Graph transformers typically lack third-order interactions, limiting their geometric understanding which is crucial for tasks like molecular geometry prediction. We propose the Triplet Graph Transformer (TGT) that enables direct communication between pairs within a 3-tuple of nodes via novel triplet attention and aggregation mechanisms. TGT is applied to molecular property prediction by first predicting interatomic distances from 2D graphs and then using these distances for downstream tasks. A novel three-stage training procedure and stochastic inference further improve training efficiency and model performance. Our model achieves new state-of-the-art (SOTA) results on open challenge benchmarks PCQM4Mv2 and OC20 IS2RE. We also obtain SOTA results on QM9, MOLPCBA, and LIT-PCBA molecular property prediction benchmarks via transfer learning. We also demonstrate the generality of TGT with SOTA results on the traveling salesman problem (TSP).
Supervised learning has recently garnered significant attention in the field of computational physics due to its ability to effectively extract complex patterns for tasks like solving partial differential equations, or predicting material properties. Traditionally, such datasets consist of inputs given as meshes with a large number of nodes representing the problem geometry (seen as graphs), and corresponding outputs obtained with a numerical solver. This means the supervised learning model must be able to handle large and sparse graphs with continuous node attributes. In this work, we focus on Gaussian process regression, for which we introduce the Sliced Wasserstein Weisfeiler-Lehman (SWWL) graph kernel. In contrast to existing graph kernels, the proposed SWWL kernel enjoys positive definiteness and a drastic complexity reduction, which makes it possible to process datasets that were previously impossible to handle. The new kernel is first validated on graph classification for molecular datasets, where the input graphs have a few tens of nodes. The efficiency of the SWWL kernel is then illustrated on graph regression in computational fluid dynamics and solid mechanics, where the input graphs are made up of tens of thousands of nodes.
Transformers have revolutionized performance in Natural Language Processing and Vision, paving the way for their integration with Graph Neural Networks (GNNs). One key challenge in enhancing graph transformers is strengthening the discriminative power of distinguishing isomorphisms of graphs, which plays a crucial role in boosting their predictive performances. To address this challenge, we introduce 'Topology-Informed Graph Transformer (TIGT)', a novel transformer enhancing both discriminative power in detecting graph isomorphisms and the overall performance of Graph Transformers. TIGT consists of four components: A topological positional embedding layer using non-isomorphic universal covers based on cyclic subgraphs of graphs to ensure unique graph representation: A dual-path message-passing layer to explicitly encode topological characteristics throughout the encoder layers: A global attention mechanism: And a graph information layer to recalibrate channel-wise graph features for better feature representation. TIGT outperforms previous Graph Transformers in classifying synthetic dataset aimed at distinguishing isomorphism classes of graphs. Additionally, mathematical analysis and empirical evaluations highlight our model's competitive edge over state-of-the-art Graph Transformers across various benchmark datasets.
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.
Spectral embedding is a powerful graph embedding technique that has received a lot of attention recently due to its effectiveness on Graph Transformers. However, from a theoretical perspective, the universal expressive power of spectral embedding comes at the price of losing two important invariance properties of graphs, sign and basis invariance, which also limits its effectiveness on graph data. To remedy this issue, many previous methods developed costly approaches to learn new invariants and suffer from high computation complexity. In this work, we explore a minimal approach that resolves the ambiguity issues by directly finding canonical directions for the eigenvectors, named Laplacian Canonization (LC). As a pure pre-processing method, LC is light-weighted and can be applied to any existing GNNs. We provide a thorough investigation, from theory to algorithm, on this approach, and discover an efficient algorithm named Maximal Axis Projection (MAP) that works for both sign and basis invariance and successfully canonizes more than 90% of all eigenvectors. Experiments on real-world benchmark datasets like ZINC, MOLTOX21, and MOLPCBA show that MAP consistently outperforms existing methods while bringing minimal computation overhead. Code is available at https://github.com/PKU-ML/LaplacianCanonization.
We present a novel edge-level ego-network encoding for learning on graphs that can boost Message Passing Graph Neural Networks (MP-GNNs) by providing additional node and edge features or extending message-passing formats. The proposed encoding is sufficient to distinguish Strongly Regular Graphs, a family of challenging 3-WL equivalent graphs. We show theoretically that such encoding is more expressive than node-based sub-graph MP-GNNs. In an empirical evaluation on four benchmarks with 10 graph datasets, our results match or improve previous baselines on expressivity, graph classification, graph regression, and proximity tasks -- while reducing memory usage by 18.1x in certain real-world settings.
