A commonly used paradigm for representing graphs is to use a vector that contains normalized frequencies of occurrence of certain motifs or sub-graphs. This vector representation can be used in a variety of applications, such as, for computing similarity between graphs. The graphlet kernel of [32] uses induced sub-graphs of k nodes (christened as graphlets by Przulj [28]) as motifs in the vector representation, and computes the kernel via a dot product between these vectors. One can easily show that this is a valid kernel between graphs. However, such a vector representation suffers from a few drawbacks. As k becomes larger we encounter the sparsity problem; most higher order graphlets will not occur in a given graph. This leads to diagonal dominance, that is, a given graph is similar to itself but not to any other graph in the dataset. On the other hand, since lower order graphlets tend to be more numerous, using lower values of k does not provide enough discrimination ability. We propose a smoothing technique to tackle the above problems. Our method is based on a novel extension of Kneser-Ney and Pitman-Yor smoothing techniques from natural language processing to graphs. We use the relationships between lower order and higher order graphlets in order to derive our method. Consequently, our smoothing algorithm not only respects the dependency between sub-graphs but also tackles the diagonal dominance problem by distributing the probability mass across graphlets. In our experiments, the smoothed graphlet kernel outperforms graph kernels based on raw frequency counts.
Graph kernels can be considered as special cases of convolutional kernels proposed by . In general, graph kernels can be categorized into three classes: graph kernels based on walks and paths @cite_4 , @cite_6 , @cite_24 , graph kernels based on limited-size sub-graphs @cite_7 , @cite_8 , @cite_22 and graph kernels based on subtree patterns @cite_5 . performs a relaxation on the vertices and exploit labeling information embedded in the graphs to derive their so-called Weisfeiler-Lehman kernels. However, their kernel is applicable only to labeled graphs. The sparsity problem of graphlet kernels has been addressed before. Hash kernels proposed by addresses the sparsity problem by applying a sparse projection into a lower dimensional space. The idea here is that many higher order graphlets will collide'' and therefore be mapped to the same lower dimensional representation, thus avoiding the diagonal dominance problem. Unfortunately, we find that in our experiments the hash kernel is very sensitive to the hash value used for embedding and rarely performed well as compared to the MLE estimate.
@cite_4 @cite_22 @cite_7 @cite_8 @cite_6 @cite_24 @cite_5
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We introduce a novel kernel that upgrades the Weisfeiler-Lehman and other graph kernels to effectively exploit high-dimensional and continuous vertex attributes. Graphs are first decomposed into subgraphs. Vertices of the subgraphs are then compared by a kernel that combines the similarity of their labels and the similarity of their structural role, using a suitable vertex invariant. By changing this invariant we obtain a family of graph kernels which includes generalizations of Weisfeiler-Lehman, NSPDK, and propagation kernels. We demonstrate empirically that these kernels obtain state-of-the-art results on relational data sets. In this paper, we present Deep Graph Kernels, a unified framework to learn latent representations of sub-structures for graphs, inspired by latest advancements in language modeling and deep learning. Our framework leverages the dependency information between sub-structures by learning their latent representations. We demonstrate instances of our framework on three popular graph kernels, namely Graphlet kernels, Weisfeiler-Lehman subtree kernels, and Shortest-Path graph kernels. Our experiments on several benchmark datasets show that Deep Graph Kernels achieve significant improvements in classification accuracy over state-of-the-art graph kernels. Learning from complex data is becoming increasingly important, and graph kernels have recently evolved into a rapidly developing branch of learning on structured data. However, previously proposed kernels rely on having discrete node label information. In this paper, we explore the power of continuous node-level features for propagation-based graph kernels. Specifically, propagation kernels exploit node label distributions from propagation schemes like label propagation, which naturally enables the construction of graph kernels for partially labeled graphs. In order to efficiently extract graph features from continuous node label distributions, and in general from continuous vector-valued node attributes, we utilize randomized techniques, which easily allow for deriving similarity measures based on propagated information. We show that propagation kernels utilizing locality-sensitive hashing reduce the runtime of existing graph kernels by several orders of magnitude. We evaluate the performance of various propagation kernels on real-world bioinformatics and image benchmark datasets. This paper presents a survey as well as an empirical comparison and evaluation of seven kernels on graphs and two related similarity matrices, that we globally refer to as ''kernels on graphs'' for simplicity. They are the exponential diffusion kernel, the Laplacian exponential diffusion kernel, the von Neumann diffusion kernel, the regularized Laplacian kernel, the commute-time (or resistance-distance) kernel, the random-walk-with-restart similarity matrix, and finally, a kernel first introduced in this paper (the regularized commute-time kernel) and two kernels defined in some of our previous work and further investigated in this paper (the Markov diffusion kernel and the relative-entropy diffusion matrix). The kernel-on-graphs approach is simple and intuitive. It is illustrated by applying the nine kernels to a collaborative-recommendation task, viewed as a link prediction problem, and to a semisupervised classification task, both on several databases. The methods compute proximity measures between nodes that help study the structure of the graph. Our comparisons suggest that the regularized commute-time and the Markov diffusion kernels perform best on the investigated tasks, closely followed by the regularized Laplacian kernel.