Characterising Graphs using the Heat Kernel

The heat-kernel of a graph is computed by exponentiating the Laplacian eigen-system with time. In this paper, we study the heat kernel mapping of the nodes of a graph into a vector-space. Specifically, we investigate whether the resulting point distribution can be used for the purposes of graphclustering. Our characterisation is based on the covariance matrix of the point distribution. We explore the relationship between the covariance matrix and the heat kernel, and demonstrate the eigenvalues of the covariance matrix are found be exponentiating the Laplacian eigenvalues with time. We apply the technique to images from the COIL database, and demonstrate that it leads to well defined graph clusters.