Graph neural networks (GNNs) are effective machine learning models for various graph learning problems. Despite their empirical successes, the theoretical limitations of GNNs have been revealed recently. Consequently, many GNN models have been proposed to overcome these limitations. In this survey, we provide a comprehensive overview of the expressive power of GNNs and provably powerful variants of GNNs.
The expressive power of message passing neural networks (MPNNs) is known to match the expressive power of the 1-dimensional Weisfeiler-Leman graph (1-WL) isomorphism test. To boost the expressive power of MPNNs, a number of graph neural network architectures have recently been proposed based on higher-dimensional Weisfeiler-Leman tests. In this paper we consider the two-dimensional (2-WL) test and introduce a new type of MPNNs, referred to as $\ell$-walk MPNNs, which aggregate features along walks of length $\ell$ between vertices. We show that $2$-walk MPNNs match 2-WL in expressive power. Mo…
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…
This paper studies Dropout Graph Neural Networks (DropGNNs), a new approach that aims to overcome the limitations of standard GNN frameworks. In DropGNNs, we execute multiple runs of a GNN on the input graph, with some of the nodes randomly and independently dropped in each of these runs. Then, we combine the results of these runs to obtain the final result. We prove that DropGNNs can distinguish various graph neighborhoods that cannot be separated by message passing GNNs. We derive theoretical bounds for the number of runs required to ensure a reliable distribution of dropouts, and we prove s…
We propose a new Graph Neural Network that combines recent advancements in the field. We give theoretical contributions by proving that the model is strictly more general than the Graph Isomorphism Network and the Gated Graph Neural Network, as it can approximate the same functions and deal with arbitrary edge values. Then, we show how a single node information can flow through the graph unchanged.
The University of Tokyo（东京大学）；Preferred Networks, Inc.（Preferred Networks公司）；RIKEN Center for Advanced Intelligence Project (AIP)（理化学研究所先进智能项目中心）
Graph Neural Networks (graph NNs) are a promising deep learning approach for analyzing graph-structured data. However, it is known that they do not improve (or sometimes worsen) their predictive performance as we pile up many layers and add non-lineality. To tackle this problem, we investigate the expressive power of graph NNs via their asymptotic behaviors as the layer size tends to infinity. Our strategy is to generalize the forward propagation of a Graph Convolutional Network (GCN), which is a popular graph NN variant, as a specific dynamical system. In the case of a GCN, we show that when …
Neural Tensor Networks (NTNs), which are structured to encode the degree of relationship among pairs of entities, are used in Logic Tensor Networks (LTNs) to facilitate Statistical Relational Learning (SRL) in first-order logic. In this paper, we propose Randomly Weighted Tensor Networks (RWTNs), which incorporate randomly drawn, untrained tensors into an NTN encoder network with a trained decoder network. We show that RWTNs meet or surpass the performance of traditionally trained LTNs for Semantic Image Interpretation (SII) tasks that have been used as a representative example of how LTNs uti…
Various classes of Graph Neural Networks (GNN) have been proposed and shown to be successful in a wide range of applications with graph structured data. In this paper, we propose a theoretical framework able to compare the expressive power of these GNN architectures. The current universality theorems only apply to intractable classes of GNNs. Here, we prove the first approximation guarantees for practical GNNs, paving the way for a better understanding of their generalization. Our theoretical results are proved for invariant GNNs computing a graph embedding (permutation of the nodes of the inp…
TU Dortmund University（多特蒙德工业大学）；Stanford University（斯坦福大学）
We present GNNAutoScale (GAS), a framework for scaling arbitrary message-passing GNNs to large graphs. GAS prunes entire sub-trees of the computation graph by utilizing historical embeddings from prior training iterations, leading to constant GPU memory consumption in respect to input node size without dropping any data. While existing solutions weaken the expressive power of message passing due to sub-sampling of edges or non-trainable propagations, our approach is provably able to maintain the expressive power of the original GNN. We achieve this by providing approximation error bounds of hi…
How Expressive Are Graph Neural Networks in the Presence of Node Identifiers?
