Distributed-Order Fractional Graph Operating Network
Abstract
We introduce the Distributed-order fRActional Graph Operating Network (DRAGON), a novel continuous Graph Neural Network (GNN) framework that incorporates distributed-order fractional calculus. Unlike traditional continuous GNNs that utilize integer-order or single fractional-order differential equations, DRAGON uses a learnable probability distribution over a range of real numbers for the derivative orders. By allowing a flexible and learnable superposition of multiple derivative orders, our framework captures complex graph feature updating dynamics beyond the reach of conventional models. We provide a comprehensive interpretation of our framework's capability to capture intricate dynamics through the lens of a non-Markovian graph random walk with node feature updating driven by an anomalous diffusion process over the graph. Furthermore, to highlight the versatility of the DRAGON framework, we conduct empirical evaluations across a range of graph learning tasks. The results consistently demonstrate superior performance when compared to traditional continuous GNN models. The implementation code is available at.
Benchmark Results
Tested on 9 datasets 13 baseline values reported too low
Tables show presented results in global context (best 3 papers + GCN + MLP).
Classic
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | GraphTARIF | 0.9703
±0.0019 | |
| 2 | GAT | 0.9669
±0.0014 best reported | best at pub: SignGT: Signed Attention-based Graph Transformer for Graph Representation Learning (2023) best today: Graph Transductive Sharpening: Leveraging Unlabeled Predictions in Node Classification (2026) Difference within reported uncertainty; not counted as underperformance. |
| 3 | GAT+T | 0.9660
±0.0007 | |
| 7 | SAGE | 0.9643
±0.0027 | |
| 8 | GCN | 0.9632
±0.0008 best reported | |
| 94 | F-GraphCON-nl | 0.9410
±0.0070 | This paper |
| 115 | D-GraphCON-nl | 0.9380
±0.0080 | This paper |
| 124 | D-GraphCON-l | 0.9370
±0.0040 | This paper |
| 168 | D-GRAND-nl | 0.9340
±0.0050 | This paper |
| 178 | F-GraphCON-l | 0.9330
±0.0100 | This paper |
| 210 | D-GRAND-l | 0.9310
±0.0080 | This paper |
| 207 | F-GRAND-nl | 0.9310
±0.0090 | This paper |
| 239 | F-GRAND-l | 0.9280
±0.0060 | This paper |
| 298 | GRAND-nl | 0.9240
±0.0080 | This paper |
| 310 | GRAND-l | 0.9230
±0.0090 | This paper |
| 390 | MLP | 0.9128
±0.0063 | |
| 410 | GraphCON-l | 0.9080
±0.0180 | This paper |
| 461 | GIL | 0.8960
±0.0130 | This paper |
| 489 | HGCN | 0.8820
±0.0140 | This paper |
| 505 | GraphCON-nl | 0.8550
±0.0230 | This paper |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | GraphTARIF | 0.9461
±0.0017 | |
| 2 | GCN | 0.9412
±0.0008 best reported | best at pub: Diss-l-ECT: Dissecting Graph Data with local Euler Characteristic Transforms (ICML 2024) |
| 3 | GAT | 0.9398
±0.0022 best reported | best at pub: Diss-l-ECT: Dissecting Graph Data with local Euler Characteristic Transforms (ICML 2024) best today: Graph Transductive Sharpening: Leveraging Unlabeled Predictions in Node Classification (2026) Difference within reported uncertainty; not counted as underperformance. |
| 8 | SAGE | 0.9351
±0.0006 | |
| 244 | D-GRAND-l | 0.8730
±0.0130 | This paper |
| 249 | D-GRAND-nl | 0.8710
±0.0100 | This paper |
| 250 | D-GraphCON-nl | 0.8710
±0.0100 | This paper |
