Technically,various methods of both categories exploit the link between graph data and linear algebra by representing graphs by their (normalized) adjacency matrix. Such methods are often defined or can be interpreted in terms ofwalks. On the other hand, the Weisfeiler-Leman heuristic for graph isomorphism testing has attracted great interest in machine learning [33, 34].

The isomorphism problem is a fundamental problem in network analysis, which involves capturing both low-order and high-order structural information. In terms of extracting low-order structural information, graph isomorphism algorithms analyze the structural equivalence to reduce the solver space dimension, which demonstrates its power in many applications, such as protein design, chemical pathways, and community detection. For the more commonly occurring high-order relationships in real-life scenarios, the problem of hypergraph isomorphism, which effectively captures these high-order structural relationships, cannot be straightforwardly addressed using graph isomorphism methods. Besides, the existing hypergraph kernel methods may suffer from high memory consumption or inaccurate sub-structure identification, thus yielding sub-optimal performance. In this paper, to address the abovementioned problems, we first propose the hypergraph Weisfiler-Lehman test algorithm for the hypergraph isomorphism test problem by generalizing the Weisfiler-Lehman test algorithm from graphs to hypergraphs. Secondly, based on the presented algorithm, we propose a general hypergraph Weisfieler-Lehman kernel framework and implement two instances, which are Hypergraph Weisfeiler-Lehamn Subtree Kernel and Hypergraph Weisfeiler-Lehamn Hyperedge Kernel. In order to fulfill our research objectives, a comprehensive set of experiments was meticulously designed, including seven graph classification datasets and 12 hypergraph classification datasets. Results on hypergraph classification datasets show significant improvements compared to other typical kernel-based methods, which demonstrates the effectiveness of the proposed methods. In our evaluation, we found that our proposed methods outperform the second-best method in terms of runtime, running over 80 times faster when handling complex hypergraph structures.

In this article, we propose fast subtree kernels on graphs. On graphs with n nodes and m edges and maximum degree d, these kernels comparing subtrees of height h can be computed in O(mh), whereas the classic subtree kernel by Ramon & G artner scales as O(n24dh). Key to this efficiency is the observation that the Weisfeiler-Lehman test of isomorphism from graph theory elegantly computes a subtree kernel as a byproduct. Our fast subtree kernels can deal with labeled graphs, scale up easily to large graphs and outperform state-of-the-art graph ker- nels on several classification benchmark datasets in terms of accuracy and runtime.

Graph kernels have attracted a lot of attention during the last decade, and have evolved into a rapidly developing branch of learning on structured data. During the past 20 years, the considerable research activity that occurred in the field resulted in the development of dozens of graph kernels, each focusing on specific structural properties of graphs. Graph kernels have proven successful in a wide range of domains, ranging from social networks to bioinformatics. The goal of this survey is to provide a unifying view of the literature on graph kernels. In particular, we present a comprehensive overview of a wide range of graph kernels. Furthermore, we perform an experimental evaluation of several of those kernels on publicly available datasets, and provide a comparative study. Finally, we discuss key applications of graph kernels, and outline some challenges that remain to be addressed.

Graph kernels have attracted a lot of attention during the last decade, and have evolved into a rapidly developing branch of learning on structured data. During the past 20 years, the considerable research activity that occurred in the field resulted in the development of dozens of graph kernels, each focusing on specific structural properties of graphs. Graph kernels have proven successful in a wide range of domains, ranging from social networks to bioinformatics. The goal of this survey is to provide a unifying view of the literature on graph kernels. In particular, we present a comprehensive overview of a wide range of graph kernels. Furthermore, we perform an experimental evaluation of several of those kernels on publicly available datasets, and provide a comparative study. Finally, we discuss key applications of graph kernels, and outline some challenges that remain to be addressed.

Tree data are ubiquitous because they model a large variety of situations, e.g., the architecture of plants, the secondary structure of RNA, or the hierarchy of XML files. Nevertheless, the analysis of these non-Euclidean data is difficul per se. In this paper, we focus on the subtree kernel that is a convolution kernel for tree data introduced by Vishwanathan and Smola in the early 2000's. More precisely, we investigate the influence of the weight function from a theoretical perspective and in real data applications. We establish on a 2-classes stochastic model that the performance of the subtree kernel is improved when the weight of leaves vanishes, which motivates the definition of a new weight function, learned from the data and not fixed by the user as usually done. To this end, we define a unified framework for computing the subtree kernel from ordered or unordered trees, that is particularly suitable for tuning parameters. We show through two real data classification problems the great efficiency of our approach, in particular with respect to the ones considered in the literature, which also states the high importance of the weight function. Finally, a visualization tool of the significant features is derived.

Neural networks are typically designed to deal with data in tensor forms. In this paper, we propose a novel neural network architecture accepting graphs of arbitrary structure. Given a dataset containing graphs in the form of (G,y) where G is a graph and y is its class, we aim to develop neural networks that read the graphs directly and learn a classification function. There are two main challenges: 1) how to extract useful features characterizing the rich information encoded in a graph for classification purpose, and 2) how to sequentially read a graph in a meaningful and consistent order. To address the first challenge, we design a localized graph convolution model and show its connection with two graph kernels. To address the second challenge, we design a novel SortPooling layer which sorts graph vertices in a consistent order so that traditional neural networks can be trained on the graphs. Experiments on benchmark graph classification datasets demonstrate that the proposed architecture achieves highly competitive performance with state-of-the-art graph kernels and other graph neural network methods. Moreover, the architecture allows end-to-end gradient-based training with original graphs, without the need to first transform graphs into vectors.

While state-of-the-art kernels for graphs with discrete labels scale well to graphs with thousands of nodes, the few existing kernels for graphs with continuous attributes, unfortunately, do not scale well. To overcome this limitation, we present hash graph kernels, a general framework to derive kernels for graphs with continuous attributes from discrete ones. The idea is to iteratively turn continuous attributes into discrete labels using randomized hash functions. We illustrate hash graph kernels for the Weisfeiler-Lehman subtree kernel and for the shortest-path kernel. The resulting novel graph kernels are shown to be, both, able to handle graphs with continuous attributes and scalable to large graphs and data sets. This is supported by our theoretical analysis and demonstrated by an extensive experimental evaluation.

In this article, we propose fast subtree kernels on graphs.

In this article, we propose fast subtree kernels on graphs.