Preserving Learned Propagation Operators for Graph Sparsification

Graph sparsification has traditionally focused on preserving the graph Laplacian through spectral approximation. However, many modern GNNs propagate in- formation through model-dependent or learned operators that can differ substan- tially from the input graph Laplacian. Consequently, preserving the input graph alone may be insufficient to preserve the behavior of a trained model. We pro- pose an operator-aware perspective on graph sparsification that instead preserves the learned propagation operator. We instantiate this perspective for Neural Sheaf Diffusion (NSD) through a no-retraining framework for constructing sparse sheaf operators while preserving learned propagation behavior. We provide theoreti- cal analysis connecting operator approximation to stable representations and pre- dictions, and experiments on four benchmark datasets demonstrate consistently higher sparse-model fidelity than conventional reweighting. These results suggest operator-action preservation as an effective post-hoc reweighting principle for no- retraining sparsification of trained NSD models.