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Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FROND
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Proposes FROND, a continuous GNN framework that replaces integer-order time derivatives in existing continuous GNNs (GRAND, GRAND++, GraphCON, CDE, GREAD) with Caputo fractional derivatives, giving non-Markovian memory-dependent feature updates that mitigate oversmoothing.
Problem
Integer-order continuous GNNs use Markovian (memoryless) feature dynamics that converge exponentially to stationary states, causing oversmoothing and limiting capture of long-range dependencies and fractal graph structure.
Approach
Generalize integer-order continuous GNN dynamics to Caputo fractional-order dynamics D_t^beta X = F(W, X), recovering the original model at beta=1; the fractional diffusion yields a non-Markovian random walk with algebraic (not exponential) convergence to stationarity, and is solved with fractional Adams-Bashforth-Moulton solvers.
Key results
- F-GRAND-l reaches 98.1% on Airport vs 80.5% for GRAND-l, and 92.4% on Disease vs 74.5% for GRAND-l (random splits).
- F-CDE improves CDE on large heterophilic datasets, e.g. Roman-empire 91.64 to 93.06 and Amazon-ratings 47.63 to 49.01.
- F-GRAND-l maintains performance up to 128 layers on Cora, while GRAND-l degrades with depth.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace GRAND-l's integer-order dynamics with fractional-order F-GRAND-l dynamics | GRAND-l · Cora | accuracy: 83.6$\pm$1.0 -> 84.8$\pm$1.1 | Helped |
| Replace GRAND-l's integer-order dynamics with fractional-order F-GRAND-l dynamics | GRAND-l · Citeseer | accuracy: 73.4$\pm$0.5 -> 74.0$\pm$1.5 | Helped |
| Replace GRAND-l's integer-order dynamics with fractional-order F-GRAND-l dynamics | GRAND-l · Pubmed | accuracy: 78.8$\pm$1.7 -> 79.4$\pm$1.5 | Helped |
| Replace GRAND-l's integer-order dynamics with fractional-order F-GRAND-l dynamics | GRAND-l · CoauthorCS | accuracy: 92.9$\pm$0.4 -> 93.0$\pm$0.3 | Helped |
| Replace GRAND-l's integer-order dynamics with fractional-order F-GRAND-l dynamics | GRAND-l · Computer | accuracy: 83.7$\pm$1.2 -> 84.4$\pm$1.5 | Helped |
| Replace GRAND-l's integer-order dynamics with fractional-order F-GRAND-l dynamics | GRAND-l · Photo | accuracy: 92.3$\pm$0.9 -> 92.8$\pm$0.6 | Helped |
| Replace GRAND-l's integer-order dynamics with fractional-order F-GRAND-l dynamics | GRAND-l · CoauthorPhy | accuracy: 93.5$\pm$0.9 -> 94.5$\pm$0.4 | Helped |
| Replace GRAND-l's integer-order dynamics with fractional-order F-GRAND-l dynamics | GRAND-l · ogbn-arxiv | accuracy: 71.9$\pm$0.2 -> 72.6$\pm$0.1 | Helped |
| Replace GRAND-l's integer-order dynamics with fractional-order F-GRAND-l dynamics | GRAND-l · Airport | accuracy: 80.5$\pm$9.6 -> 98.1$\pm$0.2 | Helped |
| Replace GRAND-l's integer-order dynamics with fractional-order F-GRAND-l dynamics | GRAND-l · Disease | accuracy: 74.5$\pm$3.4 -> 92.4$\pm$3.9 | Helped |
| Extend CDE with fractional-order dynamics (F-CDE) | CDE · Roman-empire | accuracy: 91.64$\pm$0.28 -> 93.06$\pm$0.55 | Helped |
| Extend CDE with fractional-order dynamics (F-CDE) | CDE · Wiki-cooc | accuracy: 97.99$\pm$0.38 -> 98.73$\pm$0.68 | Helped |
Surprising findings
- Smaller beta (more memory) is preferred on tree-structured/fractal datasets, while larger beta is optimal for Cora.
Limitations
- Fractional order beta must be tuned per dataset via hyperparameter search
- Main experiments use only the basic predictor solver for beta<1, not the more accurate variants
Experimental setup
Datasets: Cora Citeseer Pubmed CoauthorCS Computer Photo CoauthorPhy ogbn-arxiv Airport Disease Fake-NewsNet Roman-empire
Benchmarks: Cora Citeseer Pubmed CoauthorCS Computer Photo CoauthorPhy ogbn-arxiv Airport Disease Fake-NewsNet Roman-empire
Metrics: classification accuracy
Baselines: GCN GAT HGCN GIL GRAND GRAND-nl GRAND-l CDE
Builds on: GRAND GRAND++ GraphCON CDE GREAD
Closest papers by meaning
- Distributed-Order Fractional Graph Operating Network (NeurIPS 2024 · similarity 0.93)
- Coupling Graph Neural Networks with Fractional Order Continuous Dynamics: A Robustness Study (AAAI 2024 · similarity 0.90)
- Hyperbolic-PDE GNN: Spectral Graph Neural Networks in the Perspective of A System of Hyperbolic Partial Differential Equations (ICML 2025 · similarity 0.86)
- Neural Variable-Order Fractional Differential Equation Networks (AAAI 2025 · similarity 0.86)
- A Dynamical Systems-Inspired Pruning Strategy for Addressing Oversmoothing in Graph Attention Networks (ICML 2025 · similarity 0.85)
- When Graph Neural Networks Meet Dynamic Mode Decomposition (ICLR 2025 · similarity 0.85)
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