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 complete hierarchy of SWL with strictly growing expressivity. Concretely, we prove that any node-based subgraph GNN falls into one of the six SWL equivalence classes, among which $\mathsf{SSWL}$ achieves the maximal expressive power. We also study how these equivalence classes differ in terms of their practical expressiveness such as encoding graph distance and biconnectivity. Furthermore, we give a tight expressivity upper bound of all SWL algorithms by establishing a close relation with localized versions of WL and Folklore WL (FWL) tests. Our results provide insights into the power of existing subgraph GNNs, guide the design of new architectures, and point out their limitations by revealing an inherent gap with the 2-FWL test. Finally, experiments demonstrate that $\mathsf{SSWL}$-inspired subgraph GNNs can significantly outperform prior architectures on multiple benchmarks despite great simplicity.
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…
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…
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…
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…
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…
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）
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…
HWL-HIN: A Hypergraph-Level Hypergraph Isomorphism Network as Powerful as the Hypergraph Weisfeiler-Lehman Test with Application to Higher-Order Network Robustness
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…
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。
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…
In recent years, Graph Neural Network (GNN) has bloomly progressed for its power in processing graph-based data. Most GNNs follow a message passing scheme, and their expressive power is mathematically limited by the discriminative ability of the Weisfeiler-Lehman (WL) test. Following Tinhofer's research on compact graphs, we propose a variation of the message passing scheme, called the Weisfeiler-Lehman-Tinhofer GNN (WLT-GNN), that theoretically breaks through the limitation of the WL test. In addition, we conduct comparative experiments and ablation studies on several well-known datasets. The…
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…
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…
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 lear…
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…
TU Dortmund University（多特蒙德工业大学）；RWTH Aachen University（亚琛工业大学）；McGill University（麦吉尔大学）；MILA（MILA（蒙特利尔学习算法研究所））
In recent years, graph neural networks (GNNs) have emerged as a powerful neural architecture to learn vector representations of nodes and graphs in a supervised, end-to-end fashion. Up to now, GNNs have only been evaluated empirically -- showing promising results. The following work investigates GNNs from a theoretical point of view and relates them to the $1$-dimensional Weisfeiler-Leman graph isomorphism heuristic ($1$-WL). We show that GNNs have the same expressiveness as the $1$-WL in terms of distinguishing non-isomorphic (sub-)graphs. Hence, both algorithms also have the same shortcoming…
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…
Institute for Mathematical and Computational Engineering, PUC Chile（智利天主教大学数学与计算工程研究所）；IMFD Chile（智利数据基础研究所）；CENIA Chile（智利国家人工智能中心）；Mila Quebec AI Institute（魁北克人工智能研究所Mila）；McGill University（麦吉尔大学）；RWTH Aachen University（亚琛工业大学）；Universidad Adolfo Ibáñez（阿道夫·伊瓦涅斯大学）
Knowledge graphs, modeling multi-relational data, improve numerous applications such as question answering or graph logical reasoning. Many graph neural networks for such data emerged recently, often outperforming shallow architectures. However, the design of such multi-relational graph neural networks is ad-hoc, driven mainly by intuition and empirical insights. Up to now, their expressivity, their relation to each other, and their (practical) learning performance is poorly understood. Here, we initiate the study of deriving a more principled understanding of multi-relational graph neural net…
Graph neural networks (GNNs) are deep learning architectures for machine learning problems on graphs. It has recently been shown that the expressiveness of GNNs can be characterised precisely by the combinatorial Weisfeiler-Leman algorithms and by finite variable counting logics. The correspondence has even led to new, higher-order GNNs corresponding to the WL algorithm in higher dimensions. The purpose of this paper is to explain these descriptive characterisations of GNNs.
