Xiyuan Wang

Xiyuan Wang, Haotong Yang, Muhan Zhang

In this work, we propose a novel link prediction model and further boost it by studying graph incompleteness. First, we introduce MPNN-then-SF, an innovative architecture leveraging structural feature (SF) to guide MPNN's representation pooling, with its implementation, namely Neural Common Neighbor (NCN). NCN exhibits superior expressiveness and scalability compared with existing models, which can be classified into two categories: SF-then-MPNN, augmenting MPNN's input with SF, and SF-and-MPNN, decoupling SF and MPNN. Second, we investigate the impact of graph incompleteness -- the phenomenon that some links are unobserved in the input graph -- on SF, like the common neighbor. Through dataset visualization, we observe that incompleteness reduces common neighbors and induces distribution shifts, significantly affecting model performance. To address this issue, we propose to use a link prediction model to complete the common neighbor structure. Combining this method with NCN, we propose Neural Common Neighbor with Completion (NCNC). NCN and NCNC outperform recent strong baselines by large margins, and NCNC further surpasses state-of-the-art models in standard link prediction benchmarks. Our code is available at https://github.com/GraphPKU/NeuralCommonNeighbor.

Cai Zhou, Xiyuan Wang, Muhan Zhang · mit

Node-level random walk has been widely used to improve Graph Neural Networks. However, there is limited attention to random walk on edge and, more generally, on $k$-simplices. This paper systematically analyzes how random walk on different orders of simplicial complexes (SC) facilitates GNNs in their theoretical expressivity. First, on $0$-simplices or node level, we establish a connection between existing positional encoding (PE) and structure encoding (SE) methods through the bridge of random walk. Second, on $1$-simplices or edge level, we bridge edge-level random walk and Hodge $1$-Laplacians and design corresponding edge PE respectively. In the spatial domain, we directly make use of edge level random walk to construct EdgeRWSE. Based on the spectral analysis of Hodge $1$-Laplcians, we propose Hodge1Lap, a permutation equivariant and expressive edge-level positional encoding. Third, we generalize our theory to random walk on higher-order simplices and propose the general principle to design PE on simplices based on random walk and Hodge Laplacians. Inter-level random walk is also introduced to unify a wide range of simplicial networks. Extensive experiments verify the effectiveness of our random walk-based methods.

Xiyuan Wang, Muhan Zhang

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 between the expressive power of spectral GNNs and Graph Isomorphism (GI) testing, the latter of which is often used to characterize spatial GNNs' expressive power. Moreover, we study the difference in empirical performance among different spectral GNNs with the same expressive power from an optimization perspective, and motivate the use of an orthogonal basis whose weight function corresponds to the graph signal density in the spectrum. Inspired by the analysis, we propose JacobiConv, which uses Jacobi basis due to its orthogonality and flexibility to adapt to a wide range of weight functions. JacobiConv deserts nonlinearity while outperforming all baselines on both synthetic and real-world datasets.

Yang Hu, Xiyuan Wang, Zhouchen Lin et al.

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 approach which can directly obtain node pair (link) representations based on \textit{two-dimensional Weisfeiler-Lehman (2-WL) tests}. 2-WL tests directly use links (2-tuples) as message passing units instead of nodes, and thus can directly obtain link representations. We theoretically analyze the expressive power of 2-WL tests to discriminate non-isomorphic links, and prove their superior link discriminating power than 1-WL. Based on different 2-WL variants, we propose a series of novel 2-WL-GNN models for link prediction. Experiments on a wide range of real-world datasets demonstrate their competitive performance to state-of-the-art baselines and superiority over plain 1-WL-GNNs.

