# Bridging Distance and Spectral Positional Encodings via Anchor-Based Diffusion Geometry Approximation

arXiv: https://arxiv.org/abs/2601.04517 · HTML: https://mlsanity.com/paper/2601.04517/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 35 references that are in the map.
- First posted on arXiv 8 Jan 2026.

## Builds on (most cited first, 20 of 20)

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- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
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- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
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- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Recipe for a General, Powerful, Scalable Graph Transformer](https://mlsanity.com/paper/2205.12454/) (2022)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Modeling polypharmacy side effects with graph convolutional networks](https://mlsanity.com/paper/1802.00543/) (2018)
- [Graph Positional Encoding via Random Feature Propagation](https://mlsanity.com/paper/2303.02918/) (2023)
- [Generalized Laplacian Positional Encoding for Graph Representation Learning](https://mlsanity.com/paper/2210.15956/) (2022)
- [Distance Encoding: Design Provably More Powerful Neural Networks for Graph Representation Learning](https://mlsanity.com/paper/2009.00142/) (2020)
- [Towards Interpretable Drug-Drug Interaction Prediction: A Graph-Based Approach with Molecular and Network-Level Explanations](https://mlsanity.com/paper/2507.09173/) (2025)
- [MetaMolGen: A Neural Graph Motif Generation Model for De Novo Molecular Design](https://mlsanity.com/paper/2504.15587/) (2025)
- [A Multi-Scale Graph Neural Process with Cross-Drug Co-Attention for Drug-Drug Interactions Prediction](https://mlsanity.com/paper/2509.15256/) (2025)
- [Resolving Node Identifiability in Graph Neural Processes via Laplacian Spectral Encodings](https://mlsanity.com/paper/2511.19037/) (2025)

## Abstract (arXiv)

Molecular graph learning benefits from positional signals that capture both local neighborhoods and global topology. Two widely used families are spectral encodings derived from Laplacian or diffusion operators and anchor-based distance encodings built from shortest-path information, yet their precise relationship is poorly understood. We interpret distance encodings as a low-rank surrogate of diffusion geometry and derive an explicit trilateration map that reconstructs truncated diffusion coordinates from transformed anchor distances and anchor spectral positions, with pointwise and Frobenius-gap guarantees on random regular graphs. On DrugBank molecular graphs using a shared GNP-based DDI prediction backbone, a distance-driven Nystr"om scheme closely recovers diffusion geometry, and both Laplacian and distance encodings substantially outperform a no-encoding baseline.