Heat Kernel Embeddings, Differential Geometry and Graph Structure

Edwin Hancock,Hewayda Elghawalby

doi:10.3390/axioms4030275

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.

Highlights

Kernel embeddings allow similarity data to be embedded into a vector space using an inner product characterisation of the distance or dissimilarity between patterns
We have investigated whether we can use the heat kernel to provide a geometric characterisation of graphs that can be used for the purposes of graph matching and clustering
Performing a Young–Householder decomposition on the heat kernel maps the nodes of the graph to points in the manifold providing a matrix of embedding coordinates

Summary

Introduction

Kernel embeddings allow similarity data to be embedded into a vector space using an inner product characterisation of the distance or dissimilarity between patterns. They have found widespread use in pattern recognition, machine learning and data mining. The heat kernel is an important analytical tool in physics, where it can be used to model diffusions on discrete structures Because it is determined by the Laplacian matrix, it has been the subject of intense study in spectral graph theory [2] and spectral geometry [3,4]. If the nodes of a graph are viewed as residing on a manifold, the Laplacian matrix may be regarded as the discrete approximation to the Laplace–Beltrami curvature operator for the manifold

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