Institute for Logic Language and Computation（逻辑语言与计算研究所）；University of Amsterdam（阿姆斯特丹大学）；Department of Computer Science（计算机科学系）；University of Oxford（牛津大学）；Lehrstuhl Informatik 7（信息学7系）；RWTH Aachen University（亚琛工业大学）
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…
Amazon Science（亚马逊科学）；IBM Research（IBM研究院）；Indian Institute of Science（印度科学院）
Graph neural networks (GNNs) are able to achieve promising performance on multiple graph downstream tasks such as node classification and link prediction. Comparatively lesser work has been done to design GNNs which can operate directly for community detection on graphs. Traditionally, GNNs are trained on a semi-supervised or self-supervised loss function and then clustering algorithms are applied to detect communities. However, such decoupled approaches are inherently sub-optimal. Designing an unsupervised loss function to train a GNN and extract communities in an integrated manner is a funda…
Many problems in computer vision and machine learning can be cast as learning on hypergraphs that represent higher-order relations. Recent approaches for hypergraph learning extend graph neural networks based on message passing, which is simple yet fundamentally limited in modeling long-range dependencies and expressive power. On the other hand, tensor-based equivariant neural networks enjoy maximal expressiveness, but their application has been limited in hypergraphs due to heavy computation and strict assumptions on fixed-order hyperedges. We resolve these problems and present Equivariant Hy…
Arizona State University（亚利桑那州立大学）；University of Virginia（弗吉尼亚大学）；Singapore Management University（新加坡管理大学）
Inspired by the immense success of deep learning, graph neural networks (GNNs) are widely used to learn powerful node representations and have demonstrated promising performance on different graph learning tasks. However, most real-world graphs often come with high-dimensional and sparse node features, rendering the learned node representations from existing GNN architectures less expressive. In this paper, we propose \textit{Feature Interaction-aware Graph Neural Networks (FI-GNNs)}, a plug-and-play GNN framework for learning node representations encoded with informative feature interactions.…
Washington University in St. Louis（圣路易斯华盛顿大学）；Washington University School of Medicine（华盛顿大学医学院）；Peking University（北京大学）；Beijing Institute for General Artificial Intelligence（北京通用人工智能研究院）
The most popular design paradigm for Graph Neural Networks (GNNs) is 1-hop message passing -- aggregating information from 1-hop neighbors repeatedly. However, the expressive power of 1-hop message passing is bounded by the Weisfeiler-Lehman (1-WL) test. Recently, researchers extended 1-hop message passing to K-hop message passing by aggregating information from K-hop neighbors of nodes simultaneously. However, there is no work on analyzing the expressive power of K-hop message passing. In this work, we theoretically characterize the expressive power of K-hop message passing. Specifically, we …
Full-Spectrum Graph Neural Networks: Expressive and Scalable
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore（数学科学学院，物理与数学科学学院，南洋理工大学，新加坡637371，新加坡）
Institute for Artificial Intelligence, Peking University（北京大学人工智能研究院）；Beijing Institute for General Artificial Intelligence（北京通用人工智能研究院）
Spectral Graph Neural Network is a kind of Graph Neural Network (GNN) based on graph signal filters. Some models able to learn arbitrary spectral filters have emerged recently. However, few works analyze the expressive power of spectral GNNs. This paper studies spectral GNNs' expressive power theoretically. We first prove that even spectral GNNs without nonlinearity can produce arbitrary graph signals and give two conditions for reaching universality. They are: 1) no multiple eigenvalues of graph Laplacian, and 2) no missing frequency components in node features. We also establish a connection…
Towards Expressive Spectral-Temporal Graph Neural Networks for Time Series Forecasting
Griffith University（格里菲斯大学）；Monash University（莫纳什大学）；University of Stuttgart（斯图加特大学）；Alibaba Group（阿里巴巴集团）；University of New South Wales（新南威尔士大学）；Emory University（埃默里大学）；Squirrel Ai Learning
We study and compare different Graph Neural Network extensions that increase the expressive power of GNNs beyond the Weisfeiler-Leman test. We focus on (i) GNNs based on higher order WL methods, (ii) GNNs that preprocess small substructures in the graph, (iii) GNNs that preprocess the graph up to a small radius, and (iv) GNNs that slightly perturb the graph to compute an embedding. We begin by presenting a simple improvement for this last extension that strictly increases the expressive power of this GNN variant. Then, as our main result, we compare the expressiveness of these extensions to ea…
The expressive power of graph neural network formalisms is commonly measured by their ability to distinguish graphs. For many formalisms, the k-dimensional Weisfeiler-Leman (k-WL) graph isomorphism test is used as a yardstick. In this paper we consider the expressive power of kth-order invariant (linear) graph networks (k-IGNs). It is known that k-IGNs are expressive enough to simulate k-WL. This means that for any two graphs that can be distinguished by k-WL, one can find a k-IGN which also distinguishes those graphs. The question remains whether k-IGNs can distinguish more graphs than k-WL. …
University of Oxford（牛津大学）；RWTH Aachen University（亚琛工业大学）
