Geometric characterization and clustering of graphs using heat kernel embeddings Bai Xiao a, * , Edwin R. Hancock b , Richard C. Wilson b a Intelligence Recognition and Image Processing Lab, School of Computer Science and Engineering, Beihang University, 37 Xueyuan Road, Beijing 100191, China b Department of Computer Science, University of York, York Y010 5DD, UK article info Article history: Received 22 August 2007 Received in revised form 19 May 2009 Accepted 20 May 2009 Keywords: Graph spectra Kernel methods Graph embedding Differential geometry Graph clustering abstract In this paper, we investigate the use of heat kernels as a means of embedding the individual nodes of a graph in a vector space. The reason for turning to the heat kernel is that it encapsulates information con- cerning 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. In this paper, we explore how graphs can be characterized in a geometric manner using embeddings into a vector space obtained from the heat kernel. We explore two different embedding strategies. The first of these is a direct method in which the matrix of embedding co-ordinates is obtained by performing a Young–Householder decompo- sition on the heat kernel. The second method is indirect and involves performing a low-distortion embed- ding by applying multidimensional scaling to the geodesic distances between nodes. We show how the required geodesic distances can be computed using parametrix expansion of the heat kernel. Once the nodes of the graph are embedded using one of the two alternative methods, we can characterize them in a geometric manner using the distribution of the node co-ordinates. We investigate several alternative methods of characterization, including spatial moments for the embedded points, the Laplacian spectrum for the Euclidean distance matrix and scalar curvatures computed from the difference in geodesic and Euclidean distances. We experiment with the resulting algorithms on the COIL database. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction 1.1. Related literature One of the problems that arises in the manipulation of large amounts of graph data is that of characterizing the topological structure of individual graphs. Perhaps the most elegant way of achieving this is to use the spectrum of the Laplacian matrix [1,2]. For instance Shokoufandeh et al. [3] have used topological spectra to index tree structures, Luo et al. [4] have used the spec- trum of the adjacency matrix to construct pattern spaces for graphs, and Wilson and Hancock [5] have used algebraic graph the- ory to construct permutation invariant polynomials from the eigenvectors of the Laplacian matrix. One way of viewing these methods is that of constructing a low-dimensional feature space that captures the metric structure of the graphs under study. An interesting alternative to using topological information to characterize graphs is to embed the nodes of a graph in a vector space and to study the properties of the point distribution that re- sults from the embedding. This is a problem that arises in a num- ber of areas including manifold learning theory and graph drawing. In the mathematics literature, there is a considerable body of work aimed at understanding how graphs can be embedded on mani- folds [6]. In the pattern analysis community, there has recently been renewed interest in the use of embedding methods motivated by graph theory. One of the best known of these is ISOMAP [7]. Here a neighborhood ball is used to convert data points into a graph, and Dijkstra’s algorithm is used to compute the short- est(geodesic) distances between nodes. By applying multidimen- sional scaling (MDS) to the matrix of geodesic distances the manifold is reconstructed. Each graph is embedded into a vector space. Under the embedding each node becomes a point, and the graph can be characterized by the distribution of points (i.e. nodes) in the space. Embedding algorithms have been demonstrated to locate well formed manifolds for a number of complex data-sets. Related algorithms include locally linear embedding which is a variant of PCA that restricts the complexity of the input data using a nearest neighbor graph [8], and the Laplacian eigenmap that con- structs an adjacency weight matrix for the data points and projects the data onto the principal eigenvectors of the associated Laplacian matrix (the degree matrix minus the weight matrix) [9]. Collec- tively, these methods are sometimes referred to as manifold learn- ing theory. The spectrum of the Laplacian matrix has been widely studied in spectral graph theory [2] and has proved to be a versatile mathemat- ical tool that can be put to many practical uses including routing [10], indexing [3], clustering [11] and graph matching [12,13]. 0262-8856/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.imavis.2009.05.011 * Corresponding author. Tel.: +86 10 82314681. E-mail addresses: baixiao.buaa@googlemail.com (B. Xiao), erh@cs.york.ac.uk (E.R. Hancock), wilson@cs.york.ac.uk (R.C. Wilson). Image and Vision Computing 28 (2010) 1003–1021 Contents lists available at ScienceDirect Image and Vision Computing journal homepage: www.elsevier.com/locate/imavis