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

Bingheng Li, Erlin Pan, Zhao Kang

Abstract

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.

Benchmark Results

Tested on 4 datasets 9 baseline values reported too low

Tables show presented results in global context (best 3 papers + GCN + MLP).

Heterophilic Graphs

Texas

Global leaderboard →

#	Model	Accuracy ↑	Paper
1	P^2GNN Hyb global	0.9672 ±0.0147	$P^2$GNN: Two Prototype Sets to boost GNN Performance
2	DoG code ↗ ext data GNN global	0.9660 ±0.0210	Diffusion on Graph: Augmentation of Graph Structure for Node Classification code ↗
3	GPF-plus code ↗ ext data GNN global	0.9583 ±0.0419	ProG: A Graph Prompt Learning Benchmark code ↗
5	ACM-GCN GNN baseline actual #5 vs paper #91 86 places worse	0.9557 ±0.0360 best reported	actual #5 vs paper #91 86 places worse paper 0.8784 ±0.0440 best at pub: From Node Interaction to Hop Interaction: New Effective and Scalable Graph Learning Paradigm (2022) best today: $P^2$GNN: Two Prototype Sets to boost GNN Performance (2026) paper 0.8784 ±0.0440 Δ today 0.0773
34	MLP Trad global	0.9230 ±0.0070	Design your own universe: a physics-informed agnostic method for enhancing graph neural networks
39	GGCN GNN baseline actual #39 vs paper #148 109 places worse	0.9213 ±0.0305 best reported	actual #39 vs paper #148 109 places worse paper 0.8486 ±0.0455 best at pub: Understanding convolution on graphs via energies (TMLR 2022) best today: Graph Wave Networks (WWW 2025) paper 0.8486 ±0.0455 Δ today 0.0727
40	H2GCN GNN baseline actual #40 vs paper #148 108 places worse	0.9189 ±0.0393 best reported	actual #40 vs paper #148 108 places worse paper 0.8486 ±0.0723 best at pub: How Curvature Enhance the Adaptation Power of Framelet GCNs (2023) best today: GrokFormer: Graph Fourier Kolmogorov-Arnold Transformers (ICML 2024) Difference within reported uncertainty; not counted as underperformance. paper 0.8486 ±0.0723 Δ at pub 0.0104 Δ today 0.0703
53	GraphSAGE code ↗ GNN global	0.9000 ±0.0365	Graph Neural Networks Need Cluster-Normalize-Activate Modules code ↗
104	WRGAT GNN baseline actual #104 vs paper #173 69 places worse	0.8698 best reported	actual #104 vs paper #173 69 places worse paper 0.8362 ±0.0550 best at pub: From Node Interaction to Hop Interaction: New Effective and Scalable Graph Learning Paradigm (2022) best today: Global-Local Graph Neural Networks for Node-Classification (2024) Difference within reported uncertainty; not counted as underperformance. paper 0.8362 ±0.0550 Δ today 0.0336
141	GloGNN GNN baseline actual #141 vs paper #167 26 places worse	0.8524 ±0.0490 best reported	actual #141 vs paper #167 26 places worse paper 0.8405 ±0.0490 best at pub: Finding Global Homophily in Graph Neural Networks When Meeting Heterophily (ICML 2022) best today: $P^2$GNN: Two Prototype Sets to boost GNN Performance (2026) Difference within reported uncertainty; not counted as underperformance. paper 0.8405 ±0.0490 Δ at pub 0.0027 Δ today 0.0119
211	GAT GNN baseline actual #211 vs paper #576 365 places worse	0.8162 ±0.0645 best reported	actual #211 vs paper #576 365 places worse paper 0.5216 ±0.0663 best at pub: A Quasi-Wasserstein Loss for Learning Graph Neural Networks (WWW 2023) best today: Equivariance Everywhere All At Once: A Recipe for Graph Foundation Models (2025) paper 0.5216 ±0.0663 Δ at pub 0.2784 Δ today 0.2946
276	MixHop GNN baseline	0.7784 ±0.0773	This paper
325	LINKX GNN baseline	0.7460 ±0.0837	This paper

