Graph Convolution with Degree-Potential and Spectral Calibration

Graph neural networks (GNNs) are widely used for node classification, aggregating neighborhood information through rules in which degree influences message weighting. Degree-aware methods have extended these rules through adaptive aggregation and structural augmentation. We investigate how relative source–receiver degrees can guide shared and local message calibration on the existing sparse graph. On PubMed, matching receiver-degree distributions reveals larger GCN–mean message changes in neighborhoods with greater degree contrast. This motivates degree-potential calibrated graph convolution (DPC-GCN), which combines shared and centered receiver-dependent corrections. To examine additional spectral flexibility after degree calibration, we introduce signed spectral calibration (SSC), yielding DPC-GCN-S. One shared scalar per layer learns the direction and strength of state adjustment, allowing preservation, smoothing, or sharpening. Our analysis explains the separation of shared and local corrections and shows that identical label spectral energies can admit opposite beneficial local calibration directions. Across 16 benchmarks, including three large-scale graphs, DPC-GCN matches or exceeds the best baseline mean on all eight homophilous datasets; DPC-GCN-S ranks first on four of five heterophilous datasets among the evaluated methods. An anonymized implementation is available at https://anonymous.4open.science/r/DPCGCN-4ED8.