Xiyuan Wang, Muhan Zhang

We introduce PyTorch Geometric High Order (PyGHO), a library for High Order Graph Neural Networks (HOGNNs) that extends PyTorch Geometric (PyG). Unlike ordinary Message Passing Neural Networks (MPNNs) that exchange messages between nodes, HOGNNs, encompassing subgraph GNNs and k-WL GNNs, encode node tuples, a method previously lacking a standardized framework and often requiring complex coding. PyGHO's main objective is to provide an unified and user-friendly interface for various HOGNNs. It accomplishes this through streamlined data structures for node tuples, comprehensive data processing utilities, and a flexible suite of operators for high-order GNN methodologies. In this work, we present a detailed in-depth of PyGHO and compare HOGNNs implemented with PyGHO with their official implementation on real-world tasks. PyGHO achieves up to $50\%$ acceleration and reduces the code needed for implementation by an order of magnitude. Our library is available at \url{https://github.com/GraphPKU/PygHO}.

Zian Li, Xiyuan Wang, Yinan Huang et al.

Graph Neural Networks (GNNs) are often used for tasks involving the 3D geometry of a given graph, such as molecular dynamics simulation. While incorporating Euclidean distance into Message Passing Neural Networks (referred to as Vanilla DisGNN) is a straightforward way to learn the geometry, it has been demonstrated that Vanilla DisGNN is geometrically incomplete. In this work, we first construct families of novel and symmetric geometric graphs that Vanilla DisGNN cannot distinguish even when considering all-pair distances, which greatly expands the existing counterexample families. Our counterexamples show the inherent limitation of Vanilla DisGNN to capture symmetric geometric structures. We then propose $k$-DisGNNs, which can effectively exploit the rich geometry contained in the distance matrix. We demonstrate the high expressive power of $k$-DisGNNs from three perspectives: 1. They can learn high-order geometric information that cannot be captured by Vanilla DisGNN. 2. They can unify some existing well-designed geometric models. 3. They are universal function approximators from geometric graphs to scalars (when $k\geq 2$) and vectors (when $k\geq 3$). Most importantly, we establish a connection between geometric deep learning (GDL) and traditional graph representation learning (GRL), showing that those highly expressive GNN models originally designed for GRL can also be applied to GDL with impressive performance, and that existing complicated, equivariant models are not the only solution. Experiments verify our theory. Our $k$-DisGNNs achieve many new state-of-the-art results on MD17.

Xiyuan Wang, Pan Li, Muhan Zhang

In this paper, we study using graph neural networks (GNNs) for \textit{multi-node representation learning}, where a representation for a set of more than one node (such as a link) is to be learned. Existing GNNs are mainly designed to learn single-node representations. When used for multi-node representation learning, a common practice is to directly aggregate the single-node representations obtained by a GNN. In this paper, we show a fundamental limitation of such an approach, namely the inability to capture the dependence among multiple nodes in the node set. A straightforward solution is to distinguish target nodes from others. Formalizing this idea, we propose \text{labeling trick}, which first labels nodes in the graph according to their relationships with the target node set before applying a GNN and then aggregates node representations obtained in the labeled graph for multi-node representations. Besides node sets in graphs, we also extend labeling tricks to posets, subsets and hypergraphs. Experiments verify that the labeling trick technique can boost GNNs on various tasks, including undirected link prediction, directed link prediction, hyperedge prediction, and subgraph prediction. Our work explains the superior performance of previous node-labeling-based methods and establishes a theoretical foundation for using GNNs for multi-node representation learning.

Yanbo Wang, Xiyuan Wang, Quan Gan et al.

We introduce Griffin, the first foundation model attemptation designed specifically for Relational Databases (RDBs). Unlike previous smaller models focused on single RDB tasks, Griffin unifies the data encoder and task decoder to handle diverse tasks. Additionally, we enhance the architecture by incorporating a cross-attention module and a novel aggregator. Griffin utilizes pretraining on both single-table and RDB datasets, employing advanced encoders for categorical, numerical, and metadata features, along with innovative components such as cross-attention modules and enhanced message-passing neural networks (MPNNs) to capture the complexities of relational data. Evaluated on large-scale, heterogeneous, and temporal graphs extracted from RDBs across various domains (spanning over 150 million nodes), Griffin demonstrates superior or comparable performance to individually trained models, excels in low-data scenarios, and shows strong transferability with similarity and diversity in pretraining across new datasets and tasks, highlighting its potential as a universally applicable foundation model for RDBs. Code available at https://github.com/yanxwb/Griffin.