| 258 | D-GraphCON-l | 0.8690
±0.0100 | This paper |
| 284 | F-GraphCON-l | 0.8620
±0.0080 | This paper |
| 314 | GraphCON-l | 0.8490
±0.0050 | This paper |
| 320 | MLP | 0.8463 | |
| 332 | F-GRAND-l | 0.8440
±0.0150 | This paper |
| 348 | F-GRAND-nl | 0.8410
±0.0090 | This paper |
| 360 | GRAND-l | 0.8370
±0.0120 | This paper |
| 364 | F-GraphCON-nl | 0.8360
±0.0220 | This paper |
| 394 | GRAND-nl | 0.8240
±0.0210 | This paper |
| 426 | HGCN | 0.8060
±0.0180 | This paper |
| 433 | GraphCON-nl | 0.7920
±0.0110 | This paper |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | GrokFormer | 0.9831
±0.0018 | |
| 2 | PolyFormer | 0.9808
±0.0027 | |
| 3 | SGFormer | 0.9796
±0.0081 | |
| 6 | GAT | 0.9782
±0.0028 best reported | best at pub: Clarify Confused Nodes via Separated Learning (2023) best today: GrokFormer: Graph Fourier Kolmogorov-Arnold Transformers (ICML 2024) |
| 9 | GCN | 0.9774
±0.0035 best reported | best at pub: SelfGNN: Self-supervised Graph Neural Networks without explicit negative sampling (WWW 2021) best today: GrokFormer: Graph Fourier Kolmogorov-Arnold Transformers (ICML 2024) |
| 17 | SAGE | 0.9739
±0.0018 | |
| 58 | MLP | 0.9597
±0.0007 | |
| 172 | D-GRAND-l | 0.9460
±0.0020 | This paper |
| 176 | F-GRAND-l | 0.9450
±0.0040 | This paper |
| 185 | D-GRAND-nl | 0.9430
±0.0060 | This paper |
| 186 | D-GraphCON-l | 0.9430
±0.0050 | This paper |
| 195 | F-GraphCON-l | 0.9410
±0.0050 | This paper |
| 199 | D-GraphCON-nl | 0.9400
±0.0040 | This paper |
| 202 | F-GRAND-nl | 0.9390
±0.0050 | This paper |
| 203 | GraphCON-l | 0.9390
±0.0040 | This paper |
| 217 | GRAND-l | 0.9350
±0.0090 | This paper |
| 234 | GraphCON-nl | 0.9310
±0.0030 | This paper |
| 243 | F-GraphCON-nl | 0.9300
±0.0060 | This paper |
| 267 | GRAND-nl | 0.9140
±0.0130 | This paper |
| 271 | HGCN | 0.9080
±0.0150 | This paper |
Heterophilic Graphs
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | P^2GNN | 0.9672
±0.0147 | |
| 2 | DoG | 0.9660
±0.0210 | |
| 3 | GPF-plus | 0.9583
±0.0419 | |
| 34 | MLP | 0.9230
±0.0070 | |
| 50 | D-GREAD | 0.9054
±0.0325 | This paper |
| 53 | GraphSAGE | 0.9000
±0.0365 | |
| 57 | GREAD | 0.8973
±0.0449 best reported | best at pub: Graph Neural Reaction Diffusion Models Submitted to the editors June 2023. Accepted in March 2024. (2024) Difference within reported uncertainty; not counted as underperformance. |
| 64 | F-GREAD | 0.8946
±0.0374 | This paper |
| 107 | D-GRAND | 0.8649
±0.0320 | This paper |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | P^2GNN | 0.9541
±0.0272 | |
| 2 | ACM-GCN | 0.9480
±0.0380 | |
| 3 | RF-GCN | 0.9429 | |
| 24 | MLP | 0.9136
±0.0070 | |
| 26 | D-GRAND | 0.9000
±0.0467 | This paper |
| 30 | D-GREAD | 0.8946
±0.0426 | This paper |
| 52 | GREAD | 0.8703
±0.0495 best reported | best at pub: Graph Neural Reaction Diffusion Models Submitted to the editors June 2023. Accepted in March 2024. (2024) best today: Graph Navier–Stokes Networks (2026) Difference within reported uncertainty; not counted as underperformance. |
| 55 | F-GREAD | 0.8689
±0.0416 | This paper |
| 75 | BLEND | 0.8595
±0.0682 | This paper |
| 95 | Sheaf | 0.8486
±0.0471 | This paper |
| 112 | GRAFF | 0.8405
±0.0610 best reported | best at pub: Understanding convolution on graphs via energies (TMLR 2022) best today: Graph Neural Reaction Diffusion Models Submitted to the editors June 2023. Accepted in March 2024. (2024) Difference within reported uncertainty; not counted as underperformance. |
