  This paper presents a comprehensive theoretical analysis of the graph
p-Laplacian regularized framelet network (pL-UFG) to establish a solid
understanding of its properties. We conduct a convergence analysis on pL-UFG,
addressing the gap in the understanding of its asymptotic behaviors. Further by
investigating the generalized Dirichlet energy of pL-UFG, we demonstrate that
the Dirichlet energy remains non-zero throughout convergence, ensuring the
avoidance of over-smoothing issues. Additionally, we elucidate the energy
dynamic perspective, highlighting the synergistic relationship between the
implicit layer in pL-UFG and graph framelets. This synergy enhances the model's
adaptability to both homophilic and heterophilic data. Notably, we reveal that
pL-UFG can be interpreted as a generalized non-linear diffusion process,
thereby bridging the gap between pL-UFG and differential equations on the
graph. Importantly, these multifaceted analyses lead to unified conclusions
that offer novel insights for understanding and implementing pL-UFG, as well as
other graph neural network (GNN) models. Finally, based on our dynamic
analysis, we propose two novel pL-UFG models with manually controlled energy
dynamics. We demonstrate empirically and theoretically that our proposed models
not only inherit the advantages of pL-UFG but also significantly reduce
computational costs for training on large-scale graph datasets.
