Graph Structure Refinement with Energy-based Contrastive Learning
Abstract
Graph Neural Networks (GNNs) have recently gained widespread attention as a successful tool for analyzing graph-structured data. However, imperfect graph structure with noisy links lacks enough robustness and may damage graph representations, therefore limiting the GNNs' performance in practical tasks. Moreover, existing generative architectures fail to fit discriminative graph-related tasks. To tackle these issues, we introduce an unsupervised method based on a joint of generative training and discriminative training to learn graph structure and representation, aiming to improve the discriminative performance of generative models. We propose an Energy-based Contrastive Learning (ECL) guided Graph Structure Refinement (GSR) framework, denoted as ECL-GSR. To our knowledge, this is the first work to combine energy-based models with contrastive learning for GSR. Specifically, we leverage ECL to approximate the joint distribution of sample pairs, which increases the similarity between representations of positive pairs while reducing the similarity between negative ones. Refined structure is produced by augmenting and removing edges according to the similarity metrics among node representations. Extensive experiments demonstrate that ECL-GSR outperforms the state-of-the-art on eight benchmark datasets in node classification. ECL-GSR achieves faster training with fewer samples and memories against the leading baseline, highlighting its simplicity and efficiency in downstream tasks.
Benchmark Results
Tested on 5 datasets 16 baseline values reported too low
Tables show presented results in global context (best 3 papers + GCN + MLP).
TU Dortmund
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | MSH-GNN | 0.9910
±0.0030 | |
| 2 | ID-GNN+BEC | 0.9830
±0.0360 | |
| 3 | SupCosine | 0.9830
±0.0250 | |
| 39 | GIN | 0.9330
±0.0290 | |
| 51 | PSCN | 0.9263
±0.0421 best reported | best at pub: Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs (2017) best today: Quaternion Graph Neural Networks (2020) Difference within reported uncertainty; not counted as underperformance. |
| 63 | GCN | 0.9220
±0.0440 best reported | best at pub: Breaking the Limits of Message Passing Graph Neural Networks (ICML 2021) best today: Depth-Adaptive Graph Neural Networks via Learnable Bakry-Émery Curvature (KDD 2025) |
| 97 | ECL-GSR | 0.9145
±0.0275 | This paper |
| 183 | GAT | 0.9010
±0.0580 best reported | best at pub: Breaking the Limits of Message Passing Graph Neural Networks (ICML 2021) best today: The Expressive Power of Graph Neural Networks: A Survey (2023) |
| 207 | GFN-light | 0.8989
±0.0714 | This paper |
| 301 | CapsGNN | 0.8867
±0.0688 best reported | best at pub: Towards Better Graph Representation Learning with Parameterized Decomposition & Filtering (ICML 2023) Difference within reported uncertainty; not counted as underperformance. |
| 403 | MLP | 0.8660
±0.0495 | |
| 425 | DGCNN | 0.8583
±0.0116 | This paper |
| 661 | HGP-SL | 0.7807
±0.1085 | This paper |
| 718 | SAGE | 0.7490
±0.0870 best reported | best at pub: Fast Graph Representation Learning with PyTorch Geometric (ICLR 2019) Difference within reported uncertainty; not counted as underperformance. |
| 764 | VIB-GSL | 0.6863
±0.0515 | This paper |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | WL-MLP | 0.9792 | |
| 2 | MSH-GNN | 0.9640
±0.0070 | |
| 3 | CoCN | 0.8722
±0.0013 | |
| 17 | ECL-GSR | 0.8300
±0.0143 | This paper |
| 23 | GCN | 0.8260
±0.0220 best reported | best at pub: Scalable Expressiveness through Preprocessed Graph Perturbations (2024) |
| 30 | GIN | 0.8220
±0.0210 | |
| 67 | GFN-light | 0.8134
±0.0173 | This paper |
| 137 | SAGE | 0.7970
±0.0170 best reported | best at pub: Fast Graph Representation Learning with PyTorch Geometric (ICLR 2019) |
| 140 | CapsGNN | 0.7962
±0.0091 | This paper |
| 151 | MLP | 0.7920
±0.0240 | |
| 155 | GAT | 0.7908
±0.0136 | Improvement within reported uncertainty; not highlighted. |
