Muhan Zhang

Ling Yang, Ye Tian, Minkai Xu et al.

GNN-to-MLP distillation aims to utilize knowledge distillation (KD) to learn computationally-efficient multi-layer perceptron (student MLP) on graph data by mimicking the output representations of teacher GNN. Existing methods mainly make the MLP to mimic the GNN predictions over a few class labels. However, the class space may not be expressive enough for covering numerous diverse local graph structures, thus limiting the performance of knowledge transfer from GNN to MLP. To address this issue, we propose to learn a new powerful graph representation space by directly labeling nodes' diverse local structures for GNN-to-MLP distillation. Specifically, we propose a variant of VQ-VAE to learn a structure-aware tokenizer on graph data that can encode each node's local substructure as a discrete code. The discrete codes constitute a codebook as a new graph representation space that is able to identify different local graph structures of nodes with the corresponding code indices. Then, based on the learned codebook, we propose a new distillation target, namely soft code assignments, to directly transfer the structural knowledge of each node from GNN to MLP. The resulting framework VQGraph achieves new state-of-the-art performance on GNN-to-MLP distillation in both transductive and inductive settings across seven graph datasets. We show that VQGraph with better performance infers faster than GNNs by 828x, and also achieves accuracy improvement over GNNs and stand-alone MLPs by 3.90% and 28.05% on average, respectively. Code: https://github.com/YangLing0818/VQGraph.

Haotong Yang, Zhouchen Lin, Muhan Zhang

Most knowledge graphs (KGs) are incomplete, which motivates one important research topic on automatically complementing knowledge graphs. However, evaluation of knowledge graph completion (KGC) models often ignores the incompleteness -- facts in the test set are ranked against all unknown triplets which may contain a large number of missing facts not included in the KG yet. Treating all unknown triplets as false is called the closed-world assumption. This closed-world assumption might negatively affect the fairness and consistency of the evaluation metrics. In this paper, we study KGC evaluation under a more realistic setting, namely the open-world assumption, where unknown triplets are considered to include many missing facts not included in the training or test sets. For the currently most used metrics such as mean reciprocal rank (MRR) and Hits@K, we point out that their behavior may be unexpected under the open-world assumption. Specifically, with not many missing facts, their numbers show a logarithmic trend with respect to the true strength of the model, and thus, the metric increase could be insignificant in terms of reflecting the true model improvement. Further, considering the variance, we show that the degradation in the reported numbers may result in incorrect comparisons between different models, where stronger models may have lower metric numbers. We validate the phenomenon both theoretically and experimentally. Finally, we suggest possible causes and solutions for this problem. Our code and data are available at https://github.com/GraphPKU/Open-World-KG .

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.

Zehao Dong, Muhan Zhang, Fuhai Li et al.

Optimization of directed acyclic graph (DAG) structures has many applications, such as neural architecture search (NAS) and probabilistic graphical model learning. Encoding DAGs into real vectors is a dominant component in most neural-network-based DAG optimization frameworks. Currently, most DAG encoders use an asynchronous message passing scheme which sequentially processes nodes according to the dependency between nodes in a DAG. That is, a node must not be processed until all its predecessors are processed. As a result, they are inherently not parallelizable. In this work, we propose a Parallelizable Attention-based Computation structure Encoder (PACE) that processes nodes simultaneously and encodes DAGs in parallel. We demonstrate the superiority of PACE through encoder-dependent optimization subroutines that search the optimal DAG structure based on the learned DAG embeddings. Experiments show that PACE not only improves the effectiveness over previous sequential DAG encoders with a significantly boosted training and inference speed, but also generates smooth latent (DAG encoding) spaces that are beneficial to downstream optimization subroutines. Our source code is available at \url{https://github.com/zehao-dong/PACE}

Yanbo Wang, Muhan Zhang

Research on the theoretical expressiveness of Graph Neural Networks (GNNs) has developed rapidly, and many methods have been proposed to enhance the expressiveness. However, most methods do not have a uniform expressiveness measure except for a few that strictly follow the $k$-dimensional Weisfeiler-Lehman ($k$-WL) test hierarchy, leading to difficulties in quantitatively comparing their expressiveness. Previous research has attempted to use datasets for measurement, but facing problems with difficulty (any model surpassing 1-WL has nearly 100% accuracy), granularity (models tend to be either 100% correct or near random guess), and scale (only several essentially different graphs involved). To address these limitations, we study the realized expressive power that a practical model instance can achieve using a novel expressiveness dataset, BREC, which poses greater difficulty (with up to 4-WL-indistinguishable graphs), finer granularity (enabling comparison of models between 1-WL and 3-WL), a larger scale (consisting of 800 1-WL-indistinguishable graphs that are non-isomorphic to each other). We synthetically test 23 models with higher-than-1-WL expressiveness on BREC. Our experiment gives the first thorough measurement of the realized expressiveness of those state-of-the-art beyond-1-WL GNN models and reveals the gap between theoretical and realized expressiveness. Dataset and evaluation codes are released at: https://github.com/GraphPKU/BREC.

