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Converse and Collision-Based Achievability for Node Localization with Hybrid Distance-Spectral Graph Positional Encodings

arXiv · AI, language, vision and robotics · article · Aug 31, 2026 · UTC

Graph positional encodings are widely used in graph neural networks and graph Transformers, yet it remains unclear when the code itself can identify nodes. We study a hybrid distance-spectral encoding that combines anchor-distance profiles with quantized low-frequency Laplacian-energy coordinates. Treating the encoding as an observation map yields a simplex-refined converse, an exact collision factorization \(κ_H=κ_Dκ_{S|D}\), and the collision information \(I_H=-\logκ_D-\logκ_{S|D}\). On random regular graphs, the criterion is made explicit through a bounded-correlation Gaussian-wave surrogat

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