Cornell

Global leaderboard →

#	Model	Accuracy ↑	Paper
1	P^2GNN Hyb global	0.9541 ±0.0272	$P^2$GNN: Two Prototype Sets to boost GNN Performance
2	ACM-GCN GNN baseline actual #2 vs paper #87 85 places worse	0.9480 ±0.0380 best reported	actual #2 vs paper #87 85 places worse paper 0.8514 ±0.0607 best at pub: From Node Interaction to Hop Interaction: New Effective and Scalable Graph Learning Paradigm (2022) best today: Heterophily-informed Message Passing (TMLR 2025) paper 0.8514 ±0.0607 Δ today 0.0966
3	RF-GCN GNN global	0.9429	Simplifying Graph Convolutional Networks with Redundancy-Free Neighbors
15	GGCN GNN baseline actual #15 vs paper #79 64 places worse	0.9230 ±0.0367 best reported	actual #15 vs paper #79 64 places worse paper 0.8568 ±0.0663 best at pub: Understanding convolution on graphs via energies (TMLR 2022) best today: $P^2$GNN: Two Prototype Sets to boost GNN Performance (2026) Difference within reported uncertainty; not counted as underperformance. paper 0.8568 ±0.0663 Δ today 0.0662
24	MLP Trad global	0.9136 ±0.0070	Node-oriented Spectral Filtering for Graph Neural Networks
68	H2GCN GNN baseline actual #68 vs paper #136 68 places worse	0.8623 ±0.0471 best reported	actual #68 vs paper #136 68 places worse paper 0.8270 ±0.0528 best at pub: Graph-Coupled Oscillator Networks (ICML 2022) best today: $P^2$GNN: Two Prototype Sets to boost GNN Performance (2026) Difference within reported uncertainty; not counted as underperformance. paper 0.8270 ±0.0528 Δ today 0.0353
74	GloGNN GNN baseline	0.8595 ±0.0510	This paper
123	LINKX GNN baseline actual #123 vs paper #203 80 places worse	0.8346 ±0.0061 best reported	actual #123 vs paper #203 80 places worse paper 0.7784 ±0.0581 best at pub: GPNet: Simplifying Graph Neural Networks via Multi-channel Geometric Polynomials (2022) best today: Differentiable Cluster Graph Neural Network (2024) Difference within reported uncertainty; not counted as underperformance. paper 0.7784 ±0.0581 Δ today 0.0562
138	SAGE GNN global	0.8269	Language Models as Messengers: Enhancing Message Passing in Heterophilic Graph Learning
158	GAT GNN baseline actual #158 vs paper #358 200 places worse	0.8190 ±0.0540 best reported	actual #158 vs paper #358 200 places worse paper 0.6189 ±0.0505 best at pub: Restructuring Graphs for Higher Homophily via Adaptive Spectral Clustering (AAAI 2022) paper 0.6189 ±0.0505 Δ at pub 0.2001
160	WRGAT GNN baseline	0.8162 ±0.0390	This paper
183	MixHop GNN baseline actual #183 vs paper #249 66 places worse	0.7950 ±0.0630 best reported	actual #183 vs paper #249 66 places worse paper 0.7351 ±0.0634 best at pub: Label-Wise Graph Convolutional Network for Heterophilic Graphs (2021) Difference within reported uncertainty; not counted as underperformance. paper 0.7351 ±0.0634 Δ at pub 0.0599

Actor

Global leaderboard →

#	Model	Accuracy ↑	Paper
1	DiGGR Hyb global	0.4535 ±0.0353	Disentangled Generative Graph Representation Learning
2	TRIGON code ↗ Hyb global	0.4402 ±0.0028	Dynamic Triangulation-Based Graph Rewiring for Graph Neural Networks code ↗
3	DoG code ↗ ext data GNN global	0.4320 ±0.0120	Diffusion on Graph: Augmentation of Graph Structure for Node Classification code ↗
16	ACM-GCN GNN baseline actual #16 vs paper #173 157 places worse	0.4198 ±0.0089 best reported	actual #16 vs paper #173 157 places worse paper 0.3628 ±0.0109 best at pub: ES-GNN: Generalizing Graph Neural Networks Beyond Homophily with Edge Splitting (2022) best today: $P^2$GNN: Two Prototype Sets to boost GNN Performance (2026) paper 0.3628 ±0.0109 Δ at pub 0.0032 Δ today 0.0570
38	GCNII GNN baseline actual #38 vs paper #114 76 places worse	0.4082 ±0.0179 best reported	actual #38 vs paper #114 76 places worse paper 0.3744 ±0.0130 best at pub: Feature Expansion for Graph Neural Networks (ICML 2023) best today: $P^2$GNN: Two Prototype Sets to boost GNN Performance (2026) paper 0.3744 ±0.0130 Δ at pub 0.0117 Δ today 0.0338
42	MLP Trad global	0.4026 ±0.0108	Heterophily-Aware Graph Attention Network
58	GloGNN GNN baseline actual #58 vs paper #106 48 places worse	0.3959 ±0.0029 best reported	actual #58 vs paper #106 48 places worse paper 0.3770 ±0.0140 best at pub: SignGT: Signed Attention-based Graph Transformer for Graph Representation Learning (2023) Difference within reported uncertainty; not counted as underperformance. paper 0.3770 ±0.0140 Δ at pub 0.0189
62	GGCN GNN baseline actual #62 vs paper #113 51 places worse	0.3929 ±0.0048 best reported	actual #62 vs paper #113 51 places worse paper 0.3754 ±0.0156 best at pub: Feature Transportation Improves Graph Neural Networks (AAAI 2023) best today: $P^2$GNN: Two Prototype Sets to boost GNN Performance (2026) Difference within reported uncertainty; not counted as underperformance. paper 0.3754 ±0.0156 Δ at pub 0.0027 Δ today 0.0175
68	GraphSAGE code ↗ GNN global	0.3899 ±0.0085	A Quasi-Wasserstein Loss for Learning Graph Neural Networks code ↗
71	H2GCN GNN baseline actual #71 vs paper #198 127 places worse	0.3885 ±0.0117 best reported	actual #71 vs paper #198 127 places worse paper 0.3570 ±0.0100 best at pub: ES-GNN: Generalizing Graph Neural Networks Beyond Homophily with Edge Splitting (2022) best today: $P^2$GNN: Two Prototype Sets to boost GNN Performance (2026) paper 0.3570 ±0.0100 Δ at pub 0.0270 Δ today 0.0315
75	WRGAT GNN baseline actual #75 vs paper #158 83 places worse	0.3860 ±0.0110 best reported	actual #75 vs paper #158 83 places worse paper 0.3653 ±0.0077 best at pub: ES-GNN: Generalizing Graph Neural Networks Beyond Homophily with Edge Splitting (2022) paper 0.3653 ±0.0077 Δ at pub 0.0207
162	MixHop GNN baseline actual #162 vs paper #297 135 places worse	0.3648 ±0.0120 best reported	actual #162 vs paper #297 135 places worse paper 0.3222 ±0.0234 best at pub: Towards Deep Attention in Graph Neural Networks: Problems and Remedies (ICML 2023) paper 0.3222 ±0.0234 Δ at pub 0.0426
180	LINKX GNN baseline	0.3610 ±0.0155	This paper
182	GAT GNN baseline actual #182 vs paper #406 224 places worse	0.3605 ±0.0035 best reported	actual #182 vs paper #406 224 places worse paper 0.2744 ±0.0089 best at pub: Hierarchical Graph Transformer with Adaptive Node Sampling (NeurIPS 2022) paper 0.2744 ±0.0089 Δ at pub 0.0861