Fanxu Meng, Haotong Yang, Yiding Wang et al. · pku

The human brain is naturally equipped to comprehend and interpret visual information rapidly. When confronted with complex problems or concepts, we use flowcharts, sketches, and diagrams to aid our thought process. Leveraging this inherent ability can significantly enhance logical reasoning. However, current Large Language Models (LLMs) do not utilize such visual intuition to help their thinking. Even the most advanced version language models (e.g., GPT-4V and LLaVA) merely align images into textual space, which means their reasoning processes remain purely verbal. To mitigate such limitations, we present a Chain of Images (CoI) approach, which can convert complex language reasoning problems to simple pattern recognition by generating a series of images as intermediate representations. Furthermore, we have developed a CoI evaluation dataset encompassing 15 distinct domains where images can intuitively aid problem-solving. Based on this dataset, we aim to construct a benchmark to assess the capability of future multimodal large-scale models to leverage images for reasoning. In supporting our CoI reasoning, we introduce a symbolic multimodal large language model (SyMLLM) that generates images strictly based on language instructions and accepts both text and image as input. Experiments on Geometry, Chess and Common Sense tasks sourced from the CoI evaluation dataset show that CoI improves performance significantly over the pure-language Chain of Thoughts (CoT) baselines. The code is available at https://github.com/GraphPKU/CoI.

Zuoyu Yan, Junru Zhou, Liangcai Gao et al. · pku

We investigate the enhancement of graph neural networks' (GNNs) representation power through their ability in substructure counting. Recent advances have seen the adoption of subgraph GNNs, which partition an input graph into numerous subgraphs, subsequently applying GNNs to each to augment the graph's overall representation. Despite their ability to identify various substructures, subgraph GNNs are hindered by significant computational and memory costs. In this paper, we tackle a critical question: Is it possible for GNNs to count substructures both \textbf{efficiently} and \textbf{provably}? Our approach begins with a theoretical demonstration that the distance to rooted nodes in subgraphs is key to boosting the counting power of subgraph GNNs. To avoid the need for repetitively applying GNN across all subgraphs, we introduce precomputed structural embeddings that encapsulate this crucial distance information. Experiments validate that our proposed model retains the counting power of subgraph GNNs while achieving significantly faster performance.

Jiarui Feng, Yixin Chen, Fuhai Li et al.

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 first formally differentiate two different kernels of K-hop message passing which are often misused in previous works. We then characterize the expressive power of K-hop message passing by showing that it is more powerful than 1-WL and can distinguish almost all regular graphs. Despite the higher expressive power, we show that K-hop message passing still cannot distinguish some simple regular graphs and its expressive power is bounded by 3-WL. To further enhance its expressive power, we introduce a KP-GNN framework, which improves K-hop message passing by leveraging the peripheral subgraph information in each hop. We show that KP-GNN can distinguish many distance regular graphs which could not be distinguished by previous distance encoding or 3-WL methods. Experimental results verify the expressive power and effectiveness of KP-GNN. KP-GNN achieves competitive results across all benchmark datasets.

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}.

Haoteng Yin, Muhan Zhang, Jianguo Wang et al.

Subgraph-based graph representation learning (SGRL) has recently emerged as a powerful tool in many prediction tasks on graphs due to its advantages in model expressiveness and generalization ability. Most previous SGRL models face computational challenges associated with the high cost of subgraph extraction for each training or test query. Recently, SUREL was proposed to accelerate SGRL, which samples random walks offline and joins these walks online as a proxy of subgraph for representation learning. Thanks to the reusability of sampled walks across different queries, SUREL achieves state-of-the-art performance in terms of scalability and prediction accuracy. However, SUREL still suffers from high computational overhead caused by node duplication in sampled walks. In this work, we propose a novel framework SUREL+ that upgrades SUREL by using node sets instead of walks to represent subgraphs. This set-based representation eliminates repeated nodes by definition but can also be irregular in size. To address this issue, we design a customized sparse data structure to efficiently store and access node sets and provide a specialized operator to join them in parallel batches. SUREL+ is modularized to support multiple types of set samplers, structural features, and neural encoders to complement the structural information loss after the reduction from walks to sets. Extensive experiments have been performed to validate SUREL+ in the prediction tasks of links, relation types, and higher-order patterns. SUREL+ achieves 3-11$\times$ speedups of SUREL while maintaining comparable or even better prediction performance; compared to other SGRL baselines, SUREL+ achieves $\sim$20$\times$ speedups and significantly improves the prediction accuracy.

Yu Huang, Min Zhou, Menglin Yang et al.

