Simplified PCNet with Robustness

Bingheng Li, Xuanting Xie, Haoxiang Lei, Ruiyi Fang, Zhao Kang

Abstract

Graph Neural Networks (GNNs) have garnered significant attention for their success in learning the representation of homophilic or heterophilic graphs. However, they cannot generalize well to real-world graphs with different levels of homophily. In response, the Possion-Charlier Network (PCNet), the previous work, allows graph representation to be learned from heterophily to homophily. Although PCNet alleviates the heterophily issue, there remain some challenges in further improving the efficacy and efficiency. In this paper, we simplify PCNet and enhance its robustness. We first extend the filter order to continuous values and reduce its parameters. Two variants with adaptive neighborhood sizes are implemented. Theoretical analysis shows our model's robustness to graph structure perturbations or adversarial attacks. We validate our approach through semi-supervised learning tasks on various datasets representing both homophilic and heterophilic graphs.

Benchmark Results

Tested on 4 datasets 8 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 ↗
11	SPCNet-L GNN ★ proposed	0.9426 ±0.0131	This paper
15	SPCNet-D GNN ★ proposed	0.9393 ±0.0128	This paper
17	EvenNet GNN baseline	0.9377 ±0.0173	This paper
20	OptBasisGNN GNN baseline	0.9348 ±0.0168	This paper
21	JacobiConv GNN baseline	0.9344 ±0.0213	This paper
22	BernNet GNN baseline actual #22 vs paper #30 8 places worse	0.9328 ±0.0148 best reported	actual #22 vs paper #30 8 places worse paper 0.9262 ±0.0137 best at pub: A Quasi-Wasserstein Loss for Learning Graph Neural Networks (WWW 2023) Difference within reported uncertainty; not counted as underperformance. paper 0.9262 ±0.0137 Δ at pub 0.0066
23	ChebNetII GNN baseline	0.9328 ±0.0147	This paper
25	PDE-GCN GNN baseline	0.9324 ±0.0203	This paper
28	GPRGNN GNN baseline actual #28 vs paper #29 1 place worse	0.9295 ±0.0131 best reported	actual #28 vs paper #29 1 place worse paper 0.9291 ±0.0132 best at pub: Node-oriented Spectral Filtering for Graph Neural Networks (2022) best today: GrokFormer: Graph Fourier Kolmogorov-Arnold Transformers (ICML 2024) Difference within reported uncertainty; not counted as underperformance. paper 0.9291 ±0.0132 Δ at pub 0.0001 Δ today 0.0004
34	MLP Trad baseline actual #34 vs paper #104 70 places worse	0.9230 ±0.0070 best reported	actual #34 vs paper #104 70 places worse paper 0.8681 ±0.0224 best at pub: Design your own universe: a physics-informed agnostic method for enhancing graph neural networks (2024) paper 0.8681 ±0.0224 Δ at pub 0.0549
46	APPNP GNN baseline actual #46 vs paper #49 3 places worse	0.9131 ±0.0197 best reported	actual #46 vs paper #49 3 places worse paper 0.9064 ±0.0170 best at pub: A Quasi-Wasserstein Loss for Learning Graph Neural Networks (WWW 2023) Difference within reported uncertainty; not counted as underperformance. paper 0.9064 ±0.0170 Δ at pub 0.0067
47	TWIRLS GNN baseline	0.9131 ±0.0336	This paper
53	GraphSAGE code ↗ GNN global	0.9000 ±0.0365	Graph Neural Networks Need Cluster-Normalize-Activate Modules code ↗
70	GCNII GNN baseline actual #70 vs paper #237 167 places worse	0.8918 ±0.0443 best reported	actual #70 vs paper #237 167 places worse paper 0.8046 ±0.0591 best at pub: Convolutional Neural Networks on Graphs with Chebyshev Approximation, Revisited (NeurIPS 2022) best today: Piecewise Constant Spectral Graph Neural Network (TMLR 2025) Difference within reported uncertainty; not counted as underperformance. paper 0.8046 ±0.0591 Δ today 0.0872
110	ChebNet GNN baseline	0.8628 ±0.0262	This paper
170	ARMA GNN baseline	0.8397 ±0.0377	This paper
220	EGNN GNN baseline	0.8134 ±0.0156	This paper
257	GCN GNN baseline actual #257 vs paper #289 32 places worse	0.7933 ±0.0447 best reported	actual #257 vs paper #289 32 places worse paper 0.7697 ±0.0397 best at pub: Restructuring Graphs for Higher Homophily via Adaptive Spectral Clustering (AAAI 2022) best today: Graph Wave Networks (WWW 2025) Difference within reported uncertainty; not counted as underperformance. paper 0.7697 ±0.0397 Δ at pub 0.0143 Δ today 0.0236