This work generalizes graph neural networks (GNNs) beyond those based on the Weisfeiler-Lehman (WL) algorithm, graph Laplacians, and diffusions. Our approach, denoted Relational Pooling (RP), draws from the theory of finite partial exchangeability to provide a framework with maximal representation power for graphs. RP can work with existing graph representation models and, somewhat counterintuitively, can make them even more powerful than the original WL isomorphism test. Additionally, RP allows architectures like Recurrent Neural Networks and Convolutional Neural Networks to be used in a theo…
University of Antwerp（安特卫普大学）；Pontificia Universidad Católica de Chile（智利天主教大学）；IMFD（智利数学与计算机科学研究所）
Characterizing the separation power of graph neural networks (GNNs) provides an understanding of their limitations for graph learning tasks. Results regarding separation power are, however, usually geared at specific GNN architectures, and tools for understanding arbitrary GNN architectures are generally lacking. We provide an elegant way to easily obtain bounds on the separation power of GNNs in terms of the Weisfeiler-Leman (WL) tests, which have become the yardstick to measure the separation power of GNNs. The crux is to view GNNs as expressions in a procedural tensor language describing th…
ETH Zurich（苏黎世联邦理工学院）；SIB Swiss Institute of Bioinformatics（瑞士生物信息学研究所）；KU Leuven（鲁汶大学）；Helmholtz Munich（亥姆霍兹慕尼黑中心）；Technical University of Munich（慕尼黑工业大学）
Graph neural networks (GNNs) are a powerful architecture for tackling graph learning tasks, yet have been shown to be oblivious to eminent substructures such as cycles. We present TOGL, a novel layer that incorporates global topological information of a graph using persistent homology. TOGL can be easily integrated into any type of GNN and is strictly more expressive (in terms the Weisfeiler--Lehman graph isomorphism test) than message-passing GNNs. Augmenting GNNs with TOGL leads to improved predictive performance for graph and node classification tasks, both on synthetic data sets, which can…
The message-passing scheme is the core of graph representation learning. While most existing message-passing graph neural networks (MPNNs) are permutation-invariant in graph-level representation learning and permutation-equivariant in node- and edge-level representation learning, their expressive power is commonly limited by the 1-Weisfeiler-Lehman (1-WL) graph isomorphism test. Recently proposed expressive graph neural networks (GNNs) with specially designed complex message-passing mechanisms are not practical. To bridge the gap, we propose a plug-in Equivariant Distance ENcoding (EDEN) for M…
Tsinghua University（清华大学）；Johns Hopkins University（约翰斯·霍普金斯大学）；Microsoft Research Asia（微软亚洲研究院）
The invariance to permutations of the adjacency matrix, i.e., graph isomorphism, is an overarching requirement for Graph Neural Networks (GNNs). Conventionally, this prerequisite can be satisfied by the invariant operations over node permutations when aggregating messages. However, such an invariant manner may ignore the relationships among neighboring nodes, thereby hindering the expressivity of GNNs. In this work, we devise an efficient permutation-sensitive aggregation mechanism via permutation groups, capturing pairwise correlations between neighboring nodes. We prove that our approach is …
On the Computational Capability of Graph Neural Networks: A Circuit Complexity Bound Perspective
The University of Hong Kong（香港大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；The Simons Institute for the Theory of Computing at UC Berkeley（加州大学伯克利分校西蒙斯理论计算研究所）；UC Berkeley（加州大学伯克利分校）
Operational Research Center（运筹学研究中心）；Massachusetts Institute of Technology（麻省理工学院）；Institute for Applied Computational Science（应用计算科学研究所）；Harvard University（哈佛大学）；Harvard School of Engineering and Applied Sciences（哈佛工程与应用科学学院）
Graph neural networks (GNNs) are a powerful tool to learn representations on graphs by iteratively aggregating features from node neighbourhoods. Many variant models have been proposed, but there is limited understanding on both how to compare different architectures and how to construct GNNs systematically. Here, we propose a hierarchy of GNNs based on their aggregation regions. We derive theoretical results about the discriminative power and feature representation capabilities of each class. Then, we show how this framework can be utilized to systematically construct arbitrarily powerful GNN…
Recently, the Weisfeiler-Lehman (WL) graph isomorphism test was used to measure the expressiveness of graph neural networks (GNNs), showing that the neighborhood aggregation GNNs were at most as powerful as 1-WL test in distinguishing graph structures. There were also improvements proposed in analogy to $k$-WL test ($k>1$). However, the aggregators in these GNNs are far from injective as required by the WL test, and suffer from weak distinguishing strength, making it become expressive bottlenecks. In this paper, we improve the expressiveness by exploring powerful aggregators. We reformulate ag…
