Graph Transformers (GTs) facilitate the comprehension of complex relationships on graph-structured data by leveraging self-attention of the possible pairs of nodes. The structural information or inductive bias of the input graph is provided as positional encodings to the GT. The positional encodings are mostly Euclidean and are not able to capture the complex hierarchical relationships of the corresponding nodes. To address the limitation, we introduce a novel and efficient framework, HyPE, that generates learnable positional encodings in the non-Euclidean hyperbolic space that capture the intricate hierarchical relationships of the underlying graphs. Unlike existing methods, HyPE can generate a set of hyperbolic positional encodings, empowering us to explore diverse options for the optimal selection of PEs for specific downstream tasks. Additionally, we repurpose the generated hyperbolic positional encodings to mitigate the impact of oversmoothing in deep Graph Neural Networks (GNNs). Furthermore, we provide extensive theoretical underpinnings to offer insights into the working mechanism of the HyPE framework. Comprehensive experiments on four molecular benchmarks, including the four large-scale Open Graph Benchmark (OGB) datasets, substantiate the effectiveness of hyperbolic positional encodings in enhancing the performance of Graph Transformers. We also consider Coauthor and Copurchase networks to establish the efficacy of HyPE in controlling oversmoothing in deep GNNs.

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