# Permutation Equivariant Graph Framelets for Heterophilous Graph Learning

arXiv: https://arxiv.org/abs/2306.04265 · HTML: https://mlsanity.com/paper/2306.04265/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 61 references that are in the map.
- First posted on arXiv 7 Jun 2023.

## Builds on (most cited first, 30 of 34)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (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)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Graph Convolutional Neural Networks for Web-Scale Recommender Systems](https://mlsanity.com/paper/1806.01973/) (2018)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [MixHop: Higher-Order Graph Convolutional Architectures via Sparsified Neighborhood Mixing](https://mlsanity.com/paper/1905.00067/) (2019)
- [Deep Learning on Graphs: A Survey](https://mlsanity.com/paper/1812.04202/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Large Scale Learning on Non-Homophilous Graphs: New Benchmarks and Strong Simple Methods](https://mlsanity.com/paper/2110.14446/) (2021)
- [Revisiting Graph Neural Networks: All We Have is Low-Pass Filters](https://mlsanity.com/paper/1905.09550/) (2019)
- [A Survey on Oversmoothing in Graph Neural Networks](https://mlsanity.com/paper/2303.10993/) (2023)
- [Is Homophily a Necessity for Graph Neural Networks?](https://mlsanity.com/paper/2106.06134/) (2021)
- [Graph Neural Networks for Graphs with Heterophily: A Survey](https://mlsanity.com/paper/2202.07082/) (2022)
- [Graph Learning: A Survey](https://mlsanity.com/paper/2105.00696/) (2021)
- [A Survey on The Expressive Power of Graph Neural Networks](https://mlsanity.com/paper/2003.04078/) (2020)
- [Breaking the Limit of Graph Neural Networks by Improving the Assortativity of Graphs with Local Mixing Patterns](https://mlsanity.com/paper/2106.06586/) (2021)
- [DiffWire: Inductive Graph Rewiring via the Lov'asz Bound](https://mlsanity.com/paper/2206.07369/) (2022)
- [FairDrop: Biased Edge Dropout for Enhancing Fairness in Graph Representation Learning](https://mlsanity.com/paper/2104.14210/) (2021)
- [Simplifying approach to Node Classification in Graph Neural Networks](https://mlsanity.com/paper/2111.06748/) (2021)
- [Graph Neural Networks: Taxonomy, Advances and Trends](https://mlsanity.com/paper/2012.08752/) (2020)
- [Beyond One-hot Encoding: lower dimensional target embedding](https://mlsanity.com/paper/1806.10805/) (2018)
- [SpeqNets: Sparsity-aware Permutation-equivariant Graph Networks](https://mlsanity.com/paper/2203.13913/) (2022)
- [Fast Haar Transforms for Graph Neural Networks](https://mlsanity.com/paper/1907.04786/) (2019)
- [Homophily-oriented Heterogeneous Graph Rewiring](https://mlsanity.com/paper/2302.06299/) (2023)

## Built on by (most cited first, 19 of 19)

- [SpeGCL: Self-supervised Graph Spectrum Contrastive Learning without Positive Samples](https://mlsanity.com/paper/2410.10365/) (2024)
- [Uncertainty-Aware Graph Neural Networks: A Multi-Hop Evidence Fusion Approach](https://mlsanity.com/paper/2506.13083/) (2025)
- [BP-SGCN: Behavioral Pseudo-Label Informed Sparse Graph Convolution Network for Pedestrian and Heterogeneous Trajectory Prediction](https://mlsanity.com/paper/2502.14676/) (2025)
- [HAGNN: Hybrid Aggregation for Heterogeneous Graph Neural Networks](https://mlsanity.com/paper/2307.01636/) (2023)
- [Rethinking Spectral Graph Neural Networks with Spatially Adaptive Filtering](https://mlsanity.com/paper/2401.09071/) (2024)
- [SympGNNs: Symplectic Graph Neural Networks for identifiying high-dimensional Hamiltonian systems and node classification](https://mlsanity.com/paper/2408.16698/) (2024)
- [Flexible Diffusion Scopes with Parameterized Laplacian for Heterophilic Graph Learning](https://mlsanity.com/paper/2409.09888/) (2024)
- [Derivation of Back-propagation for Graph Convolutional Networks using Matrix Calculus and its Application to Explainable Artificial Intelligence](https://mlsanity.com/paper/2408.01408/) (2024)
- [MALADY: Multiclass Active Learning with Auction Dynamics on Graphs](https://mlsanity.com/paper/2409.09475/) (2024)
- [PromptGCN: Bridging Subgraph Gaps in Lightweight GCNs](https://mlsanity.com/paper/2410.10089/) (2024)
- [Graph Signal Adaptive Message Passing](https://mlsanity.com/paper/2410.17629/) (2024)
- [Data-Adaptive Graph Framelets with Generalized Vanishing Moments for Graph Machine Learning](https://mlsanity.com/paper/2309.03537/) (2023)
- [A Hierarchical Framework with Spatio-Temporal Consistency Learning for Emergence Detection in Complex Adaptive Systems](https://mlsanity.com/paper/2401.10300/) (2024)
- [Generative-Contrastive Heterogeneous Graph Neural Network](https://mlsanity.com/paper/2404.02810/) (2024)
- [Heterophilous Distribution Propagation for Graph Neural Networks](https://mlsanity.com/paper/2405.20640/) (2024)
- [Adaptive Backdoor Attacks with Reasonable Constraints on Graph Neural Networks](https://mlsanity.com/paper/2503.09049/) (2025)
- [Learning from Heterophilic Graphs: A Spectral Theory Perspective on the Impact of Self-Loops and Parallel Edges](https://mlsanity.com/paper/2509.13139/) (2025)
- [DGTEN: A Robust Deep Gaussian based Graph Neural Network for Dynamic Trust Evaluation with Uncertainty-Quantification Support](https://mlsanity.com/paper/2510.07620/) (2025)
- [Group-Invariant Unsupervised Skill Discovery: Symmetry-aware Skill Representations for Generalizable Behavior](https://mlsanity.com/paper/2601.14000/) (2026)

## Abstract (arXiv)

The nature of heterophilous graphs is significantly different from that of homophilous graphs, which causes difficulties in early graph neural network models and suggests aggregations beyond the 1-hop neighborhood. In this paper, we develop a new way to implement multi-scale extraction via constructing Haar-type graph framelets with desired properties of permutation equivariance, efficiency, and sparsity, for deep learning tasks on graphs. We further design a graph framelet neural network model PEGFAN (Permutation Equivariant Graph Framelet Augmented Network) based on our constructed graph framelets. The experiments are conducted on a synthetic dataset and 9 benchmark datasets to compare performance with other state-of-the-art models. The result shows that our model can achieve the best performance on certain datasets of heterophilous graphs (including the majority of heterophilous datasets with relatively larger sizes and denser connections) and competitive performance on the remaining.