Weishuo Ma, Yanbo Wang, Xiyuan Wang et al.

Recent advancements in graph neural networks (GNNs) for link prediction have introduced sophisticated training techniques and model architectures. However, reliance on outdated baselines may exaggerate the benefits of these new approaches. To tackle this issue, we systematically explore Graph Autoencoders (GAEs) by applying model-agnostic tricks in recent methods and tuning hyperparameters. We find that a well-tuned GAE can match the performance of recent sophisticated models while offering superior computational efficiency on widely-used link prediction benchmarks. Our approach delivers substantial performance gains on datasets where structural information dominates and feature data is limited. Specifically, our GAE achieves a state-of-the-art Hits@100 score of 78.41\% on the ogbl-ppa dataset. Furthermore, we examine the impact of various tricks to uncover the reasons behind our success and to guide the design of future methods. Our study emphasizes the critical need to update baselines for a more accurate assessment of progress in GNNs for link prediction. Our code is available at https://github.com/GraphPKU/Refined-GAE.

Yewei Liu, Xiyuan Wang, Muhan Zhang

We propose an entirely new meta-learning framework for network pruning. It is a general framework that can be theoretically applied to almost all types of networks with all kinds of pruning and has great generality and transferability. Experiments have shown that it can achieve outstanding results on many popular and representative pruning tasks (including both CNNs and Transformers). Unlike all prior works that either rely on fixed, hand-crafted criteria to prune in a coarse manner, or employ learning to prune ways that require special training during each pruning and lack generality. Our framework can learn complex pruning rules automatically via a neural network (metanetwork) and has great generality that can prune without any special training. More specifically, we introduce the newly developed idea of metanetwork from meta-learning into pruning. A metanetwork is a network that takes another network as input and produces a modified network as output. In this paper, we first establish a bijective mapping between neural networks and graphs, and then employ a graph neural network as our metanetwork. We train a metanetwork that learns the pruning strategy automatically and can transform a network that is hard to prune into another network that is much easier to prune. Once the metanetwork is trained, our pruning needs nothing more than a feedforward through the metanetwork and some standard finetuning to prune at state-of-the-art. Our code is available at https://github.com/Yewei-Liu/MetaPruning.

Xiaohui Zhang, Yanbo Wang, Xiyuan Wang et al.

Temporal graphs are widespread in real-world applications such as social networks, as well as trade and transportation networks. Predicting dynamic links within these evolving graphs is a key problem. Many memory-based methods use temporal interaction histories to generate node embeddings, which are then combined to predict links. However, these approaches primarily focus on individual node representations, often overlooking the inherently pairwise nature of link prediction. While some recent methods attempt to capture pairwise features, they tend to be limited by high computational complexity arising from repeated embedding calculations, making them unsuitable for large-scale datasets like the Temporal Graph Benchmark (TGB). To address the critical need for models that combine strong expressive power with high computational efficiency for link prediction on large temporal graphs, we propose Temporal Neural Common Neighbor (TNCN). Our model achieves this balance by adapting the powerful pairwise modeling principles of Neural Common Neighbor (NCN) to an efficient temporal architecture. TNCN improves upon NCN by efficiently preserving and updating temporal neighbor dictionaries for each node and by using multi-hop common neighbors to learn more expressive pairwise representations. TNCN achieves new state-of-the-art performance on Review from five large-scale real-world TGB datasets, 6 out of 7 datasets in the transductive setting and 3 out of 7 in the inductive setting on small- to medium-scale datasets. Additionally, TNCN demonstrates excellent scalability, outperforming prominent GNN baselines by up to 30.3 times in speed on large datasets. Our code is available at https://github.com/GraphPKU/TNCN.

