# PC-Conv: Unifying Homophily and Heterophily with Two-fold Filtering

arXiv: https://arxiv.org/abs/2312.14438 · HTML: https://mlsanity.com/paper/2312.14438/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 42 references that are in the map.
- First posted on arXiv 22 Dec 2023.

## Builds on (most cited first, 29 of 29)

- [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)
- [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)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [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)
- [Diffusion Improves Graph Learning](https://mlsanity.com/paper/1911.05485/) (2019)
- [Large Scale Learning on Non-Homophilous Graphs: New Benchmarks and Strong Simple Methods](https://mlsanity.com/paper/2110.14446/) (2021)
- [BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation](https://mlsanity.com/paper/2106.10994/) (2021)
- [Graph Neural Networks with convolutional ARMA filters](https://mlsanity.com/paper/1901.01343/) (2019)
- [Finding Global Homophily in Graph Neural Networks When Meeting Heterophily](https://mlsanity.com/paper/2205.07308/) (2022)
- [Interpreting and Unifying Graph Neural Networks with An Optimization Framework](https://mlsanity.com/paper/2101.11859/) (2021)
- [PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/2108.01938/) (2021)
- [Dirichlet Energy Constrained Learning for Deep Graph Neural Networks](https://mlsanity.com/paper/2107.02392/) (2021)
- [Breaking the Limit of Graph Neural Networks by Improving the Assortativity of Graphs with Local Mixing Patterns](https://mlsanity.com/paper/2106.06586/) (2021)
- [Is Heterophily A Real Nightmare For Graph Neural Networks To Do Node Classification?](https://mlsanity.com/paper/2109.05641/) (2021)
- [Multi-view Contrastive Graph Clustering](https://mlsanity.com/paper/2110.11842/) (2021)
- [Break the Ceiling: Stronger Multi-scale Deep Graph Convolutional Networks](https://mlsanity.com/paper/1906.02174/) (2019)
- [Graph Neural Networks Inspired by Classical Iterative Algorithms](https://mlsanity.com/paper/2103.06064/) (2021)
- [Dissecting the Diffusion Process in Linear Graph Convolutional Networks](https://mlsanity.com/paper/2102.10739/) (2021)
- [Graph Convolutional Networks using Heat Kernel for Semi-supervised Learning](https://mlsanity.com/paper/2007.16002/) (2020)
- [Node Dependent Local Smoothing for Scalable Graph Learning](https://mlsanity.com/paper/2110.14377/) (2021)
- [Beyond Homophily: Reconstructing Structure for Graph-agnostic Clustering](https://mlsanity.com/paper/2305.02931/) (2023)
- [Finding the Missing-half: Graph Complementary Learning for Homophily-prone and Heterophily-prone Graphs](https://mlsanity.com/paper/2306.07608/) (2023)
- [Feature Overcorrelation in Deep Graph Neural Networks: A New Perspective](https://mlsanity.com/paper/2206.07743/) (2022)
- [Self-supervised Consensus Representation Learning for Attributed Graph](https://mlsanity.com/paper/2108.04822/) (2021)

