This document provides the foundations behind the functionality provided by the @math G library (this https URL), focusing on the basic operations the library provides: subsumption, refinement of directed labeled graphs, and distance similarity assessment between directed labeled graphs. @math G development was initially supported by the National Science Foundation, by the EAGER grant IIS-1551338.
Another related area is that of kernels for structured representations, which allow the application of techniques such as Support Vector Machines to structured data. Typically, kernels for graphs are based on the idea of finding common substructures between two graphs. For example, @cite_37 present a kernel for graphs based on random walks. @cite_5 also studied how to encapsulate their similarity measure for Description Logics into a kernel. For a survey on kernels for structured data the reader is referred to @cite_31 .
@cite_5 @cite_37 @cite_31
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Positive definite kernels between labeled graphs have recently been proposed. They enable the application of kernel methods, such as support vector machines, to the analysis and classification of graphs, for example, chemical compounds. These graph kernels are obtained by marginalizing a kernel between paths with respect to a random walk model on the graph vertices along the edges. We propose two extensions of these graph kernels, with the double goal to reduce their computation time and increase their relevance as measure of similarity between graphs. First, we propose to modify the label of each vertex by automatically adding information about its environment with the use of the Morgan algorithm. Second, we suggest a modification of the random walk model to prevent the walk from coming back to a vertex that was just visited. These extensions are then tested on benchmark experiments of chemical compounds classification, with promising results. 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. Recently, kernel methods have become a popular tool for machine learning and data mining. As most ‘real-world’ data is structured, research in kernel methods has begun investigating kernels for various kinds of structured data. One of the most widely used tools for modeling structured data are graphs. In this paper we study the trade-off between expressivity and efficiency of graph kernels. First, we motivate the need for this discussion by showing that fully general graph kernels can not even be approximated efficiently. We also discuss generalizations of graph kernels defined in literature and show that they are either not positive definite or not very useful. Finally, we propose a new graph kernel based on subtree patterns. We argue that while a little more computationally expensive, this kernel is more expressive than kernels based on walks. In this paper, we present a hyper graph kernel computed using substructure isomorphism tests. Measuring the isomorphisms between hyper graphs straightforwardly tends to be elusive since a hyper graph may exhibit varying relational orders. We thus transform a hyper graph into a directed line graph. This not only accurately reflects the multiple relationships exhibited by the hyper graph but is also easier to manipulate isomorphism tests. To locate the isomorphisms between hyper graphs through their directed line graphs, we propose a new directed Weisfeiler-Lehman isomorphism test for directed graphs. The new isomorphism test precisely reflects the structure of the directed edges. By identifying the isomorphic substructures of directed graphs, the hyper graph kernel for a pair of hyper graphs is computed by counting the number of pair wise isomorphic substructures from their directed line graphs. We show that our kernel limits tottering that arises in the existing walk and sub tree based (hyper)graph kernels. Experiments on challenging (hyper)graph datasets demonstrate the effectiveness of our kernel.