Information-Theoretic Limits of Node Localization under Hybrid Graph Positional Encodings

TL;DR

This paper investigates the fundamental limits of node localization in graphs using hybrid positional encodings, revealing how structural information and encoding parameters influence identifiability and the effectiveness of graph learning models.

Contribution

It introduces an information-theoretic framework for understanding node localization limits with hybrid encodings, connecting structural graph properties with encoding parameters.

Findings

Impossibility regimes are characterized by anchor number, spectral dimension, and quantization level.

Empirical results on random and real-world graphs support the theoretical predictions.

Identifiability varies significantly across different real-world graphs, highlighting graph-dependent resolution.

Abstract

Positional encoding has become a standard component in graph learning, especially for graph Transformers and other models that must distinguish structurally similar nodes, yet its fundamental identifiability remains poorly understood. In this work, we study node localization under a hybrid positional encoding that combines anchor-distance profiles with quantized low-frequency spectral features. We cast localization as an observation-map problem whose difficulty is controlled by the number of distinct codes induced by the encoding and establish an information-theoretic converse identifying an impossibility regime jointly governed by the anchor number, spectral dimension, and quantization level. Experiments further support this picture: on random $3$-regular graphs, the empirical crossover is well organized by the predicted scaling, while on two real-world DDI graphs identifiability is strongly graph-dependent, with DrugBank remaining highly redundant under the tested encodings and the Decagon-derived graph becoming nearly injective under sufficiently rich spectral information. Overall, these results suggest that positional encoding should be understood not merely as a heuristic architectural component, but as a graph-dependent structural resolution mechanism.