Cai Zhou, Xiyuan Wang, Muhan Zhang · mit

In this paper, we propose the first framework that enables solving graph learning tasks of all levels (node, edge and graph) and all types (generation, regression and classification) using one formulation. We first formulate prediction tasks including regression and classification into a generic (conditional) generation framework, which enables diffusion models to perform deterministic tasks with provable guarantees. We then propose Latent Graph Diffusion (LGD), a generative model that can generate node, edge, and graph-level features of all categories simultaneously. We achieve this goal by embedding the graph structures and features into a latent space leveraging a powerful encoder and decoder, then training a diffusion model in the latent space. LGD is also capable of conditional generation through a specifically designed cross-attention mechanism. Leveraging LGD and the ``all tasks as generation'' formulation, our framework is capable of solving graph tasks of various levels and types. We verify the effectiveness of our framework with extensive experiments, where our models achieve state-of-the-art or highly competitive results across a wide range of generation and regression tasks.

Zian Li, Xiyuan Wang, Shijia Kang et al.

Invariant models, one important class of geometric deep learning models, are capable of generating meaningful geometric representations by leveraging informative geometric features in point clouds. These models are characterized by their simplicity, good experimental results and computational efficiency. However, their theoretical expressive power still remains unclear, restricting a deeper understanding of the potential of such models. In this work, we concentrate on characterizing the theoretical expressiveness of a wide range of invariant models under fully-connected conditions. We first rigorously characterize the expressiveness of the most classic invariant model, message-passing neural networks incorporating distance (DisGNN), restricting its unidentifiable cases to be only highly symmetric point clouds. We then prove that GeoNGNN, the geometric counterpart of one of the simplest subgraph graph neural networks, can effectively break these corner cases' symmetry and thus achieve E(3)-completeness. By leveraging GeoNGNN as a theoretical tool, we further prove that: 1) most subgraph GNNs developed in traditional graph learning can be seamlessly extended to geometric scenarios with E(3)-completeness; 2) DimeNet, GemNet and SphereNet, three well-established invariant models, are also all capable of achieving E(3)-completeness. Our theoretical results fill the gap in the expressive power of invariant models, contributing to a rigorous and comprehensive understanding of their capabilities.

Haotong Yang, Xiyuan Wang, Qian Tao et al.

Recent research on integrating Large Language Models (LLMs) with Graph Neural Networks (GNNs) typically follows two approaches: LLM-centered models, which convert graph data into tokens for LLM processing, and GNN-centered models, which use LLMs to encode text features into node and edge representations for GNN input. LLM-centered models often struggle to capture graph structures effectively, while GNN-centered models compress variable-length textual data into fixed-size vectors, limiting their ability to understand complex semantics. Additionally, GNN-centered approaches require converting tasks into a uniform, manually-designed format, restricting them to classification tasks and preventing language output. To address these limitations, we introduce a new architecture that deeply integrates GNN with LLM, featuring three key innovations: (1) Structure-Aware Transformers, which incorporate GNN's message-passing capabilities directly into LLM's transformer layers, allowing simultaneous processing of textual and structural information and generating outputs from both GNN and LLM; (2) Graph-Text Cross-Attention, which processes full, uncompressed text from graph nodes and edges, ensuring complete semantic integration; and (3) GNN-LLM Twin Predictor, enabling LLM's flexible autoregressive generation alongside GNN's scalable one-pass prediction. GL-Fusion achieves outstand performance on various tasks. Notably, it achieves state-of-the-art performance on OGBN-Arxiv and OGBG-Code2.