Recent advancements in graph learning have revolutionized the way to understand and analyze data with complex structures. Notably, Graph Neural Networks (GNNs), i.e. neural network architectures designed for learning graph representations, have become a popular paradigm. With these models being usually characterized by intuition-driven design or highly intricate components, placing them within the theoretical analysis framework to distill the core concepts, helps understand the key principles that drive the functionality better and guide further development. Given this surge in interest, this article provides a comprehensive summary of the theoretical foundations and breakthroughs concerning the approximation and learning behaviors intrinsic to prevalent graph learning models. Encompassing discussions on fundamental aspects such as expressiveness power, generalization, optimization, and unique phenomena such as over-smoothing and over-squashing, this piece delves into the theoretical foundations and frontier driving the evolution of graph learning. In addition, this article also presents several challenges and further initiates discussions on possible solutions.

Jiarui Feng, Lecheng Kong, Hao Liu et al.

Message passing neural networks (MPNNs) have emerged as the most popular framework of graph neural networks (GNNs) in recent years. However, their expressive power is limited by the 1-dimensional Weisfeiler-Lehman (1-WL) test. Some works are inspired by $k$-WL/FWL (Folklore WL) and design the corresponding neural versions. Despite the high expressive power, there are serious limitations in this line of research. In particular, (1) $k$-WL/FWL requires at least $O(n^k)$ space complexity, which is impractical for large graphs even when $k=3$; (2) The design space of $k$-WL/FWL is rigid, with the only adjustable hyper-parameter being $k$. To tackle the first limitation, we propose an extension, $(k,t)$-FWL. We theoretically prove that even if we fix the space complexity to $O(n^k)$ (for any $k\geq 2$) in $(k,t)$-FWL, we can construct an expressiveness hierarchy up to solving the graph isomorphism problem. To tackle the second problem, we propose $k$-FWL+, which considers any equivariant set as neighbors instead of all nodes, thereby greatly expanding the design space of $k$-FWL. Combining these two modifications results in a flexible and powerful framework $(k,t)$-FWL+. We demonstrate $(k,t)$-FWL+ can implement most existing models with matching expressiveness. We then introduce an instance of $(k,t)$-FWL+ called Neighborhood$^2$-FWL (N$^2$-FWL), which is practically and theoretically sound. We prove that N$^2$-FWL is no less powerful than 3-WL, and can encode many substructures while only requiring $O(n^2)$ space. Finally, we design its neural version named N$^2$-GNN and evaluate its performance on various tasks. N$^2$-GNN achieves record-breaking results on ZINC-Subset (0.059), outperforming previous SOTA results by 10.6%. Moreover, N$^2$-GNN achieves new SOTA results on the BREC dataset (71.8%) among all existing high-expressive GNN methods.

Hao Liu, Jiarui Feng, Lecheng Kong et al.

Designing a single model to address multiple tasks has been a long-standing objective in artificial intelligence. Recently, large language models have demonstrated exceptional capability in solving different tasks within the language domain. However, a unified model for various graph tasks remains underexplored, primarily due to the challenges unique to the graph learning domain. First, graph data from different areas carry distinct attributes and follow different distributions. Such discrepancy makes it hard to represent graphs in a single representation space. Second, tasks on graphs diversify into node, link, and graph tasks, requiring distinct embedding strategies. Finally, an appropriate graph prompting paradigm for in-context learning is unclear. We propose \textbf{One for All (OFA)}, the first general framework that can use a single graph model to address the above challenges. Specifically, OFA proposes text-attributed graphs to unify different graph data by describing nodes and edges with natural language and uses language models to encode the diverse and possibly cross-domain text attributes to feature vectors in the same embedding space. Furthermore, OFA introduces the concept of nodes-of-interest to standardize different tasks with a single task representation. For in-context learning on graphs, OFA introduces a novel graph prompting paradigm that appends prompting substructures to the input graph, which enables it to address varied tasks without fine-tuning. We train the OFA model using graph data from multiple domains (including citation networks, molecular graphs, knowledge graphs, etc.) simultaneously and evaluate its ability in supervised, few-shot, and zero-shot learning scenarios. OFA performs well across different tasks, making it the first general-purpose across-domains classification model on graphs.

Lecheng Kong, Yixin Chen, Muhan Zhang

Graph Neural Networks (GNNs) have recently been applied to graph learning tasks and achieved state-of-the-art (SOTA) results. However, many competitive methods run GNNs multiple times with subgraph extraction and customized labeling to capture information that is hard for normal GNNs to learn. Such operations are time-consuming and do not scale to large graphs. In this paper, we propose an efficient GNN framework called Geodesic GNN (GDGNN) that requires only one GNN run and injects conditional relationships between nodes into the model without labeling. This strategy effectively reduces the runtime of subgraph methods. Specifically, we view the shortest paths between two nodes as the spatial graph context of the neighborhood around them. The GNN embeddings of nodes on the shortest paths are used to generate geodesic representations. Conditioned on the geodesic representations, GDGNN can generate node, link, and graph representations that carry much richer structural information than plain GNNs. We theoretically prove that GDGNN is more powerful than plain GNNs. We present experimental results to show that GDGNN achieves highly competitive performance with SOTA GNN models on various graph learning tasks while taking significantly less time.

