Kernels for structured data are commonly obtained by decomposing objects into their parts and adding up the similarities between all pairs of parts measured by a base kernel. Assignment kernels are based on an optimal bijection between the parts and have proven to be an effective alternative to the established convolution kernels. We explore how the base kernel can be learned as part of the classification problem. We build on the theory of valid assignment kernels derived from hierarchies defined on the parts. We show that the weights of this hierarchy can be optimized via multiple kernel learning. We apply this result to learn vertex similarities for the Weisfeiler-Lehman optimal assignment kernel for graph classification. We present first experimental results which demonstrate the feasibility and effectiveness of the approach.
Current GNN architectures use a vertex neighborhood aggregation scheme, which limits their discriminative power to that of the 1-dimensional Weisfeiler-Lehman (WL) graph isomorphism test. Here, we propose a novel graph convolution operator that is based on the 2-dimensional WL test. We formally show that the resulting 2-WL-GNN architecture is more discriminative than existing GNN approaches. This theoretical result is complemented by experimental studies using synthetic and real data. On multiple common graph classification benchmarks, we demonstrate that the proposed model is competitive with…
Most state-of-the-art graph kernels only take local graph properties into account, i.e., the kernel is computed with regard to properties of the neighborhood of vertices or other small substructures. On the other hand, kernels that do take global graph propertiesinto account may not scale well to large graph databases. Here we propose to start exploring the space between local and global graph kernels, striking the balance between both worlds. Specifically, we introduce a novel graph kernel based on the $k$-dimensional Weisfeiler-Lehman algorithm. Unfortunately, the $k$-dimensional Weisfeiler-…
Preferred Networks, Inc.（Preferred Networks 公司）；The University of Tokyo（东京大学）
A graph neural network (GNN) is a good choice for predicting the chemical properties of molecules. Compared with other deep networks, however, the current performance of a GNN is limited owing to the "curse of depth." Inspired by long-established feature engineering in the field of chemistry, we expanded an atom representation using Weisfeiler-Lehman (WL) embedding, which is designed to capture local atomic patterns dominating the chemical properties of a molecule. In terms of representability, we show WL embedding can replace the first two layers of ReLU GNN -- a normal embedding and a hidden…
Graph neural networks are designed to learn functions on graphs. Typically, the relevant target functions are invariant with respect to actions by permutations. Therefore the design of some graph neural network architectures has been inspired by graph-isomorphism algorithms. The classical Weisfeiler-Lehman algorithm (WL) -- a graph-isomorphism test based on color refinement -- became relevant to the study of graph neural networks. The WL test can be generalized to a hierarchy of higher-order tests, known as $k$-WL. This hierarchy has been used to characterize the expressive power of graph neur…
Recent work shows that the expressive power of Graph Neural Networks (GNNs) in distinguishing non-isomorphic graphs is exactly the same as that of the Weisfeiler-Lehman (WL) graph test. In particular, they show that the WL test can be simulated by GNNs. However, those simulations involve neural networks for the 'combine' function of size polynomial or even exponential in the number of graph nodes $n$, as well as feature vectors of length linear in $n$. We present an improved simulation of the WL test on GNNs with \emph{exponentially} lower complexity. In particular, the neural network implemen…
Current neural architecture search (NAS) strategies focus only on finding a single, good, architecture. They offer little insight into why a specific network is performing well, or how we should modify the architecture if we want further improvements. We propose a Bayesian optimisation (BO) approach for NAS that combines the Weisfeiler-Lehman graph kernel with a Gaussian process surrogate. Our method optimises the architecture in a highly data-efficient manner: it is capable of capturing the topological structures of the architectures and is scalable to large graphs, thus making the high-dimen…
University of Bonn（波恩大学）；Fraunhofer IAIS（弗劳恩霍夫智能分析与信息系统研究所）
The Weisfeiler-Lehman graph kernels are among the most prevalent graph kernels due to their remarkable time complexity and predictive performance. Their key concept is based on an implicit comparison of neighborhood representing trees with respect to equality (i.e., isomorphism). This binary valued comparison is, however, arguably too rigid for defining suitable similarity measures over graphs. To overcome this limitation, we propose a generalization of Weisfeiler-Lehman graph kernels which takes into account the similarity between trees rather than equality. We achieve this using a specifical…
National Institute for Mathematical Sciences（国家数学科学研究所）；KAIST（韩国科学技术院）
This paper presents the Persistent Weisfeiler-Lehman Random walk scheme (abbreviated as PWLR) for graph representations, a novel mathematical framework which produces a collection of explainable low-dimensional representations of graphs with discrete and continuous node features. The proposed scheme effectively incorporates normalized Weisfeiler-Lehman procedure, random walks on graphs, and persistent homology. We thereby integrate three distinct properties of graphs, which are local topological features, node degrees, and global topological invariants, while preserving stability from graph pe…
In this paper we present a novel graph kernel framework inspired the by the Weisfeiler-Lehman (WL) isomorphism tests. Any WL test comprises a relabelling phase of the nodes based on test-specific information extracted from the graph, for example the set of neighbours of a node. We defined a novel relabelling and derived two kernels of the framework from it. The novel kernels are very fast to compute and achieve state-of-the-art results on five real-world datasets.