Xiyuan Wang, Pan Li, Muhan Zhang

Graph is a fundamental data structure to model interconnections between entities. Set, on the contrary, stores independent elements. To learn graph representations, current Graph Neural Networks (GNNs) primarily use message passing to encode the interconnections. In contrast, this paper introduces a novel graph-to-set conversion method that bijectively transforms interconnected nodes into a set of independent points and then uses a set encoder to learn the graph representation. This conversion method holds dual significance. Firstly, it enables using set encoders to learn from graphs, thereby significantly expanding the design space of GNNs. Secondly, for Transformer, a specific set encoder, we provide a novel and principled approach to inject graph information losslessly, different from all the heuristic structural/positional encoding methods adopted in previous graph transformers. To demonstrate the effectiveness of our approach, we introduce Point Set Transformer (PST), a transformer architecture that accepts a point set converted from a graph as input. Theoretically, PST exhibits superior expressivity for both short-range substructure counting and long-range shortest path distance tasks compared to existing GNNs. Extensive experiments further validate PST's outstanding real-world performance. Besides Transformer, we also devise a Deepset-based set encoder, which achieves performance comparable to representative GNNs, affirming the versatility of our graph-to-set method.

Xiyuan Wang, Yewei Liu, Lexi Pang et al.

Diffusion models have gained popularity in graph generation tasks; however, the extent of their expressivity concerning the graph distributions they can learn is not fully understood. Unlike models in other domains, popular backbones for graph diffusion models, such as Graph Transformers, do not possess universal expressivity to accurately model the distribution scores of complex graph data. Our work addresses this limitation by focusing on the frequency of specific substructures as a key characteristic of target graph distributions. When evaluating existing models using this metric, we find that they fail to maintain the distribution of substructure counts observed in the training set when generating new graphs. To address this issue, we establish a theoretical connection between the expressivity of Graph Neural Networks (GNNs) and the overall performance of graph diffusion models, demonstrating that more expressive GNN backbones can better capture complex distribution patterns. By integrating advanced GNNs into the backbone architecture, we achieve significant improvements in substructure generation.

Xiyuan Wang, Muhan Zhang

Recent advances in Graph Neural Networks (GNNs) have explored the potential of random noise as an input feature to enhance expressivity across diverse tasks. However, naively incorporating noise can degrade performance, while architectures tailored to exploit noise for specific tasks excel yet lack broad applicability. This paper tackles these issues by laying down a theoretical framework that elucidates the increased sample complexity when introducing random noise into GNNs without careful design. We further propose Equivariant Noise GNN (ENGNN), a novel architecture that harnesses the symmetrical properties of noise to mitigate sample complexity and bolster generalization. Our experiments demonstrate that using noise equivariantly significantly enhances performance on node-level, link-level, subgraph, and graph-level tasks and achieves comparable performance to models designed for specific tasks, thereby offering a general method to boost expressivity across various graph tasks.

Junru Zhou, Cai Zhou, Xiyuan Wang et al. · mit

Graph neural networks (GNNs) have achieved remarkable success in a variety of machine learning tasks over graph data. Existing GNNs usually rely on message passing, i.e., computing node representations by gathering information from the neighborhood, to build their underlying computational graphs. They are known fairly limited in expressive power, and often fail to capture global characteristics of graphs. To overcome the issue, a popular solution is to use Laplacian eigenvectors as additional node features, as they contain global positional information of nodes, and can serve as extra node identifiers aiding GNNs to separate structurally similar nodes. For such an approach, properly handling the orthogonal group symmetry among eigenvectors with equal eigenvalue is crucial for its stability and generalizability. However, using a naive orthogonal group invariant encoder for each separate eigenspace may not keep the full expressivity in the Laplacian eigenvectors. Moreover, computing such invariants inevitably entails a hard split of Laplacian eigenvalues according to their numerical identity, which suffers from great instability when the graph structure is perturbed. In this paper, we propose a novel method exploiting Laplacian eigenvectors to generate stable and globally expressive graph representations. The main difference from previous works is that (i) our method utilizes learnable orthogonal group invariant representations for each Laplacian eigenspace, based upon powerful orthogonal group equivariant neural network layers already well studied in the literature, and that (ii) our method deals with numerically close eigenvalues in a smooth fashion, ensuring its better robustness against perturbations. Experiments on various graph learning benchmarks witness the competitive performance of our method, especially its great potential to learn global properties of graphs.

