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This paper presents a comprehensive theoretical analysis of the graphp-Laplacian regularized framelet network (pL-UFG) to establish a solidunderstanding of its properties. We conduct a convergence analysis on pL-UFG,addressing the gap in the understanding of its asymptotic behaviors. Further byinvestigating the generalized Dirichlet energy of pL-UFG, we demonstrate thatthe Dirichlet energy remains non-zero throughout convergence, ensuring theavoidance of over-smoothing issues. Additionally, we elucidate the energydynamic perspective, highlighting the synergistic relationship between theimplicit layer in pL-UFG and graph framelets. This synergy enhances the model'sadaptability to both homophilic and heterophilic data. Notably, we reveal thatpL-UFG can be interpreted as a generalized non-linear diffusion process,thereby bridging the gap between pL-UFG and differential equations on thegraph. Importantly, these multifaceted analyses lead to unified conclusionsthat offer novel insights for understanding and implementing pL-UFG, as well asother graph neural network (GNN) models. Finally, based on our dynamicanalysis, we propose two novel pL-UFG models with manually controlled energydynamics. We demonstrate empirically and theoretically that our proposed modelsnot only inherit the advantages of pL-UFG but also significantly reducecomputational costs for training on large-scale graph datasets.

Revisiting Generalized p-Laplacian Regularized Framelet GCNs: Convergence, Energy Dynamic and Training with Non-Linear Diffusion