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New Gauge-Invariant Positional Encodings for Directed Graphs.

Jiaqing Xie, Yuxin Wang· July 9, 2026 View original

Summary

This paper introduces learnable, gauge-invariant spectral positional encodings (PEs) for directed graphs, addressing challenges with magnetic Laplacians and complex eigenvectors. The method computes PEs using Hermitian block Krylov subspaces, offering efficiency and improved performance on directed graph benchmarks where direction-blind PEs fail.

Applying spectral positional encodings (PEs) to directed graphs presents significant challenges, primarily due to the computational cost of Hermitian eigendecomposition for magnetic Laplacians and the gauge ambiguity of complex eigenvectors. This research proposes a novel solution: learnable spectral PEs that are inherently gauge-invariant. The new PEs are formulated as matrix functions of a normalized magnetic operator, ensuring gauge invariance by design. They are efficiently computed within a Hermitian block Krylov subspace, relying only on sparse matrix-vector products, which significantly reduces the computational burden. The authors demonstrate that a small number of block steps are sufficient for various response families. On a directed stochastic block model where traditional direction-blind PEs fail, the magnetic Krylov PEs converge to an oracle performance, also improving heterophilous benchmarks in the undirected case.

Why it matters

For professionals working with graph neural networks on directed graphs, this innovation provides a more robust, efficient, and accurate method for incorporating positional information, leading to better model performance in complex network analysis tasks.

How to implement this in your domain

  1. 1Investigate integrating these new gauge-invariant PEs into your graph neural network architectures for directed graphs.
  2. 2Benchmark the performance and computational efficiency of Krylov subspace-based PEs against existing positional encoding methods.
  3. 3Explore how these PEs can enhance GNN applications in areas like social network analysis, knowledge graphs, or biological networks.
  4. 4Consider contributing to or utilizing open-source implementations of this technique to accelerate adoption.

Who benefits

Social NetworksBioinformaticsCybersecurityKnowledge GraphsLogistics

Key takeaways

  • Spectral PEs for directed graphs face challenges with computation and gauge invariance.
  • New learnable PEs are gauge-invariant by construction, simplifying implementation.
  • They are efficiently computed using Hermitian block Krylov subspaces.
  • The method significantly improves performance on directed graph tasks where other PEs fail.

Original post by Jiaqing Xie, Yuxin Wang

"arXiv:2607.07032v1 Announce Type: new Abstract: Spectral positional encodings (PEs) for \emph{directed} graphs face two obstacles: magnetic Laplacians require an $O(n^3)$ Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gau…"

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Originally posted by Jiaqing Xie, Yuxin Wang on X · view source

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