Computer Science and Artificial Intelligence Lab, MIT（麻省理工学院计算机科学与人工智能实验室）；Department of Chemical Engineering, MIT（麻省理工学院化学工程系）
The prediction of organic reaction outcomes is a fundamental problem in computational chemistry. Since a reaction may involve hundreds of atoms, fully exploring the space of possible transformations is intractable. The current solution utilizes reaction templates to limit the space, but it suffers from coverage and efficiency issues. In this paper, we propose a template-free approach to efficiently explore the space of product molecules by first pinpointing the reaction center -- the set of nodes and edges where graph edits occur. Since only a small number of atoms contribute to reaction cente…
ETH Zurich（苏黎世联邦理工学院）；SIB Swiss Institute of Bioinformatics（SIB瑞士生物信息学研究所）
Most graph kernels are an instance of the class of $\mathcal{R}$-Convolution kernels, which measure the similarity of objects by comparing their substructures. Despite their empirical success, most graph kernels use a naive aggregation of the final set of substructures, usually a sum or average, thereby potentially discarding valuable information about the distribution of individual components. Furthermore, only a limited instance of these approaches can be extended to continuously attributed graphs. We propose a novel method that relies on the Wasserstein distance between the node feature vec…
Chongqing University of Posts and Telecommunications（重庆邮电大学）；Beijing Institute of Technology（北京理工大学）；Hong Kong Baptist University（香港浸会大学）
Aligning users across networks using graph representation learning has been found effective where the alignment is accomplished in a low-dimensional embedding space. Yet, achieving highly precise alignment is still challenging, especially when nodes with long-range connectivity to the labeled anchors are encountered. To alleviate this limitation, we purposefully designed WL-Align which adopts a regularized representation learning framework to learn distinctive node representations. It extends the Weisfeiler-Lehman Isormorphism Test and learns the alignment in alternating phases of "across-netw…
Peking University（北京大学）；Purdue University（普渡大学）；Beijing Institute for General Artificial Intelligence（北京通用人工智能研究院）
Link prediction is one important application of graph neural networks (GNNs). Most existing GNNs for link prediction are based on one-dimensional Weisfeiler-Lehman (1-WL) test. 1-WL-GNNs first compute node representations by iteratively passing neighboring node features to the center, and then obtain link representations by aggregating the pairwise node representations. As pointed out by previous works, this two-step procedure results in low discriminating power, as 1-WL-GNNs by nature learn node-level representations instead of link-level. In this paper, we study a completely different approa…
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）
The expressive power of message passing GNNs is upper-bounded by Weisfeiler-Lehman (WL) test. To achieve high expressive GNNs beyond WL test, we propose a novel graph isomorphism test method, namely Twin-WL, which simultaneously passes node labels and node identities rather than only passes node label as WL. The identity-passing mechanism encodes complete structure information of rooted subgraph, and thus Twin-WL can offer extra power beyond WL at distinguishing graph structures. Based on Twin-WL, we implement two Twin-GNNs for graph classification via defining readout function over rooted sub…
University of California San Diego（加利福尼亚大学圣地亚哥分校）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）；The Ohio State University（俄亥俄州立大学）；Halıcıoğlu Data Science Institute（哈利奇奥卢数据科学研究所）
The Weisfeiler-Lehman (WL) test is a classical procedure for graph isomorphism testing. The WL test has also been widely used both for designing graph kernels and for analyzing graph neural networks. In this paper, we propose the Weisfeiler-Lehman (WL) distance, a notion of distance between labeled measure Markov chains (LMMCs), of which labeled graphs are special cases. The WL distance is polynomial time computable and is also compatible with the WL test in the sense that the former is positive if and only if the WL test can distinguish the two involved graphs. The WL distance captures and co…
In this paper, we propose a novel graph kernel specifically to address a challenging problem in the field of cyber-security, namely, malware detection. Previous research has revealed the following: (1) Graph representations of programs are ideally suited for malware detection as they are robust against several attacks, (2) Besides capturing topological neighbourhoods (i.e., structural information) from these graphs it is important to capture the context under which the neighbourhoods are reachable to accurately detect malicious neighbourhoods. We observe that state-of-the-art graph kernels, su…
