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Coupling Graph Neural Networks with Fractional Order Continuous Dynamics: A Robustness Study
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Studies the adversarial robustness of graph neural fractional-order ODE models (FROND, with Caputo derivative) versus integer-order graph neural ODEs (GRAND, GraphBel, GraphCON), with theoretical perturbation bounds and experiments under Metattack and graph injection attacks.
Problem
Graph neural ODEs have known robustness properties under adversarial attacks, but the robustness of graph neural fractional-order differential equation (FDE) models remains unexplored.
Approach
Replace the integer-order time derivative in GRAND, GraphBel and GraphCON with a Caputo fractional derivative (order beta in (0,1]) so feature updates carry long-term memory; prove output perturbation bounds based on the Mittag-Leffler function that increase with beta, so smaller beta gives tighter bounds.
Key results
- Under Metattack poisoning (modification) on Cora at 20% perturbation, F-GRAND reaches 69.27% vs GRAND 60.53%, and F-GraphCON 69.23% vs GraphCON 57.28%.
- Under PGD graph injection attack on Cora, F-GraphBel reaches 69.50% vs GraphBel 63.93%, and F-GRAND 56.38% vs GRAND 36.80%.
- Inference times are similar between fractional and integer-order models on Cora; the fractional models add no trainable parameters.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace GRAND dynamics with fractional-order dynamics | GRAND · Cora, PGD graph injection attack | accuracy: 56.38±6.39 vs 36.80±1.86 | Helped |
| replace GraphBel dynamics with fractional-order dynamics | GraphBel · Citeseer, TDGIA graph injection attack | accuracy: 65.52±0.55 vs 47.88±1.83 | Helped |
| replace GraphCON dynamics with fractional-order dynamics | GraphCON · Computers, PGD graph injection attack | accuracy: 91.36±0.74 vs 81.28±7.99 | Helped |
| replace GRAND dynamics with fractional-order dynamics | GRAND · Cora, clean graph injection evaluation | accuracy: 86.44±0.31 vs 85.87±0.59; 71.91±0.43 vs 72.52±0.73 | Mixed |
| increase fractional order β (ablation) | FROND models · GIA attacks | accuracy | Hurt |
| add PGD adversarial training (ablation) | FROND models · Citeseer, PGD graph injection attack | accuracy: 71.16±0.84; 71.25±0.24; 65.93±0.10 | Helped |
| add GNNGUARD preprocessing (ablation) | F-GraphCON · Citeseer, PGD graph injection attack | accuracy: 54.47±1.0 -> 62.28±1.86 | Helped |
| add GNNGUARD preprocessing (ablation) | GraphCON · Citeseer, PGD graph injection attack | accuracy: 37.71±7.0 -> 46.52±3.37 | Helped |
| add GNNGUARD preprocessing (ablation) | F-GraphCON · Pubmed, PGD graph injection attack | accuracy: 51.16±6.04 -> 83.39±3.38 | Helped |
| add GNNGUARD preprocessing (ablation) | GraphCON · Pubmed, PGD graph injection attack | accuracy: 45.85±1.97 -> 63.55±2.19 | Helped |
| replace GRAND dynamics with fractional-order dynamics | GRAND · Cora, Metattack at 0% perturbation | accuracy: 81.25±0.89 vs 82.24±1.82 | Hurt |
Surprising findings
- On clean data, several fractional variants are slightly below their integer-order counterparts (e.g., F-GraphBel 77.55% vs GraphBel 79.07% on Cora clean in inductive setting).
Limitations
- Theoretical analysis covers only topology and input perturbations; functional (parameter) perturbations are left for future work.
- Only the Caputo derivative with beta in (0,1] and the predictor-only fractional Adams-Bashforth-Moulton solver (no corrector) are used.
Experimental setup
Datasets: Cora Citeseer PubMed Computers
Benchmarks: Cora Citeseer PubMed Computers
Metrics: node classification accuracy
Baselines: GRAND GraphBel GraphCON GCN GAT
Builds on: FROND GRAND GraphBel GraphCON Metattack
Closest papers by meaning
- Distributed-Order Fractional Graph Operating Network (NeurIPS 2024 · similarity 0.90)
- Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FROND (ICLR 2024 · similarity 0.90)
- Uncertainty Modeling in Graph Neural Networks via Stochastic Differential Equations (ICLR 2025 · similarity 0.88)
- Efficient Training of Neural Fractional-Order Differential Equation via Adjoint Backpropagation (AAAI 2025 · similarity 0.87)
- Neural Variable-Order Fractional Differential Equation Networks (AAAI 2025 · similarity 0.87)
- Signed Graph Neural Ordinary Differential Equation for Modeling Continuous-Time Dynamics (AAAI 2024 · similarity 0.87)
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