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.

Hao Liu, Jiarui Feng, Lecheng Kong et al.

Graph Neural Networks (GNNs) have become popular in Graph Representation Learning (GRL). One fundamental application is few-shot node classification. Most existing methods follow the meta learning paradigm, showing the ability of fast generalization to few-shot tasks. However, recent works indicate that graph contrastive learning combined with fine-tuning can significantly outperform meta learning methods. Despite the empirical success, there is limited understanding of the reasons behind it. In our study, we first identify two crucial advantages of contrastive learning compared to meta learning, including (1) the comprehensive utilization of graph nodes and (2) the power of graph augmentations. To integrate the strength of both contrastive learning and meta learning on the few-shot node classification tasks, we introduce a new paradigm: Contrastive Few-Shot Node Classification (COLA). Specifically, COLA employs graph augmentations to identify semantically similar nodes, which enables the construction of meta-tasks without the need for label information. Therefore, COLA can utilize all nodes to construct meta-tasks, further reducing the risk of overfitting. Through extensive experiments, we validate the essentiality of each component in our design and demonstrate that COLA achieves new state-of-the-art on all tasks.

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.

Muhan Zhang

In the realm of deep learning, the self-attention mechanism has substantiated its pivotal role across a myriad of tasks, encompassing natural language processing and computer vision. Despite achieving success across diverse applications, the traditional self-attention mechanism primarily leverages linear transformations for the computation of query, key, and value (QKV), which may not invariably be the optimal choice under specific circumstances. This paper probes into a novel methodology for QKV computation-implementing a specially-designed neural network structure for the calculation. Utilizing a modified Marian model, we conducted experiments on the IWSLT 2017 German-English translation task dataset and juxtaposed our method with the conventional approach. The experimental results unveil a significant enhancement in BLEU scores with our method. Furthermore, our approach also manifested superiority when training the Roberta model with the Wikitext-103 dataset, reflecting a notable reduction in model perplexity compared to its original counterpart. These experimental outcomes not only validate the efficacy of our method but also reveal the immense potential in optimizing the self-attention mechanism through neural network-based QKV computation, paving the way for future research and practical applications. The source code and implementation details for our proposed method can be accessed at https://github.com/ocislyjrti/NeuralAttention.

Xiyuan Wang, Muhan Zhang

Modeling molecular potential energy surface is of pivotal importance in science. Graph Neural Networks have shown great success in this field. However, their message passing schemes need special designs to capture geometric information and fulfill symmetry requirement like rotation equivariance, leading to complicated architectures. To avoid these designs, we introduce a novel local frame method to molecule representation learning and analyze its expressivity. Projected onto a frame, equivariant features like 3D coordinates are converted to invariant features, so that we can capture geometric information with these projections and decouple the symmetry requirement from GNN design. Theoretically, we prove that given non-degenerate frames, even ordinary GNNs can encode molecules injectively and reach maximum expressivity with coordinate projection and frame-frame projection. In experiments, our model uses a simple ordinary GNN architecture yet achieves state-of-the-art accuracy. The simpler architecture also leads to higher scalability. Our model only takes about 30% inference time and 10% GPU memory compared to the most efficient baselines.

Hao Liu, Muhan Zhang, Zehao Dong et al.

Rich Electronic Health Records (EHR), have created opportunities to improve clinical processes using machine learning methods. Prediction of the same patient events at different time horizons can have very different applications and interpretations; however, limited number of events in each potential time window hurts the effectiveness of conventional machine learning algorithms. We propose a novel time associated meta learning (TAML) method to make effective predictions at multiple future time points. We view time-associated disease prediction as classification tasks at multiple time points. Such closely-related classification tasks are an excellent candidate for model-based meta learning. To address the sparsity problem after task splitting, TAML employs a temporal information sharing strategy to augment the number of positive samples and include the prediction of related phenotypes or events in the meta-training phase. We demonstrate the effectiveness of TAML on multiple clinical datasets, where it consistently outperforms a range of strong baselines. We also develop a MetaEHR package for implementing both time-associated and time-independent few-shot prediction on EHR data.

Lecheng Kong, Jiarui Feng, Hao Liu et al.

