Graph Transformers without Positional Encodings

Recently, Transformers for graph representation learning have become increasingly popular, achieving state-of-the-art performance on a wide-variety of datasets, either alone or in combination with message-passing graph neural networks (MP-GNNs). Infusing graph inductive-biases in the innately structure-agnostic transformer architecture in the form of structural or positional encodings (PEs) is key to achieving these impressive results. However, designing such encodings is tricky and disparate attempts have been made to engineer such encodings including Laplacian eigenvectors, relative random-walk probabilities (RRWP), spatial encodings, centrality encodings, edge encodings etc. In this work, we argue that such encodings may not be required at all, provided the attention mechanism itself incorporates information about the graph structure. We introduce Eigenformer, which uses a novel spectrum-aware attention mechanism cognizant of the Laplacian spectrum of the graph, and empirically show that it achieves performance comparable to SOTA MP-GNN architectures and Graph Transformers on a number of standard GNN benchmark datasets, even surpassing the SOTA on some datasets. We also find that our architecture is much faster to train in terms of number of epochs, presumably due to the innate graph inductive biases.