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.
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