performing the spectral decomposition on the graph Laplacian. The heat kernel of the graph is found by exponentiating the resulting eigensystem over time. By equating the spectral heat kernel and its Gaussian form we are able to approximate the geodesic distance between nodes on a manifold. We use the resulting pattern of distances to embed the trees in a Euclidean space using multidimensional scaling. The arrangement of points in this space can be used to construct pattern vectors suitable for clustering the graphs. Here we compute a weighted proximity matrix, and from the proximity matrix a Laplacian matrix is computed. We use the eigenvalues of the Laplacian matrix to characterise the distribution of points representing the embedded nodes. Experiments on sets of shock graphs reveal the utility of the method on real-world data.