POSTECH（浦项科技大学）；Center for Geometry and Physics, Institute for Basic Science (IBS)（基础科学研究院几何与物理中心）；Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）
University of Cambridge（剑桥大学）；Imperial College London（伦敦帝国理工学院）；Twitter（推特）；UCLA（加州大学洛杉矶分校）；MPI-MIS（马克斯·普朗克智能系统研究所）；SJTU（上海交通大学）；UNSW（新南威尔士大学）
Graph Neural Networks (GNNs) are limited in their expressive power, struggle with long-range interactions and lack a principled way to model higher-order structures. These problems can be attributed to the strong coupling between the computational graph and the input graph structure. The recently proposed Message Passing Simplicial Networks naturally decouple these elements by performing message passing on the clique complex of the graph. Nevertheless, these models can be severely constrained by the rigid combinatorial structure of Simplicial Complexes (SCs). In this work, we extend recent the…
HWL-HIN: A Hypergraph-Level Hypergraph Isomorphism Network as Powerful as the Hypergraph Weisfeiler-Lehman Test with Application to Higher-Order Network Robustness
The University of Tokyo（东京大学）；Kyoto University（京都大学）；Alato University（阿尔托大学）
Graph is an usual representation of relational data, which are ubiquitous in manydomains such as molecules, biological and social networks. A popular approach to learningwith graph structured data is to make use of graph kernels, which measure the similaritybetween graphs and are plugged into a kernel machine such as a support vector machine.Weisfeiler-Lehman (WL) based graph kernels, which employ WL labeling scheme to extract subtree patterns and perform node embedding, are demonstrated to achieve great performance while being efficiently computable. However, one of the main drawbacks of agen…
University of Cambridge（剑桥大学）；Twitter（推特）；Imperial College London（帝国理工学院）；University of New South Wales（新南威尔士大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）；Shanghai Jiao Tong University（上海交通大学）；University of California, Los Angeles（加州大学洛杉矶分校）
The pairwise interaction paradigm of graph machine learning has predominantly governed the modelling of relational systems. However, graphs alone cannot capture the multi-level interactions present in many complex systems and the expressive power of such schemes was proven to be limited. To overcome these limitations, we propose Message Passing Simplicial Networks (MPSNs), a class of models that perform message passing on simplicial complexes (SCs). To theoretically analyse the expressivity of our model we introduce a Simplicial Weisfeiler-Lehman (SWL) colouring procedure for distinguishing no…
Recently, subgraph GNNs have emerged as an important direction for developing expressive graph neural networks (GNNs). While numerous architectures have been proposed, so far there is still a limited understanding of how various design paradigms differ in terms of expressive power, nor is it clear what design principle achieves maximal expressiveness with minimal architectural complexity. To address these fundamental questions, this paper conducts a systematic study of general node-based subgraph GNNs through the lens of Subgraph Weisfeiler-Lehman Tests (SWL). Our central result is to build a …
Center for Machine Learning, Sankt Augustin（机器学习中心，圣奥古斯丁）
The majority of popular graph kernels is based on the concept of Haussler's $\mathcal{R}$-convolution kernel and defines graph similarities in terms of mutual substructures. In this work, we enrich these similarity measures by considering graph filtrations: Using meaningful orders on the set of edges, which allow to construct a sequence of nested graphs, we can consider a graph at multiple granularities. For one thing, this provides access to features on different levels of resolution. Furthermore, rather than to simply compare frequencies of features in graphs, it allows for their comparison …
Attacking Graph Neural Networks with Bit Flips: Weisfeiler and Lehman Go Indifferent
University of Vienna（维也纳大学）；Doctoral School Computer Science, Faculty of Computer Science, University of Vienna（维也纳大学计算机学院计算机科学博士学院）；Christian Doppler Laboratory for Assurance and Transparency in Software Protection, Faculty of Computer Science, University of Vienna（维也纳大学计算机学院克里斯蒂安·多普勒软件保护保障与透明度实验室）；Research Network Data Science, Faculty of Computer Science, University of Vienna（维也纳大学计算机学院数据科学研究网络）
该研究提出首个专门针对图神经网络的Injectivity Bit Flip Attack，通过攻击量化消息传递神经网络的邻域聚合函数削弱其结构区分能力，仅翻转少量比特就能让高表达力的图同构网络输出随机结果，破坏力优于迁移自CNN的位翻转攻击。
The success of kernel methods has initiated the design of novel positive semidefinite functions, in particular for structured data. A leading design paradigm for this is the convolution kernel, which decomposes structured objects into their parts and sums over all pairs of parts. Assignment kernels, in contrast, are obtained from an optimal bijection between parts, which can provide a more valid notion of similarity. In general however, optimal assignments yield indefinite functions, which complicates their use in kernel methods. We characterize a class of base kernels used to compare parts th…