Transformers, renowned for their self-attention mechanism, have achieved state-of-the-art performance across various tasks in natural language processing, computer vision, time-series modeling, etc. However, one of the challenges with deep Transformer models is the oversmoothing problem, where representations across layers converge to indistinguishable values, leading to significant performance degradation. We interpret the original self-attention as a simple graph filter and redesign it from a graph signal processing (GSP) perspective. We propose a graph-filter-based self-attention (GFSA) to learn a general yet effective one, whose complexity, however, is slightly larger than that of the original self-attention mechanism. We demonstrate that GFSA improves the performance of Transformers in various fields, including computer vision, natural language processing, graph-level tasks, speech recognition, and code classification.
We present a novel framework to overcome the limitations of equivariant architectures in learning functions with group symmetries. In contrary to equivariant architectures, we use an arbitrary base model such as an MLP or a transformer and symmetrize it to be equivariant to the given group by employing a small equivariant network that parameterizes the probabilistic distribution underlying the symmetrization. The distribution is end-to-end trained with the base model which can maximize performance while reducing sample complexity of symmetrization. We show that this approach ensures not only equivariance to given group but also universal approximation capability in expectation. We implement our method on various base models, including patch-based transformers that can be initialized from pretrained vision transformers, and test them for a wide range of symmetry groups including permutation and Euclidean groups and their combinations. Empirical tests show competitive results against tailored equivariant architectures, suggesting the potential for learning equivariant functions for diverse groups using a non-equivariant universal base architecture. We further show evidence of enhanced learning in symmetric modalities, like graphs, when pretrained from non-symmetric modalities, like vision. Code is available at https://github.com/jw9730/lps.
This work analyzes Graph Neural Networks, a generalization of Fully-Connected Deep Neural Nets on Graph structured data, when their width, that is the number of nodes in each fullyconnected layer is increasing to infinity. Infinite Width Neural Networks are connecting Deep Learning to Gaussian Processes and Kernels, both Machine Learning Frameworks with long traditions and extensive theoretical foundations. Gaussian Processes and Kernels have much less hyperparameters then Neural Networks and can be used for uncertainty estimation, making them more user friendly for applications. This works extends the increasing amount of research connecting Gaussian Processes and Kernels to Neural Networks. The Kernel and Gaussian Process closed forms are derived for a variety of architectures, namely the standard Graph Neural Network, the Graph Neural Network with Skip-Concatenate Connections and the Graph Attention Neural Network. All architectures are evaluated on a variety of datasets on the task of transductive Node Regression and Classification. Additionally, a Spectral Sparsification method known as Effective Resistance is used to improve runtime and memory requirements. Extending the setting to inductive graph learning tasks (Graph Regression/ Classification) is straightforward and is briefly discussed in 3.5.
Joint-Embedding Predictive Architectures (JEPAs) have recently emerged as a novel and powerful technique for self-supervised representation learning. They aim to learn an energy-based model by predicting the latent representation of a target signal y from the latent representation of a context signal x. JEPAs bypass the need for negative and positive samples, traditionally required by contrastive learning while avoiding the overfitting issues associated with generative pretraining. In this paper, we show that graph-level representations can be effectively modeled using this paradigm by proposing a Graph Joint-Embedding Predictive Architecture (Graph-JEPA). In particular, we employ masked modeling and focus on predicting the latent representations of masked subgraphs starting from the latent representation of a context subgraph. To endow the representations with the implicit hierarchy that is often present in graph-level concepts, we devise an alternative prediction objective that consists of predicting the coordinates of the encoded subgraphs on the unit hyperbola in the 2D plane. Through multiple experimental evaluations, we show that Graph-JEPA can learn highly semantic and expressive representations, as shown by the downstream performance in graph classification, regression, and distinguishing non-isomorphic graphs. The code is available at https://github.com/geriskenderi/graph-jepa.
