Multiple-parameter graph fractional fourier transform: Theory and applications

Graph neural networks (GNNs) excel in processing non-Euclidean data, but traditional spectral GNNs rely on static bases and fundamentally lack active spectral regulation. Although the graph fractional Fourier transform (GFRFT) introduces cross-domain modulation, it applies a uniform fractional parameter across all frequencies. This ignores frequency heterogeneity and restricts the models' adaptive capacity in graph node classification tasks. In the paper, we propose two novel types of multiple-parameter GFRFTs (MPGFRFTs) and establish their corresponding theoretical frameworks, including essential properties, computational complexity, and parameters differentiability. By assigning independent, learnable fractional parameters to distinct frequency bands, MPGFRFTs enable fine-grained spectral regulation. Then, we operationalize this mathematical framework by designing the adaptive multiple-parameter fractional spectral regulation (MPFSR) module, a plug-and-play component for mainstream spectral models. We also establish rigorous theoretical bounds on the spectral stability of this module, guaranteeing a stable and reliable convergence during the end-to-end parameters optimization. Experiments demonstrate that integrating the proposed MPFSR module alleviates the constraints of static bases and yields performance gains in node classification on complex graphs, advancing a novel paradigm for active spectral modulation in graph representation learning.