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 code ↗ GNN global	0.9480 ±0.0380	Heterophily-informed Message Passing code ↗
3	RF-GCN GNN global	0.9429	Simplifying Graph Convolutional Networks with Redundancy-Free Neighbors
4	SPCNet-D GNN ★ proposed	0.9425 ±0.0192	This paper
6	SPCNet-L GNN ★ proposed	0.9362 ±0.0213	This paper
9	OptBasisGNN GNN baseline	0.9299 ±0.0185	This paper
10	JacobiConv GNN baseline	0.9295 ±0.0246	This paper
16	ChebNetII GNN baseline	0.9230 ±0.0148	This paper
17	BernNet GNN baseline	0.9213 ±0.0164	This paper
18	EvenNet GNN baseline	0.9213 ±0.0172	This paper
21	APPNP GNN baseline actual #21 vs paper #22 1 place worse	0.9181 ±0.0196 best reported	actual #21 vs paper #22 1 place worse paper 0.9152 ±0.0181 best at pub: LON-GNN: Spectral GNNs with Learnable Orthonormal Basis (2023) Difference within reported uncertainty; not counted as underperformance. paper 0.9152 ±0.0181 Δ at pub 0.0029
22	GPRGNN GNN baseline actual #22 vs paper #22	0.9157 ±0.0196	actual #22 vs paper #22 paper 0.9157 ±0.0196 prev best: LON-GNN: Spectral GNNs with Learnable Orthonormal Basis (2023) Improvement within reported uncertainty; not highlighted. paper 0.9157 ±0.0196 Δ at pub 0.0020
24	MLP Trad baseline actual #24 vs paper #109 85 places worse	0.9136 ±0.0070 best reported	actual #24 vs paper #109 85 places worse paper 0.8415 ±0.0305 best at pub: Node-oriented Spectral Filtering for Graph Neural Networks (2022) paper 0.8415 ±0.0305 Δ at pub 0.0721
27	TWIRLS GNN baseline	0.8983 ±0.0229	This paper
28	PDE-GCN GNN baseline	0.8973 ±0.0135	This paper
35	GCNII GNN baseline actual #35 vs paper #109 74 places worse	0.8918 ±0.0396 best reported	actual #35 vs paper #109 74 places worse paper 0.8426 ±0.0213 best at pub: Convolutional Neural Networks on Graphs with Chebyshev Approximation, Revisited (NeurIPS 2022) best today: $P^2$GNN: Two Prototype Sets to boost GNN Performance (2026) Difference within reported uncertainty; not counted as underperformance. paper 0.8426 ±0.0213 Δ today 0.0492
82	ARMA GNN baseline	0.8562 ±0.0213	This paper
85	ChebNet GNN baseline actual #85 vs paper #115 30 places worse	0.8533 ±0.0104 best reported	actual #85 vs paper #115 30 places worse paper 0.8391 ±0.0217 best at pub: Node-oriented Spectral Filtering for Graph Neural Networks (2022) Difference within reported uncertainty; not counted as underperformance. paper 0.8391 ±0.0217 Δ at pub 0.0142
117	EGNN GNN baseline actual #117 vs paper #155 38 places worse	0.8380 ±0.0460 best reported	actual #117 vs paper #155 38 places worse paper 0.8209 ±0.0116 best at pub: Dirichlet Energy Enhancement of Graph Neural Networks by Framelet Augmentation (2023) Difference within reported uncertainty; not counted as underperformance. paper 0.8209 ±0.0116 Δ at pub 0.0171
138	SAGE GNN global	0.8269	Language Models as Messengers: Enhancing Message Passing in Heterophilic Graph Learning
170	GCN GNN baseline actual #170 vs paper #322 152 places worse	0.8082 ±0.0360 best reported	actual #170 vs paper #322 152 places worse paper 0.6578 ±0.0416 best at pub: Restructuring Graphs for Higher Homophily via Adaptive Spectral Clustering (AAAI 2022) best today: Bayesian Neighborhood Adaptation for Graph Neural Networks (TMLR 2025) paper 0.6578 ±0.0416 Δ at pub 0.1342 Δ today 0.1504

