PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

Geometric Characterizations of Graphs Using Heat Kernel Embeddings
Hewayda ElGhawalby and Edwin Hancock
In: IMA Conference on the Mathematics of Surfaces 2009, Sept 7-9, 2009, York, UK.

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.

EPrint Type:Conference or Workshop Item (Oral)
Project Keyword:Project Keyword UNSPECIFIED
Subjects:Learning/Statistics & Optimisation
Theory & Algorithms
ID Code:6867
Deposited By:Edwin Hancock
Deposited On:08 April 2010