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PC-Conv: Unifying Homophily and Heterophily with Two-Fold Filtering
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Proposes PC-Conv, a spectral graph convolution combining homophilic low-pass aggregation with heterophilic graph-heat-kernel aggregation via two-fold filtering, approximated exactly by Poisson-Charlier polynomials, instantiated as PCNet for node classification.
Problem
GNNs perform well either on homophilic or heterophilic graphs but not both, because they neglect homophily within heterophilic graphs and heterophily within homophilic graphs, and real graphs have mixed levels of homophily.
Approach
A two-fold filter that first applies a heterophilic aggregation derived from an extended graph heat kernel on the complementary graph (e^{t L~}, capturing global information with odd orders pushed away and even orders aggregated) and then a local low-pass filter, with the product approximated exactly by Poisson-Charlier polynomials whose coefficients are learnable.
Key results
- PCNet reaches 95.57% on Texas and 93.83% on Cornell in the polynomial-based comparison, the best among listed methods on all six datasets in that table.
- PCNet achieves the top average rank (1.88) across eight datasets in the heterophilic GNN comparison, outperforming GloGNN++ (3.00) and GGCN.
- Under sparse splits, PCNet reports 52.08% on Cornell versus 42.19% for the runner-up ChebNetII in the semi-supervised table.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use PCNet instead of the strongest polynomial-based baseline | node classification methods · Cora, CiteSeer, PubMed, Texas, Actor, Cornell | accuracy: 90.02 vs. 88.98; 81.76 vs. 80.78; 91.30 vs. 89.94; 95.57 vs. 93.77; 42.72 vs. 41.75; 93.83 vs. 92.95 | Helped |
| use generalized Laplacian normalization instead of traditional Laplacian normalization (η=0.5) (ablation) | PCNet · Cora, CiteSeer, PubMed, Texas, Actor, Cornell | accuracy: 82.33$_{\pm\text{0.83}}$, 69.01$_{\pm\text{1.08}}$, 79.73 $_{\pm\text{0.97}}$, 57.55$_{\pm\text{2.66}}$, 33.28$_{\pm\tex | Helped |
| add identity mapping to the PC-filter combinations (ablation) | PCNet · Cora, CiteSeer, PubMed, Texas, Actor, Cornell | accuracy: 89.92$_{\pm\text{0.95}}$, 81.01$_{\pm\text{0.63}}$, 90.96$_{\pm\text{0.35}}$, 95.25$_{\pm\text{2.30}}$, 42.11$_{\pm\text | Helped |
Surprising findings
- Plain MLP outperforms GCN and GAT by more than 20% on Texas and Cornell, which the authors attribute to the importance of identity mapping for heterophilic graphs.
Limitations
- The homophilic low-pass component is the simplest possible (first-order GCN-style), and the authors acknowledge limited knowledge of homophily.
- Heterophilic heat kernel relies on a complementary graph with hyperparameter p constrained by alpha_1 and t, requiring tuning.
- Efficiency is argued theoretically; no empirical runtime comparison is reported.
Experimental setup
Datasets: Cora CiteSeer PubMed Texas Cornell Wisconsin Actor Penn94
Benchmarks: Cora CiteSeer PubMed Texas Actor Cornell Wisconsin Penn94
Metrics: mean accuracy average rank
Baselines: MLP GCN ChebNet ARMA APPNP GPRGNN BernNet ChebNetII GCNII TWIRLS EGNN PDE-GCN
Builds on: GPRGNN BernNet JacobiConv ChebNetII GCN
Closest papers by meaning
- HeroFilter: Adaptive Spectral Graph Filter for Varying Heterophilic Relations (NeurIPS 2025 · similarity 0.90)
- Homophily-Related: Adaptive Hybrid Graph Filter for Multi-View Graph Clustering (AAAI 2024 · similarity 0.90)
- Refining Latent Homophilic Structures over Heterophilic Graphs for Robust Graph Convolution Networks (AAAI 2024 · similarity 0.90)
- Understanding Heterophily for Graph Neural Networks (ICML 2024 · similarity 0.89)
- When Hypergraph Meets Heterophily: New Benchmark Datasets and Baseline (AAAI 2025 · similarity 0.89)
- What Is Missing For Graph Homophily? Disentangling Graph Homophily For Graph Neural Networks (NeurIPS 2024 · similarity 0.88)
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