A Spectral Interpretation of Redundancy in a Graph Reservoir

Reservoir computing has been successfully applied to graphs as a preprocessing method to improve the training efficiency of Graph Neural Networks (GNNs). However, a common issue that arises when repeatedly applying layer operators on graphs is over-smoothing, which consists in the convergence of graph signals toward low-frequency components of the graph Laplacian. This work revisits the definition of the reservoir in the Multiresolution Reservoir Graph Neural Network (MRGNN), a spectral reservoir model, and proposes a variant based on a Fairing algorithm originally introduced in the field of surface design in computer graphics. This algorithm provides a pass-band spectral filter that allows smoothing without shrinkage, and it can be adapted to the graph setting through the Laplacian operator. Given its spectral formulation, this method naturally connects to GNN architectures for tasks where smoothing, when properly controlled, can be beneficial,such as graph classification. The core contribution of the paper lies in the theoretical analysis of the algorithm from a random walks perspective. In particular, it shows how tuning the spectral coefficients can be interpreted as modulating the contribution of redundant random walks. Exploratory experiments based on the MRGNN architecture illustrate the potential of this approach and suggest promising directions for future research.