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Graph transformers have been competitive on graph classification tasks, but they fail to outperform Graph Neural Networks (GNNs) on node classification, which is a common task performed on large-scale graphs for industrial applications. Meanwhile, existing GNN architectures are limited in their ability to perform equally well on both homophilious and heterophilious graphs as their inductive biases are generally tailored to only one setting. To address these issues, we propose GOAT, a scalable global graph transformer. In GOAT, each node conceptually attends to all the nodes in the graph and homophily/heterophily relationships can be learnt adaptively from the data. We provide theoretical justification for our approximate global self-attention scheme, and show it to be scalable to large-scale graphs. We demonstrate the competitiveness of GOAT on both heterophilious and homophilious graphs with millions of nodes. 

Most state-of-the-art Graph Neural Networks (GNNs) can be defined as a form of graph convolution which can be realized by message passing between direct neighbors or beyond. To scale such GNNs to large graphs, various neighbor-, layer-, or subgraph-sampling techniques are proposed to alleviate the "neighbor explosion" problem by considering only a small subset of messages passed to the nodes in a mini-batch. However, sampling-based methods are difficult to apply to GNNs that utilize many-hops-away or global context each layer, show unstable performance for different tasks and datasets, and do not speed up model inference. We propose a principled and fundamentally different approach, VQ-GNN, a universal framework to scale up any convolution-based GNNs using Vector Quantization (VQ) without compromising the performance. In contrast to sampling-based techniques, our approach can effectively preserve all the messages passed to a mini-batch of nodes by learning and updating a small number of quantized reference vectors of global node representations, using VQ within each GNN layer. Our framework avoids the "neighbor explosion" problem of GNNs using quantized representations combined with a low-rank version of the graph convolution matrix. We show that such a compact low-rank version of the gigantic convolution matrix is sufficient both theoretically and experimentally. In company with VQ, we design a novel approximated message passing algorithm and a nontrivial back-propagation rule for our framework. Experiments on various types of GNN backbones demonstrate the scalability and competitive performance of our framework on large-graph node classification and link prediction benchmarks.