The recent Long-Range Graph Benchmark (LRGB, Dwivedi et al. 2022) introduced a set of graph learning tasks strongly dependent on long-range interaction between vertices. Empirical evidence suggests that on these tasks Graph Transformers significantly outperform Message Passing GNNs (MPGNNs). In this paper, we carefully reevaluate multiple MPGNN baselines as well as the Graph Transformer GPS (Ramp\'a\v{s}ek et al. 2022) on LRGB. Through a rigorous empirical analysis, we demonstrate that the reported performance gap is overestimated due to suboptimal hyperparameter choices. It is noteworthy that across multiple datasets the performance gap completely vanishes after basic hyperparameter optimization. In addition, we discuss the impact of lacking feature normalization for LRGB's vision datasets and highlight a spurious implementation of LRGB's link prediction metric. The principal aim of our paper is to establish a higher standard of empirical rigor within the graph machine learning community.
Graph transformers need strong inductive biases to derive meaningful attention scores. Yet, current methods often fall short in capturing longer ranges, hierarchical structures, or community structures, which are common in various graphs such as molecules, social networks, and citation networks. This paper presents a Hierarchical Distance Structural Encoding (HDSE) method to model node distances in a graph, focusing on its multi-level, hierarchical nature. We introduce a novel framework to seamlessly integrate HDSE into the attention mechanism of existing graph transformers, allowing for simultaneous application with other positional encodings. To apply graph transformers with HDSE to large-scale graphs, we further propose a high-level HDSE that effectively biases the linear transformers towards graph hierarchies. We theoretically prove the superiority of HDSE over shortest path distances in terms of expressivity and generalization. Empirically, we demonstrate that graph transformers with HDSE excel in graph classification, regression on 7 graph-level datasets, and node classification on 11 large-scale graphs, including those with up to a billion nodes.
We propose an extension of the Contextual Graph Markov Model, a deep and probabilistic machine learning model for graphs, to model the distribution of edge features. Our approach is architectural, as we introduce an additional Bayesian network mapping edge features into discrete states to be used by the original model. In doing so, we are also able to build richer graph representations even in the absence of edge features, which is confirmed by the performance improvements on standard graph classification benchmarks. Moreover, we successfully test our proposal in a graph regression scenario where edge features are of fundamental importance, and we show that the learned edge representation provides substantial performance improvements against the original model on three link prediction tasks. By keeping the computational complexity linear in the number of edges, the proposed model is amenable to large-scale graph processing.
We introduce DeSCo, a scalable neural deep subgraph counting pipeline, designed to accurately predict both the count and occurrence position of queries on target graphs post single training. Firstly, DeSCo uses a novel canonical partition and divides the large target graph into small neighborhood graphs, greatly reducing the count variation while guaranteeing no missing or double-counting. Secondly, neighborhood counting uses an expressive subgraph-based heterogeneous graph neural network to accurately count in each neighborhood. Finally, gossip propagation propagates neighborhood counts with learnable gates to harness the inductive biases of motif counts. DeSCo is evaluated on eight real-world datasets from various domains. It outperforms state-of-the-art neural methods with 137x improvement in the mean squared error of count prediction, while maintaining the polynomial runtime complexity. Our open source project is at https://github.com/fuvty/DeSCo.
Recently, Graph Neural Networks (GNNs) have shown promising performance in tasks on dynamic graphs such as node classification, link prediction and graph regression. However, few work has studied the temporal edge regression task which has important real-world applications. In this paper, we explore the application of GNNs to edge regression tasks in both static and dynamic settings, focusing on predicting food and agriculture trade values between nations. We introduce three simple yet strong baselines and comprehensively evaluate one static and three dynamic GNN models using the UN Trade dataset. Our experimental results reveal that the baselines exhibit remarkably strong performance across various settings, highlighting the inadequacy of existing GNNs. We also find that TGN outperforms other GNN models, suggesting TGN is a more appropriate choice for edge regression tasks. Moreover, we note that the proportion of negative edges in the training samples significantly affects the test performance. The companion source code can be found at: https://github.com/scylj1/GNN_Edge_Regression.
Graph regression is a fundamental task that has gained significant attention in various graph learning tasks. However, the inference process is often not easily interpretable. Current explanation techniques are limited to understanding Graph Neural Network (GNN) behaviors in classification tasks, leaving an explanation gap for graph regression models. In this work, we propose a novel explanation method to interpret the graph regression models (XAIG-R). Our method addresses the distribution shifting problem and continuously ordered decision boundary issues that hinder existing methods away from being applied in regression tasks. We introduce a novel objective based on the graph information bottleneck theory (GIB) and a new mix-up framework, which can support various GNNs and explainers in a model-agnostic manner. Additionally, we present a self-supervised learning strategy to tackle the continuously ordered labels in regression tasks. We evaluate our proposed method on three benchmark datasets and a real-life dataset introduced by us, and extensive experiments demonstrate its effectiveness in interpreting GNN models in regression tasks.
