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 Nyström scheme closely recovers diffusion geometry, and both Laplacian and distance encodings substantially outperform a no-encoding baseline.

翻译：分子图学习受益于能够捕捉局部邻域与全局拓扑结构的位置信号。目前广泛使用的两类编码方法包括：源自拉普拉斯算子或扩散算子的谱编码，以及基于最短路径信息构建的锚点距离编码，然而二者之间的精确关系尚未得到充分理解。本文将距离编码解释为扩散几何的低秩替代，并推导出一个显式的三边测量映射，该映射能够从变换后的锚点距离与锚点谱位置中重建截断的扩散坐标，且在随机正则图上具有逐点误差与Frobenius间隙的理论保证。在使用共享的基于图神经网络的药物相互作用预测框架对DrugBank分子图进行实验时，一种基于距离驱动的Nyström方案能够高度还原扩散几何结构，且拉普拉斯编码与距离编码均显著优于无位置编码的基线模型。