While Graph Neural Networks (GNNs) recently became powerful tools in graph learning tasks, considerable efforts have been spent on improving GNNs' structural encoding ability. A particular line of work proposed subgraph GNNs that use subgraph information to improve GNNs' expressivity and achieved great success. However, such effectivity sacrifices the efficiency of GNNs by enumerating all possible subgraphs. In this paper, we analyze the necessity of complete subgraph enumeration and show that a model can achieve a comparable level of expressivity by considering a small subset of the subgraphs. We then formulate the identification of the optimal subset as a combinatorial optimization problem and propose Magnetic Graph Neural Network (MAG-GNN), a reinforcement learning (RL) boosted GNN, to solve the problem. Starting with a candidate subgraph set, MAG-GNN employs an RL agent to iteratively update the subgraphs to locate the most expressive set for prediction. This reduces the exponential complexity of subgraph enumeration to the constant complexity of a subgraph search algorithm while keeping good expressivity. We conduct extensive experiments on many datasets, showing that MAG-GNN achieves competitive performance to state-of-the-art methods and even outperforms many subgraph GNNs. We also demonstrate that MAG-GNN effectively reduces the running time of subgraph GNNs.

Haotong Yang, Fanxu Meng, Zhouchen Lin et al.

The pre-trained large language models (LLMs) have shown their extraordinary capacity to solve reasoning tasks, even on tasks that require a complex process involving multiple sub-steps. However, given the vast possible generation space of all the tasks, how the pretrained model learns the reasoning ability remains an open question. We firstly propose that an intrinsic structural constraint on the generated sequence of language-based reasoning -- we called it template-content structure (T-C structure) -- is the key to explain why LLMs can solve a large number of complex reasoning problems with limited training data by showing this structure can reduce the possible space from exponential level to linear level. Furthermore, by generalizing this structure to the hierarchical case, we demonstrate that models can achieve task composition, further reducing the space needed to learn from linear to logarithmic, thereby effectively learning on complex reasoning involving multiple steps. We provide both examples and formal theory of our T-C structure. We also experimentally validate the existence of the T-C structure in some current LLMs and its effectiveness for reasoning.

Jiarui Feng, Hao Liu, Lecheng Kong et al.

In this report, we present TAGLAS, an atlas of text-attributed graph (TAG) datasets and benchmarks. TAGs are graphs with node and edge features represented in text, which have recently gained wide applicability in training graph-language or graph foundation models. In TAGLAS, we collect and integrate more than 23 TAG datasets with domains ranging from citation graphs to molecule graphs and tasks from node classification to graph question-answering. Unlike previous graph datasets and benchmarks, all datasets in TAGLAS have a unified node and edge text feature format, which allows a graph model to be simultaneously trained and evaluated on multiple datasets from various domains. Further, we provide a standardized, efficient, and simplified way to load all datasets and tasks. We also provide useful utils like text-to-embedding conversion, and graph-to-text conversion, which can facilitate different evaluation scenarios. Finally, we also provide standard and easy-to-use evaluation utils. The project is open-sourced at https://github.com/JiaruiFeng/TAGLAS and is still under construction. Please expect more datasets/features in the future.

Muhan Zhang

In this draft, we study a novel problem, called lexical invariance, using the medium of multisets and graphs. Traditionally in the NLP domain, lexical invariance indicates that the semantic meaning of a sentence should remain unchanged regardless of the specific lexical or word-based representation of the input. For example, ``The movie was extremely entertaining'' would have the same meaning as ``The film was very enjoyable''. In this paper, we study a more challenging setting, where the output of a function is invariant to any injective transformation applied to the input lexical space. For example, multiset {1,2,3,2} is equivalent to multiset {a,b,c,b} if we specify an injective transformation that maps 1 to a, 2 to b and 3 to c. We study the sufficient and necessary conditions for a most expressive lexical invariant (and permutation invariant) function on multisets and graphs, and proves that for multisets, the function must have a form that only takes the multiset of counts of the unique elements in the original multiset as input. For example, a most expressive lexical invariant function on {a,b,c,b} must have a form that only operates on {1,1,2} (meaning that there are 1, 1, 2 unique elements corresponding to a,c,b). For graphs, we prove that a most expressive lexical invariant and permutation invariant function must have a form that only takes the adjacency matrix and a difference matrix as input, where the (i,j)th element of the difference matrix is 1 if node i and node j have the same feature and 0 otherwise. We perform synthetic experiments on TU datasets to verify our theorems.

Shi Qiu, Shaoyang Guo, Zhuo-Yang Song et al.

Current benchmarks for evaluating the reasoning capabilities of Large Language Models (LLMs) face significant limitations: task oversimplification, data contamination, and flawed evaluation items. These deficiencies necessitate more rigorous assessment methods. To address these limitations, we introduce PHYBench, a benchmark of 500 original physics problems ranging from high school to Physics Olympiad difficulty. PHYBench addresses data contamination through original content and employs a systematic curation pipeline to eliminate flawed items. Evaluations show that PHYBench activates more tokens and provides stronger differentiation between reasoning models compared to other baselines like AIME 2024, OlympiadBench and GPQA. Even the best-performing model, Gemini 2.5 Pro, achieves only 36.9% accuracy compared to human experts' 61.9%. To further enhance evaluation precision, we introduce the Expression Edit Distance (EED) Score for mathematical expression assessment, which improves sample efficiency by 204% over binary scoring. Moreover, PHYBench effectively elicits multi-step and multi-condition reasoning, providing a platform for examining models' reasoning robustness, preferences, and deficiencies. The benchmark results and dataset are publicly available at https://www.phybench.cn/.

