Relationship between Smoothing Filters and Over-smoothing in Graph Neural Networks

The thesis explores the mathematical foundations of Graph Neural Networks (GNNs), specifically focusing on the issue of over-smoothing. It draws a parallel between over-smoothing in GNNs and the effect of smoothing filters on surfaces, linking it to a fairing algorithm proposed in the literature. The treatment exploits the Dirac formalism of quantum mechanics used in spectral analysis, which allows to ease the analysis of links between various definitions of Dirichlet energy. Different methods used in literature for the analysis of over-smoothing are discussed using the proposed framework. Finally, the thesis presents an application of the fairing algorithm to a reservoir computing model, unveiling the connections with random walks theory.