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 ↗
11	OptBasisGNN GNN baseline	0.4239 ±0.0052	This paper
21	BernNet GNN baseline actual #21 vs paper #22 1 place worse	0.4179 ±0.0101 best reported	actual #21 vs paper #22 1 place worse paper 0.4171 ±0.0112 best at pub: LON-GNN: Spectral GNNs with Learnable Orthonormal Basis (2023) best today: $P^2$GNN: Two Prototype Sets to boost GNN Performance (2026) Difference within reported uncertainty; not counted as underperformance. paper 0.4171 ±0.0112 Δ at pub 0.0008 Δ today 0.0008
22	ChebNetII GNN baseline	0.4175 ±0.0107	This paper
34	JacobiConv GNN baseline	0.4117 ±0.0064	This paper
37	SPCNet-D GNN ★ proposed	0.4089 ±0.0061	This paper
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
40	EvenNet GNN baseline	0.4048 ±0.0062	This paper
41	APPNP GNN baseline actual #41 vs paper #54 13 places worse	0.4046 ±0.0064 best reported	actual #41 vs paper #54 13 places worse paper 0.3976 ±0.0049 best at pub: A Quasi-Wasserstein Loss for Learning Graph Neural Networks (WWW 2023) Difference within reported uncertainty; not counted as underperformance. paper 0.3976 ±0.0049 Δ at pub 0.0070
42	MLP Trad baseline actual #42 vs paper #43 1 place worse	0.4026 ±0.0108 best reported	actual #42 vs paper #43 1 place worse paper 0.4018 ±0.0055 best at pub: Heterophily-Aware Graph Attention Network (2023) Difference within reported uncertainty; not counted as underperformance. paper 0.4018 ±0.0055 Δ at pub 0.0008
43	SPCNet-L GNN ★ proposed	0.4023 ±0.0070	This paper
52	GPRGNN GNN baseline actual #52 vs paper #53 1 place worse	0.3992 ±0.0067 best reported	actual #52 vs paper #53 1 place worse paper 0.3991 ±0.0062 best at pub: LON-GNN: Spectral GNNs with Learnable Orthonormal Basis (2023) best today: GrokFormer: Graph Fourier Kolmogorov-Arnold Transformers (ICML 2024) Difference within reported uncertainty; not counted as underperformance. paper 0.3991 ±0.0062 Δ at pub 0.0001 Δ today 0.0001
55	PDE-GCN GNN baseline	0.3976 ±0.0074	This paper
68	GraphSAGE code ↗ GNN global	0.3899 ±0.0085	A Quasi-Wasserstein Loss for Learning Graph Neural Networks code ↗
85	TWIRLS GNN baseline	0.3813 ±0.0081	This paper
88	ChebNet GNN baseline actual #88 vs paper #114 26 places worse	0.3802 ±0.0023 best reported	actual #88 vs paper #114 26 places worse paper 0.3742 ±0.0058 best at pub: Node-oriented Spectral Filtering for Graph Neural Networks (2022) Difference within reported uncertainty; not counted as underperformance. paper 0.3742 ±0.0058 Δ at pub 0.0060
111	ARMA GNN baseline	0.3767 ±0.0054	This paper
135	GCN GNN baseline actual #135 vs paper #284 149 places worse	0.3690 best reported	actual #135 vs paper #284 149 places worse paper 0.3326 ±0.0115 best at pub: Beyond Low-frequency Information in Graph Convolutional Networks (AAAI 2021) paper 0.3326 ±0.0115 Δ at pub 0.0364
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
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
224	EGNN GNN baseline	0.3516 ±0.0064	This paper

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	0.8574 ±0.0042	This paper
10	SPCNet-D GNN ★ proposed	0.8475 ±0.0027	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
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
28	SPCNet-L GNN ★ proposed	0.8269 ±0.0022	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
38	H_2GCN GNN baseline	0.8131 ±0.0060	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