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.

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.

Yi Hu, Shijia Kang, Haotong Yang et al.

Length generalization, the ability to solve problems longer than those seen during training, remains a critical challenge for large language models (LLMs). Previous work modifies positional encodings (PEs) and data formats to improve length generalization on specific symbolic tasks such as addition and sorting. However, these approaches are fundamentally limited to special tasks, often degrading general language performance. Furthermore, they are typically evaluated on small transformers trained from scratch on single tasks and can cause performance drop when applied during post-training stage of practical LLMs with general capabilities. Hu et al., (2024) proposed Rule-Following Fine-Tuning (RFFT) to improve length generalization in the post-training stage of LLMs. Despite its compatibility with practical models and strong performance, RFFT is proposed for single tasks too, requiring re-training for each individual task with extensive examples. In this paper, we study length generalization in multi-task settings and propose Meta Rule-Following Fine-Tuning (Meta-RFFT), the first framework enabling robust cross-task length generalization. As our first contribution, we construct a large length generalization dataset containing 86 tasks spanning code execution, number processing, symbolic and logical reasoning tasks, beyond the common addition or multiplication tasks. Secondly, we show that cross-task length generalization is possible with Meta-RFFT. After training on a large number of tasks and instances, the models achieve remarkable length generalization ability on unseen tasks with minimal fine-tuning or one-shot prompting. For example, after fine-tuning on 1 to 5 digit addition, our 32B model achieves 95% accuracy on 30 digit addition, significantly outperforming the state-of-the-art reasoning models (DeepSeek-R1-671B: 72%), despite never seeing this task during RF-pretraining.

Zian Li, Muhan Zhang

Masked autoregressive models (MAR) have recently emerged as a powerful paradigm for image and video generation, combining the flexibility of masked modeling with the potential of continuous tokenizer. However, video MAR models suffer from two major limitations: the slow-start problem, caused by the lack of a structured global prior at early sampling stages, and error accumulation across the autoregression in both spatial and temporal dimensions. In this work, we propose CanvasMAR, a novel video MAR model that mitigates these issues by introducing a canvas mechanism--a blurred, global prediction of the next frame, used as the starting point for masked generation. The canvas provides global structure early in sampling, enabling faster and more coherent frame synthesis. Furthermore, we introduce compositional classifier-free guidance that jointly enlarges spatial (canvas) and temporal conditioning, and employ noise-based canvas augmentation to enhance robustness. Experiments on the BAIR and Kinetics-600 benchmarks demonstrate that CanvasMAR produces high-quality videos with fewer autoregressive steps. Our approach achieves remarkable performance among autoregressive models on Kinetics-600 dataset and rivals diffusion-based methods.

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.

Lecheng Kong, Xiyuan Wang, Yixin Chen et al.

Large Language Models (LLMs) are emerging as versatile foundation models for computational chemistry, handling bidirectional tasks like reaction prediction and retrosynthesis. However, these models often lack round-trip consistency. For instance, a state-of-the-art chemical LLM may successfully caption a molecule, yet be unable to accurately reconstruct the original structure from its own generated text. This inconsistency suggests that models are learning unidirectional memorization rather than flexible mastery. Indeed, recent work has demonstrated a strong correlation between a model's round-trip consistency and its performance on the primary tasks. This strong correlation reframes consistency into a direct target for model improvement. We therefore introduce Round-Trip Reinforcement Learning (RTRL), a novel framework that trains a model to improve its consistency by using the success of a round-trip transformation as a reward signal. We further propose an iterative variant where forward and reverse mappings alternately train each other in a self-improvement loop, a process that is highly data-efficient and notably effective with the massive amount of unlabelled data common in chemistry. Experiments demonstrate that RTRL significantly \textbf{boosts performance and consistency} over strong baselines across supervised, self-supervised, and synthetic data regimes. This work shows that round-trip consistency is not just a desirable property but a trainable objective, offering a new path toward more robust and reliable foundation models.

Zehao Dong, Muhan Zhang, Yixin Chen

Graph neural networks (GNNs) have revolutionized the field of machine learning on non-Euclidean data such as graphs and networks. GNNs effectively implement node representation learning through neighborhood aggregation and achieve impressive results in many graph-related tasks. However, most neighborhood aggregation approaches are summation-based, which can be problematic as they may not be sufficiently expressive to encode informative graph structures. Furthermore, though the graph pooling module is also of vital importance for graph learning, especially for the task of graph classification, research on graph down-sampling mechanisms is rather limited. To address the above challenges, we propose a concatenation-based graph convolution mechanism that injectively updates node representations to maximize the discriminative power in distinguishing non-isomorphic subgraphs. In addition, we design a novel graph pooling module, called WL-SortPool, to learn important subgraph patterns in a deep-learning manner. WL-SortPool layer-wise sorts node representations (i.e. continuous WL colors) to separately learn the relative importance of subtrees with different depths for the purpose of classification, thus better characterizing the complex graph topology and rich information encoded in the graph. We propose a novel Subgraph Pattern GNN (SPGNN) architecture that incorporates these enhancements. We test the proposed SPGNN architecture on many graph classification benchmarks. Experimental results show that our method can achieve highly competitive results with state-of-the-art graph kernels and other GNN approaches.

