Frequency Orders and Response Floors of Fixed Filters in Decoupled Graph Models

Decoupled graph models classify nodes by filtering node features once with a fixed filter and training only a classifier. The filter comes from a propagation scheme, adjacency powers (SGC) or personalized PageRank (PPR), and only its hyperparameter is tuned. Dataset-level metrics and block-model analyses of specific operators leave open which scheme to use and how much it can lose to no filter. Since a filter rescales each graph frequency separately, we characterize a scheme by its frequency order. PPR orders frequencies by eigenvalue and SGC by magnitude. In a linear feature model, a filter raises the signal-to-noise ratio (SNR) exactly when its squared response covaries positively with the excess of signal over noise across frequencies. Stochastic dominance between signal and noise along a scheme's order decides the whole scheme. Every PPR filter has a response floor bounding its SNR loss, whereas adjacency powers have none. On contextual stochastic block models, the criterion predicts the sign of every significant gain. On 30 real graphs with equal tuning budgets, neither scheme dominates, and PPR significantly beats SGC wherever SGC is significantly below an MLP. The results support searching both frequency orders and keeping the identity in schemes without a floor.