Let persistence ShePHerD your sheaves

Persistence-augmented graph neural networks enrich message passing with global topological information. Persistent homology (PH) tracks the birth and death of topological features along a graph filtration; RePHINE and SpectRe make this filtration learnable while keeping the coefficient system fixed. We introduce ShePHerD, which additionally learns cellular-sheaf coefficients, persists its cohomology, and decorates persistence events with sheaf-Laplacian spectra. The trivial sheaf recovers SpectRe and, without spectra, RePHINE. Beyond this case, an $8$-vertex attributed pair shows that ShePHerD detects nontrivial holonomy invisible to these methods and edge-aware $1$-WL. On ZINC, ShePHerD persistence beats these baselines and sheaf diffusion, with further gains when combined with diffusion.