| 176 | HGP-SL | 0.7806
±0.0217 | This paper |
| 191 | DGCNN | 0.7759
±0.0224 best reported | best at pub: CogDL: A Comprehensive Library for Graph Deep Learning (WWW 2021) best today: Explainable Graph Representation Learning via Graph Pattern Analysis (IJCAI 2025) |
| 205 | VIB-GSL | 0.7714
±0.0159 | This paper |
| 281 | PSCN | 0.7260
±0.0215 | 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 | |
| 53 | GraphSAGE | 0.9000
±0.0365 | |
| 155 | NodeFormer | 0.8463
±0.0347 best reported | best at pub: GrokFormer: Graph Fourier Kolmogorov-Arnold Transformers (ICML 2024) |
| 211 | GAT | 0.8162
±0.0645 best reported | best at pub: GrokFormer: Graph Fourier Kolmogorov-Arnold Transformers (ICML 2024) best today: Equivariance Everywhere All At Once: A Recipe for Graph Foundation Models (2025) |
| 257 | GCN | 0.7933
±0.0447 best reported | best at pub: Restructuring Graphs for Higher Homophily via Adaptive Spectral Clustering (AAAI 2022) best today: Graph Wave Networks (WWW 2025) |
| 273 | GEN | 0.7830
±0.0800 best reported | best at pub: ES-GNN: Generalizing Graph Neural Networks Beyond Homophily with Edge Splitting (2022) |
| 353 | ECL-GSR | 0.7297
±0.0339 | This paper |
| 391 | SUBLIME | 0.7021
±0.0232 | This paper |
| 470 | IDGL | 0.6243
±0.0609 | This paper |
| 475 | GRCN | 0.6216
±0.0705 | This paper |
| 493 | SGSR | 0.6081
±0.0487 | This paper |
| 516 | CoGSL | 0.5946
±0.0436 | This paper |
| 529 | LDS | 0.5892
±0.0432 | This paper |
| 552 | GAuG-O | 0.5690
±0.0360 | This paper |
| 567 | STABLE | 0.5524
±0.0604 | This paper |
| 593 | ProGNN | 0.4837
±0.0875 | 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 | |
| 138 | SAGE | 0.8269 | |
| 158 | GAT | 0.8190
±0.0540 best reported | best at pub: Restructuring Graphs for Higher Homophily via Adaptive Spectral Clustering (AAAI 2022) |
| 168 | GEN | 0.8100
±0.0390 best reported | best at pub: ES-GNN: Generalizing Graph Neural Networks Beyond Homophily with Edge Splitting (2022) |
| 170 | GCN | 0.8082
±0.0360 best reported | best at pub: Differentiable Cluster Graph Neural Network (2024) best today: Bayesian Neighborhood Adaptation for Graph Neural Networks (TMLR 2025) |
| 274 | ECL-GSR | 0.7127
±0.0206 | This paper |
| 282 | SUBLIME | 0.7029
±0.0351 | This paper |
| 323 | NodeFormer | 0.6577
±0.0459 best reported | best today: HopFormer: Sparse Graph Transformers with Explicit Receptive Field Control (2026) |
| 409 | GAuG-O | 0.5760
±0.0380 | This paper |
| 435 | GRCN | 0.5432
±0.0824 | This paper |
| 441 | ProGNN | 0.5405
±0.0616 | This paper |
| 465 | CoGSL | 0.5216
±0.0321 | This paper |
| 490 | IDGL | 0.5000
±0.0898 | This paper |
| 509 | LDS | 0.4787
±0.0714 | This paper |
| 532 | SGSR | 0.4432
±0.0216 | This paper |
| 536 | STABLE | 0.4403
±0.0405 | This paper |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | DiGGR | 0.4535
±0.0353 | |
| 2 | TRIGON | 0.4402
±0.0028 | |
| 3 | DoG | 0.4320
±0.0120 | |
| 42 | MLP | 0.4026
±0.0108 | |
| 68 | GraphSAGE | 0.3899
±0.0085 | |
| 119 | GEN | 0.3730
±0.0140 best reported | best at pub: ES-GNN: Generalizing Graph Neural Networks Beyond Homophily with Edge Splitting (2022) |
| 135 | GCN | 0.3690 best reported | best at pub: Beyond Low-frequency Information in Graph Convolutional Networks (AAAI 2021) |
| 136 | NodeFormer | 0.3690
±0.0100 best reported | best at pub: Even Sparser Graph Transformers (NeurIPS 2024) |
| 182 | GAT | 0.3605
±0.0035 best reported | best at pub: Hierarchical Graph Transformer with Adaptive Node Sampling (NeurIPS 2022) |
| 276 | ECL-GSR | 0.3371
±0.0096 | This paper |
| 290 | CoGSL | 0.3295
±0.0120 | This paper |
| 326 | LDS | 0.3105
±0.0131 | This paper |
| 331 | SUBLIME | 0.3079
±0.0068 | This paper |
| 343 | SGSR | 0.3023
±0.0038 | This paper |
| 347 | STABLE | 0.3018
±0.0100 | This paper |
| 354 | GRCN | 0.2997
±0.0071 | This paper |
| 392 | IDGL | 0.2816
±0.0141 | This paper |
| 421 | GAuG-O | 0.2580
±0.0100 | This paper |
| 465 | ProGNN | 0.2235
±0.0088 | This paper |