Graph neural networks (GNNs) are effective models for representation learning on relational data. However, standard GNNs are limited in their expressive power, as they cannot distinguish graphs beyond the capability of the Weisfeiler-Leman graph isomorphism heuristic. In order to break this expressiveness barrier, GNNs have been enhanced with random node initialization (RNI), where the idea is to train and run the models with randomized initial node features. In this work, we analyze the expressive power of GNNs with RNI, and prove that these models are universal, a first such result for GNNs …
Message passing neural networks (MPNNs) have become a dominant flavor of graph neural networks (GNNs) in recent years. Yet, MPNNs come with notable limitations; namely, they are at most as powerful as the 1-dimensional Weisfeiler-Leman (1-WL) test in distinguishing graphs in a graph isomorphism testing frame-work. To this end, researchers have drawn inspiration from the k-WL hierarchy to develop more expressive GNNs. However, current k-WL-equivalent GNNs are not practical for even small values of k, as k-WL becomes combinatorially more complex as k grows. At the same time, several works have f…
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…
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…
This paper studies asynchronous message passing (AMP), a new paradigm for applying neural network based learning to graphs. Existing graph neural networks use the synchronous distributed computing model and aggregate their neighbors in each round, which causes problems such as oversmoothing and limits their expressiveness. On the other hand, AMP is based on the asynchronous model, where nodes react to messages of their neighbors individually. We prove that (i) AMP can simulate synchronous GNNs and that (ii) AMP can theoretically distinguish any pair of graphs. We experimentally validate AMP's …
Institute for Advanced Study（高等研究院）；Zhejiang University（浙江大学）；Carnegie Mellon University（卡内基梅隆大学）；Massachusetts Institute of Technology（麻省理工学院）
While graph kernels (GKs) are easy to train and enjoy provable theoretical guarantees, their practical performances are limited by their expressive power, as the kernel function often depends on hand-crafted combinatorial features of graphs. Compared to graph kernels, graph neural networks (GNNs) usually achieve better practical performance, as GNNs use multi-layer architectures and non-linear activation functions to extract high-order information of graphs as features. However, due to the large number of hyper-parameters and the non-convex nature of the training procedure, GNNs are harder to …
University of Texas at Arlington（德克萨斯大学阿灵顿分校）；Tencent AI Lab（腾讯人工智能实验室）；Tsinghua University（清华大学）
The crux of molecular property prediction is to generate meaningful representations of the molecules. One promising route is to exploit the molecular graph structure through Graph Neural Networks (GNNs). It is well known that both atoms and bonds significantly affect the chemical properties of a molecule, so an expressive model shall be able to exploit both node (atom) and edge (bond) information simultaneously. Guided by this observation, we present Multi-View Graph Neural Network (MV-GNN), a multi-view message passing architecture to enable more accurate predictions of molecular properties. …
The Logical Expressiveness of Temporal GNNs via Two-Dimensional Product Logics
RPTU Kaiserslautern-Landau（亚琛工业大学凯撒斯劳滕-兰道分校）；Queen Mary University of London（伦敦女王大学）；University of Łódź（卢宾大学）；Theoretical Computer Science / Formal Methods（理论计算机科学/形式方法）；University of Kassel（卡塞尔大学）
Weizmann Institute of Science（魏茨曼科学研究所）；MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；NVIDIA Research（英伟达研究院）；Meta AI Research Centre for Artificial Intelligence（Meta人工智能研究中心）
Stanford University（斯坦福大学）；Carnegie Mellon University（卡内基梅隆大学）；Facebook AI Research（脸书人工智能研究院）；Georgia Institute of Technology（佐治亚理工学院）
With massive amounts of atomic simulation data available, there is a huge opportunity to develop fast and accurate machine learning models to approximate expensive physics-based calculations. The key quantity to estimate is atomic forces, where the state-of-the-art Graph Neural Networks (GNNs) explicitly enforce basic physical constraints such as rotation-covariance. However, to strictly satisfy the physical constraints, existing models have to make tradeoffs between computational efficiency and model expressiveness. Here we explore an alternative approach. By not imposing explicit physical co…
Message passing neural networks (MPNNs) learn the representation of graph-structured data based on graph original information, including node features and graph structures, and have shown astonishing improvement in node classification tasks. However, the expressive power of MPNNs is upper bounded by the first-order Weisfeiler-Leman test and its accuracy still has room for improvement. This work studies how to improve MPNNs' expressiveness and generalizability by fully exploiting graph original information both theoretically and empirically. It further proposes a new GNN model called INGNN (INf…
Expressive Power of Graph Neural Networks for (Mixed-Integer) Quadratic Programs
Department of Mathematics, Massachusetts Institute of Technology（麻省理工学院数学系）；Decision Intelligence Lab, Damo Academy, Alibaba US（达摩院决策智能实验室）；Department of Statistics and Data Science, University of Central Florida（中央佛罗里达大学统计与数据科学系）
Graph Neural Networks (GNNs) show strong expressive power on graph data mining, by aggregating information from neighbors and using the integrated representation in the downstream tasks. The same aggregation methods and parameters for each node in a graph are used to enable the GNNs to utilize the homophily relational data. However, not all graphs are homophilic, even in the same graph, the distributions may vary significantly. Using the same convolution over all nodes may lead to the ignorance of various graph patterns. Furthermore, many existing GNNs integrate node features and structure ide…