| 138 | SAGE | 0.8269 |
Heterophily Benchmark
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | Dir-Poly | 0.9451
±0.0022 | |
| 2 | D-CDE | 0.9387
±0.0041 | This paper |
| 3 | LargeScaleNet | 0.9358
±0.0024 | |
| 6 | F-CDE | 0.9306
±0.0055 | This paper |
| 18 | CDE | 0.9164
±0.0028 | This paper |
| 26 | GraphSAGE | 0.9106
±0.0027 | |
| 105 | Diag-NSD | 0.7750
±0.0067 | This paper |
| 107 | GBK-GNN | 0.7587
±0.0043 | This paper |
| 108 | FAGCN | 0.7571
±0.0139 best reported | best at pub: NTFormer: A Composite Node Tokenized Graph Transformer for Node Classification (2024) |
| 114 | GPR-GNN | 0.7337
±0.0068 | This paper |
| 115 | ACM-GCN | 0.7189
±0.0061 best reported | best at pub: Graph Neural Networks for Graphs with Heterophily: A Survey (2022) |
| 116 | GRAND | 0.7160
±0.0058 | This paper |
| 119 | ACMP | 0.7127
±0.0059 | This paper |
| 126 | GraphBel | 0.6947
±0.0037 | This paper |
| 129 | H2GCN | 0.6809
±0.0029 | This paper |
| 135 | MLP | 0.6665
±0.0047 | |
| 141 | ResNet | 0.6588
±0.0038 best reported | best at pub: Evolving Computation Graphs (ICML 2023) best today: Geometric Evolution Graph Convolutional Networks: Enhancing Graph Representation Learning via Ricci Flow (2026) Difference within reported uncertainty; not counted as underperformance. |
| 149 | TDE-GNN | 0.6429
±0.0058 | This paper |
| 152 | CPGNN | 0.6396
±0.0062 best reported | best at pub: Evolving Computation Graphs (ICML 2023) best today: Graph Transductive Sharpening: Leveraging Unlabeled Predictions in Node Classification (2026) Difference within reported uncertainty; not counted as underperformance. |
| 154 | GloGNN | 0.6385
±0.0049 | This paper |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | UniGAP | 0.6375
±0.0110 | |
| 2 | H2GCN + HalfHop | 0.5961
±0.0100 | |
| 3 | GraphSAGE + HalfHop | 0.5897
±0.0150 | |
| 7 | H2GCN | 0.5681
±0.0090 best reported | |
| 12 | SAGE | 0.5569
±0.0023 | |
| 47 | ACM-GCN | 0.5294
±0.0023 best reported | best at pub: Graph Neural Networks for Graphs with Heterophily: A Survey (2022) |
| 84 | MLP | 0.5058 | |
| 102 | D-CDE | 0.4943
±0.0126 | This paper |
| 103 | GPR-GNN | 0.4937
±0.0071 best reported | best at pub: Evolving Computation Graphs (ICML 2023) best today: Piecewise Constant Spectral Graph Neural Network (TMLR 2025) |
| 106 | F-CDE | 0.4901
±0.0056 | This paper |
| 127 | CDE | 0.4763
±0.0043 | This paper |
| 137 | FAGCN | 0.4632
±0.0250 | This paper |
| 140 | GBK-GNN | 0.4598
±0.0071 best reported | best at pub: Evolving Computation Graphs (ICML 2023) |
| 141 | ResNet | 0.4590
±0.0052 best reported | best at pub: Evolving Computation Graphs (ICML 2023) best today: Geometric Evolution Graph Convolutional Networks: Enhancing Graph Representation Learning via Ricci Flow (2026) Difference within reported uncertainty; not counted as underperformance. |
| 148 | GRAND | 0.4505
±0.0065 | This paper |
| 151 | ACMP | 0.4476
±0.0052 | This paper |
| 154 | CPGNN | 0.4436
±0.0035 | This paper |
| 162 | GraphBel | 0.4363
±0.0042 | This paper |
| 186 | TDE-GNN | 0.4033
±0.0137 | This paper |
| 194 | Diag-NSD | 0.3796
±0.0020 | This paper |
| 196 | GloGNN | 0.3728
±0.0066 | This paper |
| # | Model | ROC-AUC ↑ | Paper |
|---|---|---|---|