Weishuo Ma, Yanbo Wang, Xiyuan Wang et al.

Graph Neural Networks (GNNs) are powerful tools for precessing relational data but often struggle to generalize to unseen graphs, giving rise to the development of Graph Foundational Models (GFMs). However, current GFMs are challenged by the extreme heterogeneity of graph data, where each graph can possess a unique feature space, label set, and topology. To address this, two main paradigms have emerged. The first leverages Large Language Models (LLMs), but is fundamentally text-dependent, thus struggles to handle the numerical features in vast graphs. The second pre-trains a structure-based model, but the adaptation to new tasks typically requires a costly, per-graph tuning stage, creating a critical efficiency bottleneck. In this work, we move beyond these limitations and introduce \textbf{G}raph \textbf{I}n-context \textbf{L}earning \textbf{T}ransformer (GILT), a framework built on an LLM-free and tuning-free architecture. GILT introduces a novel token-based framework for in-context learning (ICL) on graphs, reframing classification tasks spanning node, edge and graph levels in a unified framework. This mechanism is the key to handling heterogeneity, as it is designed to operate on generic numerical features. Further, its ability to understand class semantics dynamically from the context enables tuning-free adaptation. Comprehensive experiments show that GILT achieves stronger few-shot performance with significantly less time than LLM-based or tuning-based baselines, validating the effectiveness of our approach.

Qian Tao, Xiyuan Wang, Muhan Zhang et al.

Graph neural networks (GNNs) have become a prevalent framework for graph tasks. Many recent studies have proposed the use of graph convolution methods over the numerous subgraphs of each graph, a concept known as subgraph graph neural networks (subgraph GNNs), to enhance GNNs' ability to distinguish non-isomorphic graphs. To maximize the expressiveness, subgraph GNNs often require each subgraph to have equal size to the original graph. Despite their impressive performance, subgraph GNNs face challenges due to the vast number and large size of subgraphs which lead to a surge in training data, resulting in both storage and computational inefficiencies. In response to this problem, this paper introduces Ego-Nets-Fit-All (ENFA), a model that uniformly takes the smaller ego nets as subgraphs, thereby providing greater storage and computational efficiency, while at the same time guarantees identical outputs to the original subgraph GNNs even taking the whole graph as subgraphs. The key is to identify and eliminate the redundant computation among subgraphs. For example, a node $v_i$ may appear in multiple subgraphs but is far away from all of their centers (the unsymmetric part between subgraphs). Therefore, its first few rounds of message passing within each subgraph can be computed once in the original graph instead of being computed multiple times within each subgraph. Such strategy enables our ENFA to accelerate subgraph GNNs in an exact way, unlike previous sampling approaches that often lose the performance. Extensive experiments across various datasets reveal that compared with the conventional subgraph GNNs, ENFA can reduce storage space by 29.0% to 84.5% and improve training efficiency by up to 1.66x.

Cai Zhou, Xiyuan Wang, Muhan Zhang

Relational pooling is a framework for building more expressive and permutation-invariant graph neural networks. However, there is limited understanding of the exact enhancement in the expressivity of RP and its connection with the Weisfeiler Lehman hierarchy. Starting from RP, we propose to explicitly assign labels to nodes as additional features to improve expressive power of message passing neural networks. The method is then extended to higher dimensional WL, leading to a novel $k,l$-WL algorithm, a more general framework than $k$-WL. Theoretically, we analyze the expressivity of $k,l$-WL with respect to $k$ and $l$ and unifies it with a great number of subgraph GNNs. Complexity reduction methods are also systematically discussed to build powerful and practical $k,l$-GNN instances. We theoretically and experimentally prove that our method is universally compatible and capable of improving the expressivity of any base GNN model. Our $k,l$-GNNs achieve superior performance on many synthetic and real-world datasets, which verifies the effectiveness of our framework.