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DiGRAF: Diffeomorphic Graph-Adaptive Activation Function
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Proposes DiGRAF, a graph-adaptive activation function that learns CPAB diffeomorphisms whose parameters are predicted per input graph by an auxiliary GNN, and reports gains over standard, learnable, and graph activations on node, graph, and molecular tasks.
Problem
Standard activation functions in GNNs are not adaptive to the input graph, and existing graph-specific activations (Max, Median, GReLU) have fixed or piecewise-linear blueprints that limit flexibility and can be non-differentiable.
Approach
Use CPAB diffeomorphisms as the activation blueprint, with velocity-field parameters produced by an additional GNN (GNN_act) followed by graph-wise pooling, so the activation is differentiable, bounded, and graph-adaptive and is trained end-to-end.
Key results
- On Cora node classification, GCN+DiGRAF reaches 82.8% accuracy versus 81.8% for the best other method (GReLU).
- On ZINC-12k, GIN+DiGRAF achieves MAE 0.1302, beating the best baseline Maxout (0.1587), a ~18% relative improvement.
- On ogbg-molhiv, DiGRAF obtains 80.28 ROC-AUC, a 4.7-point absolute gain over the best baseline (ReLU).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add the graph-adaptive activation-parameter GNN (ablation) | DiGRAF with GCN · BlogCatalog, Flickr, CiteSeer, Cora, PubMed | accuracy (%): 80.8±0.6 -> 81.6±0.8; 68.6±1.8 -> 69.6±0.6; 69.2±2.1 -> 69.5±1.4; 81.5±1.1 -> 82.8±1.1; 78.3±1.6 -> 79.3±1.4 | Helped |
| add the graph-adaptive activation-parameter GNN (ablation) | DiGRAF with GIN · ZINC-12k | MAE: 0.1382±0.0080 -> 0.1302±0.0090 | Helped |
| add the graph-adaptive activation-parameter GNN (ablation) | DiGRAF with GIN · molesol | RMSE: 0.9011±0.047 -> 0.8196±0.051 | Helped |
| add the graph-adaptive activation-parameter GNN (ablation) | DiGRAF with GIN · moltox21 | ROC-AUC: 76.37±0.49 -> 77.03±0.59 | Helped |
| add the graph-adaptive activation-parameter GNN (ablation) | DiGRAF with GIN · molbace | ROC-AUC: 78.90±1.41 -> 80.37±1.37 | Helped |
| add the graph-adaptive activation-parameter GNN (ablation) | DiGRAF with GIN · molhiv | ROC-AUC: 79.19±1.36 -> 80.28±1.44 | Helped |
| replace the strongest listed baseline activation with DiGRAF | GCN · BlogCatalog, Flickr, CiteSeer, Cora, PubMed | accuracy (%): 77.7±0.7 -> 81.6±0.8; 58.3±0.6 -> 69.6±0.6; 69.2±1.5 -> 69.5±1.4; 81.8±1.8 -> 82.8±1.1; 78.9±1.7 -> 79.3±1.4 | Helped |
| replace the strongest listed baseline activation with DiGRAF | GIN · ZINC-12k | MAE: 0.1587±0.0057 -> 0.1302±0.0090 | Helped |
| replace the strongest listed baseline activation with DiGRAF | GIN · molesol | RMSE: 1.049±0.038 -> 0.8196±0.051 | Helped |
| replace the strongest listed baseline activation with DiGRAF | GIN · moltox21 | ROC-AUC: 75.50±0.77 -> 77.03±0.59 | Helped |
| replace the strongest listed baseline activation with DiGRAF | GIN · molbace | ROC-AUC: 77.23±2.35 -> 80.37±1.37 | Helped |
| replace the strongest listed baseline activation with DiGRAF | GIN · molhiv | ROC-AUC: 75.58±1.40 -> 80.28±1.44 | Helped |
Surprising findings
- Sigmoid as an activation in GCN collapses performance badly (e.g., 32.1% on Cora versus ~80% for ReLU).
Limitations
- Restricted to activations within the class of diffeomorphisms, which may not be optimal.
- Adds an auxiliary GNN and its parameters, increasing cost relative to fixed activations (W/O Adap. variant is lighter).
Experimental setup
Datasets: BlogCatalog Flickr CiteSeer Cora PubMed ZINC-12k OGB molesol OGB moltox21 OGB molbace OGB molhiv MUTAG PTC
Benchmarks: BlogCatalog Flickr CiteSeer Cora PubMed ZINC-12k ogbg-molesol ogbg-moltox21 ogbg-molbace ogbg-molhiv MUTAG PTC
Metrics: accuracy MAE RMSE ROC-AUC
Baselines: ReLU ELU Tanh GeLU PReLU Maxout Swish Max Median GReLU
Builds on: CPAB GCN GIN GReLU DiTAC
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