| 1 | BuNN | 0.9899
±0.0016 | |
| 2 | DC-GNN | 0.9850
±0.0021 | |
| 3 | Tango -GPS | 0.9839
±0.0054 | |
| 11 | D-CDE | 0.9647
±0.0189 | This paper |
| 15 | F-CDE | 0.9604
±0.0025 | This paper |
| 18 | CDE | 0.9550
±0.0523 | This paper |
| 42 | GraphSAGE | 0.9358
±0.0005 | |
| 76 | GBK-GNN | 0.9085
±0.0058 best reported | best at pub: Evolving Computation Graphs (ICML 2023) best today: TANGO: Graph Neural Dynamics via Learned Energy and Tangential Flows (2025) |
| 81 | ACM-GCN | 0.9047
±0.0057 | This paper |
| 85 | H2GCN | 0.8995
±0.0038 | prev best: Graph Neural Networks for Graphs with Heterophily: A Survey (2022) Improvement within reported uncertainty; not highlighted. |
| 91 | FAGCN | 0.8969
±0.0060 | prev best: Evolving Computation Graphs (ICML 2023) |
| 94 | Diag-NSD | 0.8959
±0.0061 | This paper |
| 110 | GPR-GNN | 0.8624
±0.0061 best reported | best at pub: Evolving Computation Graphs (ICML 2023) best today: TANGO: Graph Neural Dynamics via Learned Energy and Tangential Flows (2025) |
| 123 | GRAND | 0.7667
±0.0098 | This paper |
| 124 | GraphBel | 0.7651
±0.0103 | This paper |
| 125 | ACMP | 0.7615
±0.0112 | This paper |
| 137 | CPGNN | 0.7127
±0.0114 | prev best: Evolving Computation Graphs (ICML 2023) |
| 148 | GloGNN | 0.6253
±0.0134 | prev best: Graph Neural Networks for Graphs with Heterophily: A Survey (2022) |
| 150 | TDE-GNN | 0.6115
±0.0224 | This paper |
| 156 | MLP | 0.5099
±0.0147 | |
| 157 | ResNet | 0.5095
±0.0112 | prev best: Evolving Computation Graphs (ICML 2023) Improvement within reported uncertainty; not highlighted. |
| # | Model | ROC-AUC ↑ | Paper |
|---|---|---|---|
| 1 | NID_GAT | 0.9703
±0.0002 | |
| 2 | NID_GCN | 0.9685
±0.0010 | |
| 3 | UniGAP | 0.8421
±0.0080 | |
| 13 | H2GCN | 0.7928
±0.0080 best reported | |
| 16 | GCN | 0.7902
±0.0060 | |
| 43 | FAGCN | 0.7724
±0.0126 best reported | best at pub: Evolving Computation Graphs (ICML 2023) best today: TANGO: Graph Neural Dynamics via Learned Energy and Tangential Flows (2025) Difference within reported uncertainty; not counted as underperformance. |
| 70 | D-CDE | 0.7553
±0.0098 | This paper |
| 75 | CDE | 0.7517
±0.0099 | This paper |
| 76 | F-CDE | 0.7517
±0.0099 | This paper |
| 81 | GBK-GNN | 0.7447
±0.0086 best reported | best at pub: Evolving Computation Graphs (ICML 2023) best today: TANGO: Graph Neural Dynamics via Learned Energy and Tangential Flows (2025) Difference within reported uncertainty; not counted as underperformance. |
| 86 | GPR-GNN | 0.7341
±0.0124 | prev best: Evolving Computation Graphs (ICML 2023) |
| 93 | ACMP | 0.7118
±0.0103 | This paper |
| 96 | MLP | 0.7082
±0.0010 | |
| 98 | GraphBel | 0.7079
±0.0099 | This paper |
| 99 | GRAND | 0.7067
±0.0128 | This paper |
| 101 | ResNet | 0.7034
±0.0076 best reported | best at pub: Evolving Computation Graphs (ICML 2023) best today: Differentiable Cluster Graph Neural Network (2024) Difference within reported uncertainty; not counted as underperformance. |
| 111 | Diag-NSD | 0.6925
±0.0115 | This paper |
| 113 | TDE-GNN | 0.6894
±0.0169 | This paper |
| 119 | GloGNN | 0.6715
±0.0192 | prev best: Evolving Computation Graphs (ICML 2023) Improvement within reported uncertainty; not highlighted. |
| 120 | CPGNN | 0.6709
±0.0263 | prev best: Evolving Computation Graphs (ICML 2023) Improvement within reported uncertainty; not highlighted. |