LINKX Benchmarks

Penn94

Global leaderboard →

#	Model	Accuracy ↑	Paper
1	DC-GNN GNN global	0.8669 ±0.0022	Differentiable Cluster Graph Neural Network
2	M2M-GNN code ↗ GNN global	0.8594 ±0.0040	Sign is Not a Remedy: Multiset-to-Multiset Message Passing for Learning on Heterophilic Graphs code ↗
3	GloGNN GNN baseline actual #3 vs paper #3	0.8574 ±0.0042	actual #3 vs paper #3 paper 0.8574 ±0.0042 prev best: Uplifting Message Passing Neural Network with Graph Original Information (2022) Improvement within reported uncertainty; not highlighted. paper 0.8574 ±0.0042 Δ at pub 0.0014
9	PCNet GNN ★ proposed	0.8475 ±0.0051	This paper
11	LINKX GNN baseline	0.8471 ±0.0052	This paper
13	GPRGNN GNN baseline actual #13 vs paper #37 24 places worse	0.8455 ±0.0054 best reported	actual #13 vs paper #37 24 places worse paper 0.8138 ±0.0016 best at pub: Clarify Confused Nodes via Separated Learning (2023) paper 0.8138 ±0.0016 Δ at pub 0.0317
17	MixHop GNN baseline actual #17 vs paper #20 3 places worse	0.8363 ±0.0054 best reported	actual #17 vs paper #20 3 places worse paper 0.8347 ±0.0071 best at pub: Uplifting Message Passing Neural Network with Graph Original Information (2022) Difference within reported uncertainty; not counted as underperformance. paper 0.8347 ±0.0071 Δ at pub 0.0016
24	H2GCN GNN baseline actual #24 vs paper #37 13 places worse	0.8320 ±0.0055 best reported	actual #24 vs paper #37 13 places worse paper 0.8131 ±0.0060 best at pub: Clarify Confused Nodes via Separated Learning (2023) paper 0.8131 ±0.0060 Δ at pub 0.0189
26	ACM-GCN GNN baseline actual #26 vs paper #30 4 places worse	0.8298 ±0.0063 best reported	actual #26 vs paper #30 4 places worse paper 0.8252 ±0.0096 best at pub: Clarify Confused Nodes via Separated Learning (2023) Difference within reported uncertainty; not counted as underperformance. paper 0.8252 ±0.0096 Δ at pub 0.0046
27	GCNII GNN baseline	0.8292 ±0.0059	This paper
29	GIN code ↗ GNN global	0.8268 ±0.0032	Mixture of Weak & Strong Experts on Graphs code ↗
31	GCN GNN baseline	0.8247 ±0.0027	This paper
36	GAT GNN baseline	0.8153 ±0.0055	This paper
51	MLP Trad baseline actual #51 vs paper #57 6 places worse	0.7534 ±0.0064 best reported	actual #51 vs paper #57 6 places worse paper 0.7361 ±0.0040 best at pub: Clarify Confused Nodes via Separated Learning (2023) paper 0.7361 ±0.0040 Δ at pub 0.0173
54	WRGAT GNN baseline	0.7432 ±0.0053	This paper

SOTA performance (not significantly different from the best) Proposed method (this paper) Baseline / comparison (this paper) Global context (other papers) Best at pub — reported result worse than the best known when published (best reported in green) Best today — OK at publication; today's leaderboard best is elsewhere