Zehao Dong, Muhan Zhang, Philip R. O. Payne et al.

The expressivity of Graph Neural Networks (GNNs) has been studied broadly in recent years to reveal the design principles for more powerful GNNs. Graph canonization is known as a typical approach to distinguish non-isomorphic graphs, yet rarely adopted when developing expressive GNNs. This paper proposes to maximize the expressivity of GNNs by graph canonization, then the power of such GNNs is studies from the perspective of model stability. A stable GNN will map similar graphs to close graph representations in the vectorial space, and the stability of GNNs is critical to generalize their performance to unseen graphs. We theoretically reveal the trade-off of expressivity and stability in graph-canonization-enhanced GNNs. Then we introduce a notion of universal graph canonization as the general solution to address the trade-off and characterize a widely applicable sufficient condition to solve the universal graph canonization. A comprehensive set of experiments demonstrates the effectiveness of the proposed method. In many popular graph benchmark datasets, graph canonization successfully enhances GNNs and provides highly competitive performance, indicating the capability and great potential of proposed method in general graph representation learning. In graph datasets where the sufficient condition holds, GNNs enhanced by universal graph canonization consistently outperform GNN baselines and successfully improve the SOTA performance up to $31\%$, providing the optimal solution to numerous challenging real-world graph analytical tasks like gene network representation learning in bioinformatics.

Yi Hu, Haotong Yang, Zhouchen Lin et al.

Large language models (LLMs) have scaled up to unlock a wide range of complex reasoning tasks with the aid of various prompting methods. However, current prompting methods generate natural language intermediate steps to help reasoning, which can cause imperfect task reduction and confusion. To mitigate such limitations, we explore code prompting, a neural symbolic prompting method with both zero-shot and few-shot versions which triggers code as intermediate steps. We conduct experiments on 7 widely-used benchmarks involving symbolic reasoning and arithmetic reasoning. Code prompting generally outperforms chain-of-thought (CoT) prompting. To further understand the performance and limitations of code prompting, we perform extensive ablation studies and error analyses, and identify several exclusive advantages of using symbolic promptings compared to natural language. We also consider the ensemble of code prompting and CoT prompting to combine the strengths of both. Finally, we show through experiments how code annotations and their locations affect code prompting.

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.

Haoteng Yin, Muhan Zhang, Yanbang Wang et al.

Subgraph-based graph representation learning (SGRL) has been recently proposed to deal with some fundamental challenges encountered by canonical graph neural networks (GNNs), and has demonstrated advantages in many important data science applications such as link, relation and motif prediction. However, current SGRL approaches suffer from scalability issues since they require extracting subgraphs for each training or test query. Recent solutions that scale up canonical GNNs may not apply to SGRL. Here, we propose a novel framework SUREL for scalable SGRL by co-designing the learning algorithm and its system support. SUREL adopts walk-based decomposition of subgraphs and reuses the walks to form subgraphs, which substantially reduces the redundancy of subgraph extraction and supports parallel computation. Experiments over six homogeneous, heterogeneous and higher-order graphs with millions of nodes and edges demonstrate the effectiveness and scalability of SUREL. In particular, compared to SGRL baselines, SUREL achieves 10$\times$ speed-up with comparable or even better prediction performance; while compared to canonical GNNs, SUREL achieves 50% prediction accuracy improvement.

Hanqing Zeng, Muhan Zhang, Yinglong Xia et al.

State-of-the-art Graph Neural Networks (GNNs) have limited scalability with respect to the graph and model sizes. On large graphs, increasing the model depth often means exponential expansion of the scope (i.e., receptive field). Beyond just a few layers, two fundamental challenges emerge: 1. degraded expressivity due to oversmoothing, and 2. expensive computation due to neighborhood explosion. We propose a design principle to decouple the depth and scope of GNNs -- to generate representation of a target entity (i.e., a node or an edge), we first extract a localized subgraph as the bounded-size scope, and then apply a GNN of arbitrary depth on top of the subgraph. A properly extracted subgraph consists of a small number of critical neighbors, while excluding irrelevant ones. The GNN, no matter how deep it is, smooths the local neighborhood into informative representation rather than oversmoothing the global graph into "white noise". Theoretically, decoupling improves the GNN expressive power from the perspectives of graph signal processing (GCN), function approximation (GraphSAGE) and topological learning (GIN). Empirically, on seven graphs (with up to 110M nodes) and six backbone GNN architectures, our design achieves significant accuracy improvement with orders of magnitude reduction in computation and hardware cost.

