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

arXiv: https://arxiv.org/abs/2305.15639 · HTML: https://mlsanity.com/paper/2305.15639/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 62 references that are in the map.
- First posted on arXiv 25 May 2023.

## Builds on (most cited first, 30 of 42)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Convolutional Networks on Graphs for Learning Molecular Fingerprints](https://mlsanity.com/paper/1509.09292/) (2015)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [FastGCN: Fast Learning with Graph Convolutional Networks via Importance Sampling](https://mlsanity.com/paper/1801.10247/) (2018)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation](https://mlsanity.com/paper/2106.10994/) (2021)
- [Graph Neural Networks for Graphs with Heterophily: A Survey](https://mlsanity.com/paper/2202.07082/) (2022)
- [How Powerful are Spectral Graph Neural Networks](https://mlsanity.com/paper/2205.11172/) (2022)
- [Traffic Graph Convolutional Recurrent Neural Network: A Deep Learning Framework for Network-Scale Traffic Learning and Forecasting](https://mlsanity.com/paper/1802.07007/) (2018)
- [Graph Wavelet Neural Network](https://mlsanity.com/paper/1904.07785/) (2019)
- [Interpreting and Unifying Graph Neural Networks with An Optimization Framework](https://mlsanity.com/paper/2101.11859/) (2021)
- [Implicit Graph Neural Networks](https://mlsanity.com/paper/2009.06211/) (2020)
- [Beltrami Flow and Neural Diffusion on Graphs](https://mlsanity.com/paper/2110.09443/) (2021)
- [Dissecting the Diffusion Process in Linear Graph Convolutional Networks](https://mlsanity.com/paper/2102.10739/) (2021)
- [How Framelets Enhance Graph Neural Networks](https://mlsanity.com/paper/2102.06986/) (2021)
- [p-Laplacian Based Graph Neural Networks](https://mlsanity.com/paper/2111.07337/) (2021)

## Built on by (most cited first, 1 of 1)

- [Unifying over-smoothing and over-squashing in graph neural networks: A physics informed approach and beyond](https://mlsanity.com/paper/2309.02769/) (2023)

## Abstract (arXiv)

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.