## Built on by (most cited first, 28 of 28)

- [Multi-Domain Graph Foundation Models: Robust Knowledge Transfer via Topology Alignment](https://mlsanity.com/paper/2502.02017/) (2025)
- [On the Benefits of Attribute-Driven Graph Domain Adaptation](https://mlsanity.com/paper/2502.06808/) (2025)
- [Node-wise Filtering in Graph Neural Networks: A Mixture of Experts Approach](https://mlsanity.com/paper/2406.03464/) (2024)
- [Beyond Redundancy: Information-aware Unsupervised Multiplex Graph Structure Learning](https://mlsanity.com/paper/2409.17386/) (2024)
- [Homophily Enhanced Graph Domain Adaptation](https://mlsanity.com/paper/2505.20089/) (2025)
- [Cooperation of Experts: Fusing Heterogeneous Information with Large Margin](https://mlsanity.com/paper/2505.20853/) (2025)
- [Provable Filter for Real-world Graph Clustering](https://mlsanity.com/paper/2403.03666/) (2024)
- [ERGNN: Spectral Graph Neural Network With Explicitly-Optimized Rational Graph Filters](https://mlsanity.com/paper/2412.19106/) (2024)
- [Structure-Centric Graph Foundation Model via Geometric Bases](https://mlsanity.com/paper/2605.08689/) (2026)
- [Enhancing the Influence of Labels on Unlabeled Nodes in Graph Convolutional Networks](https://mlsanity.com/paper/2411.02279/) (2024)
- [Simplified PCNet with Robustness](https://mlsanity.com/paper/2403.03676/) (2024)
- [Unveiling Mode Connectivity in Graph Neural Networks](https://mlsanity.com/paper/2502.12608/) (2025)
- [The Post-GCN Decade Revisited: Curvature-Stratified Evaluation of Relational Learning](https://mlsanity.com/paper/2606.06397/) (2026)
- [ScaleGNN: Towards Scalable Graph Neural Networks via Adaptive High-order Neighboring Feature Fusion](https://mlsanity.com/paper/2504.15920/) (2025)
- [Partition-wise Graph Filtering: A Unified Perspective Through the Lens of Graph Coarsening](https://mlsanity.com/paper/2505.14033/) (2025)
- [Polynomial Selection in Spectral Graph Neural Networks: An Error-Sum of Function Slices Approach](https://mlsanity.com/paper/2404.15354/) (2024)
- [Hierarchical Uncertainty-Aware Graph Neural Network](https://mlsanity.com/paper/2504.19820/) (2025)
- [A Re-node Self-training Approach for Deep Graph-based Semi-supervised Classification on Multi-view Image Data](https://mlsanity.com/paper/2510.24791/) (2025)
- [Gauge-Equivariant Graph Networks via Self-Interference Cancellation](https://mlsanity.com/paper/2511.16062/) (2025)
- [Dual-Frequency Filtering Self-aware Graph Neural Networks for Homophilic and Heterophilic Graphs](https://mlsanity.com/paper/2411.11284/) (2024)
- [H3GNNs: Harmonizing Heterophily and Homophily in GNNs via Joint Structural Node Encoding and Self-Supervised Learning](https://mlsanity.com/paper/2504.11699/) (2025)
- [Aggregation-aware MLP: An Unsupervised Approach for Graph Message-passing](https://mlsanity.com/paper/2507.20127/) (2025)
- [GraphTorque: Torque-Driven Rewiring Graph Neural Network](https://mlsanity.com/paper/2507.21422/) (2025)
- [Bridging Computational Social Science and Deep Learning: Cultural Dissemination-Inspired Graph Neural Networks](https://mlsanity.com/paper/2509.19084/) (2025)
- [Pure Node Selection for Imbalanced Graph Node Classification](https://mlsanity.com/paper/2509.23662/) (2025)
- [Persistent Tri-State Message Passing](https://mlsanity.com/paper/2601.01207/) (2026)
- [Continuous-Time Quantum Walks based Graph Neural Network](https://mlsanity.com/paper/2608.20738/) (2026)
- [Spectral Reversal: Counteracting Singular Value Bias for Graph Prompting](https://mlsanity.com/paper/2609.32143/) (2026)

## Abstract (arXiv)

Recently, many carefully crafted graph representation learning methods have achieved impressive performance on either strong heterophilic or homophilic graphs, but not both. Therefore, they are incapable of generalizing well across real-world graphs with different levels of homophily. This is attributed to their neglect of homophily in heterophilic graphs, and vice versa. In this paper, we propose a two-fold filtering mechanism to extract homophily in heterophilic graphs and vice versa. In particular, we extend the graph heat equation to perform heterophilic aggregation of global information from a long distance. The resultant filter can be exactly approximated by the Possion-Charlier (PC) polynomials. To further exploit information at multiple orders, we introduce a powerful graph convolution PC-Conv and its instantiation PCNet for the node classification task. Compared with state-of-the-art GNNs, PCNet shows competitive performance on well-known homophilic and heterophilic graphs. Our implementation is available at https://github.com/uestclbh/PC-Conv.