Graph neural networks (GNNs) have recently become the standard approach for learning with graph-structured data. Prior work has shed light into their potential, but also their limitations. Unfortunately, it was shown that standard GNNs are limited in their expressive power. These models are no more powerful than the 1-dimensional Weisfeiler-Leman (1-WL) algorithm in terms of distinguishing non-isomorphic graphs. In this paper, we propose Path Neural Networks (PathNNs), a model that updates node representations by aggregating paths emanating from nodes. We derive three different variants of the PathNN model that aggregate single shortest paths, all shortest paths and all simple paths of length up to K. We prove that two of these variants are strictly more powerful than the 1-WL algorithm, and we experimentally validate our theoretical results. We find that PathNNs can distinguish pairs of non-isomorphic graphs that are indistinguishable by 1-WL, while our most expressive PathNN variant can even distinguish between 3-WL indistinguishable graphs. The different PathNN variants are also evaluated on graph classification and graph regression datasets, where in most cases, they outperform the baseline methods.
Transformers use the dense self-attention mechanism which gives a lot of flexibility for long-range connectivity. Over multiple layers of a deep transformer, the number of possible connectivity patterns increases exponentially. However, very few of these contribute to the performance of the network, and even fewer are essential. We hypothesize that there are sparsely connected sub-networks within a transformer, called information pathways which can be trained independently. However, the dynamic (i.e., input-dependent) nature of these pathways makes it difficult to prune dense self-attention during training. But the overall distribution of these pathways is often predictable. We take advantage of this fact to propose Stochastically Subsampled self-Attention (SSA) - a general-purpose training strategy for transformers that can reduce both the memory and computational cost of self-attention by 4 to 8 times during training while also serving as a regularization method - improving generalization over dense training. We show that an ensemble of sub-models can be formed from the subsampled pathways within a network, which can achieve better performance than its densely attended counterpart. We perform experiments on a variety of NLP, computer vision and graph learning tasks in both generative and discriminative settings to provide empirical evidence for our claims and show the effectiveness of the proposed method.
Transformers for graph data are increasingly widely studied and successful in numerous learning tasks. Graph inductive biases are crucial for Graph Transformers, and previous works incorporate them using message-passing modules and/or positional encodings. However, Graph Transformers that use message-passing inherit known issues of message-passing, and differ significantly from Transformers used in other domains, thus making transfer of research advances more difficult. On the other hand, Graph Transformers without message-passing often perform poorly on smaller datasets, where inductive biases are more crucial. To bridge this gap, we propose the Graph Inductive bias Transformer (GRIT) -- a new Graph Transformer that incorporates graph inductive biases without using message passing. GRIT is based on several architectural changes that are each theoretically and empirically justified, including: learned relative positional encodings initialized with random walk probabilities, a flexible attention mechanism that updates node and node-pair representations, and injection of degree information in each layer. We prove that GRIT is expressive -- it can express shortest path distances and various graph propagation matrices. GRIT achieves state-of-the-art empirical performance across a variety of graph datasets, thus showing the power that Graph Transformers without message-passing can deliver.
Data imbalance is easily found in annotated data when the observations of certain continuous label values are difficult to collect for regression tasks. When they come to molecule and polymer property predictions, the annotated graph datasets are often small because labeling them requires expensive equipment and effort. To address the lack of examples of rare label values in graph regression tasks, we propose a semi-supervised framework to progressively balance training data and reduce model bias via self-training. The training data balance is achieved by (1) pseudo-labeling more graphs for under-represented labels with a novel regression confidence measurement and (2) augmenting graph examples in latent space for remaining rare labels after data balancing with pseudo-labels. The former is to identify quality examples from unlabeled data whose labels are confidently predicted and sample a subset of them with a reverse distribution from the imbalanced annotated data. The latter collaborates with the former to target a perfect balance using a novel label-anchored mixup algorithm. We perform experiments in seven regression tasks on graph datasets. Results demonstrate that the proposed framework significantly reduces the error of predicted graph properties, especially in under-represented label areas.