Aggregation is itself a spectral valley: SpecValley learns where the valley sits, leading four of six heterophilic graphs
Synopsis
In node classification on heterophilic graphs, the step before propagation has stayed fixed by degree normalisation, which acts spectrally as a valley whose zero is set by the normalisation rather than by the data, exactly where dense graphs' eigenvalues accumulate. SpecValley learns the valley's centre, sharpness and flank gains per graph, then feeds an ordinary ChebNet spectral filter. Across six heterophilic benchmarks it matches or exceeds the best published result on four, sits within the confidence interval of the best on Tolokers, and trails on Actor; crossing five aggregators with four filters attributes the gain to aggregation rather than to the downstream filter.
Interpretation
Aggregation stops being a fixed step written by degree normalisation and becomes a spectral valley whose position can be adjusted per graph, placed ahead of an ordinary spectral filter. Prior work designed ever more expressive convolution operators or graph structures while leaving the step before the filter as degree normalisation, whose spectral response has its zero pinned at the spectral mean, a position set by the normalisation rather than by the data. On stored splits of six heterophilic benchmarks, five aggregators were crossed with four filters alongside each graph's label-energy spectrum; on Chameleon-f, Crocodile and HM-categories every aggregator beats no aggregation under all four filters, whereas on Actor and in 16 of 20 combinations on Tolokers the parameter-free and attention aggregators fall below no aggregation.
The valley's shape is described by three learnable quantities, its centre, sharpness and flank gains, with degree normalisation as a special case. Ego-neighbour separation read spectrally yields only the affine family, so the position of the response is learnable but its shape is not, and sharpness and asymmetry remain those of degree normalisation. Training starts from the parameters corresponding to degree normalisation so the model learns departures; on Actor the learned valley stays within a few points of no aggregation where the others lose up to a larger margin, and on Tolokers it matches no aggregation under the polynomial filters and exceeds it by a few points otherwise.
Aggregation fixes how much spectral energy reaches the downstream filter, and a linear spectral filter can only reweight what aggregation supplies, not restore a band it removed. Prior spectral methods put expressivity on the filter side and assumed the preceding aggregation had already fixed the frequency profile. A spectral energy budget yields a band-wise bound and a non-creation result; in the filter-order sweep, the gap to ChebNet alone on the graphs where the valley is effective does not close with order, even at orders far beyond those used in the benchmark.
Perspective
The demonstrated setting is transductive node classification on six dense heterophilic graphs with stored, reused splits, spanning under one thousand to about forty-six thousand nodes and mean degrees from 8 to 923; the analysis is confined to the linear part of the pipeline, and the operator parameters are fitted on half of the validation set.
The authors note that crowding alone does not predict whether aggregation helps, since what matters is how much label energy the attenuated band carries, and that this quantity is not a complete predictor on HM-categories either; when attention before propagation aligns with the spectrum and when it learns node-specific weights remains open, as does why a Chebyshev filter does not reproduce the valley's gain by itself.