Xinshi Chen, Yan Zhu, Haowen Xu et al.

Dynamic graphs with ordered sequences of events between nodes are prevalent in real-world industrial applications such as e-commerce and social platforms. However, representation learning for dynamic graphs has posed great computational challenges due to the time and structure dependency and irregular nature of the data, preventing such models from being deployed to real-world applications. To tackle this challenge, we propose an efficient algorithm, Efficient Dynamic Graph lEarning (EDGE), which selectively expresses certain temporal dependency via training loss to improve the parallelism in computations. We show that EDGE can scale to dynamic graphs with millions of nodes and hundreds of millions of temporal events and achieve new state-of-the-art (SOTA) performance.

Xiang Song, Runjie Ma, Jiahang Li et al.

Graph Neural Networks (GNNs) have shown success in learning from graph structured data containing node/edge feature information, with application to social networks, recommendation, fraud detection and knowledge graph reasoning. In this regard, various strategies have been proposed in the past to improve the expressiveness of GNNs. For example, one straightforward option is to simply increase the parameter size by either expanding the hid-den dimension or increasing the number of GNN layers. However, wider hidden layers can easily lead to overfitting, and incrementally adding more GNN layers can potentially result in over-smoothing.In this paper, we present a model-agnostic methodology, namely Network In Graph Neural Network (NGNN ), that allows arbitrary GNN models to increase their model capacity by making the model deeper. However, instead of adding or widening GNN layers, NGNN deepens a GNN model by inserting non-linear feedforward neural network layer(s) within each GNN layer. An analysis of NGNN as applied to a GraphSage base GNN on ogbn-products data demonstrate that it can keep the model stable against either node feature or graph structure perturbations. Furthermore, wide-ranging evaluation results on both node classification and link prediction tasks show that NGNN works reliably across diverse GNN architectures.For instance, it improves the test accuracy of GraphSage on the ogbn-products by 1.6% and improves the hits@100 score of SEAL on ogbl-ppa by 7.08% and the hits@20 score of GraphSage+Edge-Attr on ogbl-ppi by 6.22%. And at the time of this submission, it achieved two first places on the OGB link prediction leaderboard.

Muhan Zhang, Pan Li, Yinglong Xia et al.

In this paper, we provide a theory of using graph neural networks (GNNs) for multi-node representation learning (where we are interested in learning a representation for a set of more than one node, such as link). We know that GNN is designed to learn single-node representations. When we want to learn a node set representation involving multiple nodes, a common practice in previous works is to directly aggregate the single-node representations obtained by a GNN into a joint node set representation. In this paper, we show a fundamental constraint of such an approach, namely the inability to capture the dependence between nodes in the node set, and argue that directly aggregating individual node representations does not lead to an effective joint representation for multiple nodes. Then, we notice that a few previous successful works for multi-node representation learning, including SEAL, Distance Encoding, and ID-GNN, all used node labeling. These methods first label nodes in the graph according to their relationships with the target node set before applying a GNN. Then, the node representations obtained in the labeled graph are aggregated into a node set representation. By investigating their inner mechanisms, we unify these node labeling techniques into a single and most general form -- labeling trick. We prove that with labeling trick a sufficiently expressive GNN learns the most expressive node set representations, thus in principle solves any joint learning tasks over node sets. Experiments on one important two-node representation learning task, link prediction, verified our theory. Our work explains the superior performance of previous node-labeling-based methods, and establishes a theoretical foundation of using GNNs for multi-node representation learning.

Xiaoxiao Li, Yuan Zhou, Nicha C. Dvornek et al.

Understanding how certain brain regions relate to a specific neurological disorder has been an important area of neuroimaging research. A promising approach to identify the salient regions is using Graph Neural Networks (GNNs), which can be used to analyze graph structured data, e.g. brain networks constructed by functional magnetic resonance imaging (fMRI). We propose an interpretable GNN framework with a novel salient region selection mechanism to determine neurological brain biomarkers associated with disorders. Specifically, we design novel regularized pooling layers that highlight salient regions of interests (ROIs) so that we can infer which ROIs are important to identify a certain disease based on the node pooling scores calculated by the pooling layers. Our proposed framework, Pooling Regularized-GNN (PR-GNN), encourages reasonable ROI-selection and provides flexibility to preserve either individual- or group-level patterns. We apply the PR-GNN framework on a Biopoint Autism Spectral Disorder (ASD) fMRI dataset. We investigate different choices of the hyperparameters and show that PR-GNN outperforms baseline methods in terms of classification accuracy. The salient ROI detection results show high correspondence with the previous neuroimaging-derived biomarkers for ASD.