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Distributed-Order Fractional Graph Operating Network
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Proposes DRAGON, a continuous GNN framework whose feature dynamics use a learnable distributed-order fractional derivative (learnable measure over orders in [0,1]), with a non-Markovian random-walk interpretation and improved results over integer- and single-fractional-order continuous GNNs.
Problem
Continuous GNNs use integer-order or single fractional-order differential equations, which limit the temporal/memory dynamics they can capture; single fractional orders also require extensive per-dataset tuning.
Approach
Replace a single derivative order with a learnable measure over a range of fractional orders (distributed-order fractional derivative), discretized via quadrature into a multi-term FDE solved with fractional Adams-Bashforth-Moulton or Grünwald-Letnikov; interpreted as a non-Markovian graph random walk and shown able to approximate any waiting-time distribution.
Key results
- D-GRAND-l reaches 7571 AP on Peptides-func (0.7571 +/- 0.0014) vs 0.7126 for F-GRAND-l, and 0.2461 MAE on Peptides-struct vs 0.2677 for F-GRAND-l.
- On the Airport dataset D-GRAND-l reaches 98.5% vs 98.1% for F-GRAND-l and 80.5% for GRAND-l, and D-CDE is best on several heterophilic datasets (e.g., 93.87% on Roman-empire vs 93.06% for F-CDE).
- In a viscoelastic constitutive-model fitting example, DRAGON-NN achieves MSE about ten times smaller than FROND-NN across Maxwell, Zener, and Kelvin-Voigt models.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use distributed-order derivatives in place of GRAND-l's integer-order dynamics | GRAND-l · Peptides-func test | AP: 0.6962$\pm$0.0015 -> 0.7571$\pm$0.0014 | Helped |
| Use distributed-order derivatives in place of GRAND-l's integer-order dynamics | GRAND-l · Peptides-struct test | MAE: 0.2867$\pm$0.0009 -> 0.2461$\pm$0.0014 | Helped |
| Use distributed-order derivatives instead of a single fractional order | GRAND-l · Cora | accuracy: 84.8$\pm$1.1 -> 85.1$\pm$1.3 | Helped |
| Use distributed-order derivatives instead of a single fractional order | GRAND-l · Citeseer | accuracy: 74.0$\pm$1.5 -> 74.5$\pm$1.1 | Helped |
| Use distributed-order derivatives in place of CDE's integer-order dynamics | CDE · Roman-empire | accuracy: 91.64$\pm$0.28 -> 93.87$\pm$0.41 | Helped |
| Use distributed-order derivatives instead of a single fractional order | CDE · Wiki-cooc | accuracy: 98.73$\pm$0.68 -> 98.58$\pm$0.12 | Hurt |
| Use distributed-order derivatives in place of CDE's integer-order dynamics | CDE · Minesweeper | accuracy: 95.50$\pm$5.23 -> 96.47$\pm$1.89 | Helped |
| Use distributed-order derivatives instead of a single fractional order | CDE · Questions | accuracy: 75.17$\pm$0.99 -> 75.53$\pm$0.98 | Helped |
| Use distributed-order derivatives instead of a single fractional order | CDE · Workers | accuracy: 82.68$\pm$0.86 -> 83.02$\pm$0.86 | Helped |
| Use distributed-order derivatives instead of a single fractional order | CDE · Amazon-ratings | accuracy: 49.01$\pm$0.56 -> 49.43$\pm$1.26 | Helped |
Surprising findings
- Performance of FROND varies significantly as the single fractional order changes from 0 to 1, motivating the learnable distribution.
- On some homophilic datasets (e.g., CoauthorCS, Photo) D-GRAND gains are marginal relative to F-GRAND, and F-CDE slightly beats D-CDE on Wiki-cooc.
Limitations
- Distributed-order solver adds cost: discretizing into multi-term FDE with n interpolation points (typically 10) increases complexity to O(EC + E log E) or O(EC+E^2) for the convolution-based solver.
- Experiments restrict fractional orders to [0,1] for compute and consistency with FROND; behavior for orders above 1 is only explored empirically in the appendix.
Experimental setup
Datasets: Cora Citeseer Pubmed CoauthorCS Computer Photo CoauthorPhy Airport Disease Roman-empire Wiki-cooc Minesweeper
Benchmarks: Peptides-func Peptides-struct Cora Citeseer Pubmed CoauthorCS Computer Photo CoauthorPhy Airport Disease Roman-empire
Metrics: accuracy average precision mean absolute error MSE
Baselines: GRAND-l F-GRAND-l GRAND-nl F-GRAND-nl GraphCON-l F-GraphCON-l CDE F-CDE TDE-GNN GCN GAT GCNII
Builds on: GRAND GraphCON CDE FROND TDE-GNN
Closest papers by meaning
- Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FROND (ICLR 2024 · similarity 0.93)
- Coupling Graph Neural Networks with Fractional Order Continuous Dynamics: A Robustness Study (AAAI 2024 · similarity 0.90)
- Neural Variable-Order Fractional Differential Equation Networks (AAAI 2025 · similarity 0.87)
- Hyperbolic-PDE GNN: Spectral Graph Neural Networks in the Perspective of A System of Hyperbolic Partial Differential Equations (ICML 2025 · similarity 0.86)
- A Generalized Neural Diffusion Framework on Graphs (AAAI 2024 · similarity 0.86)
- When Graph Neural Networks Meet Dynamic Mode Decomposition (ICLR 2025 · similarity 0.85)
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