The classical Weisfeiler-Leman algorithm aka color refinement is fundamental for graph learning with kernels and neural networks. Originally developed for graph isomorphism testing, the algorithm iteratively refines vertex colors. On many datasets, the stable coloring is reached after a few iterations and the optimal number of iterations for machine learning tasks is typically even lower. This suggests that the colors diverge too fast, defining a similarity that is too coarse. We generalize the concept of color refinement and propose a framework for gradual neighborhood refinement, which allow…
Beijing Normal University（北京师范大学）；Central University of Finance and Economics（中央财经大学）；University of York（约克大学）
In this paper, we propose a novel graph kernel, namely the Quantum-based Entropic Subtree Kernel (QESK), for Graph Classification. To this end, we commence by computing the Average Mixing Matrix (AMM) of the Continuous-time Quantum Walk (CTQW) evolved on each graph structure. Moreover, we show how this AMM matrix can be employed to compute a series of entropic subtree representations associated with the classical Weisfeiler-Lehman (WL) algorithm. For a pair of graphs, the QESK kernel is defined by computing the exponentiation of the negative Euclidean distance between their entropic subtree re…
In this paper, we present subgraph2vec, a novel approach for learning latent representations of rooted subgraphs from large graphs inspired by recent advancements in Deep Learning and Graph Kernels. These latent representations encode semantic substructure dependencies in a continuous vector space, which is easily exploited by statistical models for tasks such as graph classification, clustering, link prediction and community detection. subgraph2vec leverages on local information obtained from neighbourhoods of nodes to learn their latent representations in an unsupervised fashion. We demonstr…
Indian Institute of Space Science and Technology（印度空间科学技术研究院）；Litmus7 Systems Consulting Pvt Ltd（Litmus7系统咨询私人有限公司）
We describe the design of a reproducing kernel suitable for attributed graphs, in which the similarity between the two graphs is defined based on the neighborhood information of the graph nodes with the aid of a product graph formulation. We represent the proposed kernel as the weighted sum of two other kernels of which one is an R-convolution kernel that processes the attribute information of the graph and the other is an optimal assignment kernel that processes label information. They are formulated in such a way that the edges processed as part of the kernel computation have the same neighb…
Indian Institute of Technology Madras（印度理工学院马德拉斯分校）；University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）；Ohio State University（俄亥俄州立大学）
Given a graph where every node has certain attributes associated with it and some nodes have labels associated with them, Collective Classification (CC) is the task of assigning labels to every unlabeled node using information from the node as well as its neighbors. It is often the case that a node is not only influenced by its immediate neighbors but also by higher order neighbors, multiple hops away. Recent state-of-the-art models for CC learn end-to-end differentiable variations of Weisfeiler-Lehman (WL) kernels to aggregate multi-hop neighborhood information. In this work, we propose a Hig…
State Key Lab of Software Development Environment（软件开发环境国家重点实验室）；Beihang University（北京航空航天大学）
In deep neural networks, better results can often be obtained by increasing the complexity of previously developed basic models. However, it is unclear whether there is a way to boost performance by decreasing the complexity of such models. Here, based on an optimization method, we investigate the feasibility of improving graph classification performance while simplifying the model learning process. Inspired by progress in structural information assessment, we optimize the given data sample from graphs to encoding trees. In particular, we minimize the structural entropy of the transformed enco…
Can neural networks learn to compare graphs without feature engineering? In this paper, we show that it is possible to learn representations for graph similarity with neither domain knowledge nor supervision (i.e.\ feature engineering or labeled graphs). We propose Deep Divergence Graph Kernels, an unsupervised method for learning representations over graphs that encodes a relaxed notion of graph isomorphism. Our method consists of three parts. First, we learn an encoder for each anchor graph to capture its structure. Second, for each pair of graphs, we train a cross-graph attention network wh…
