Negative-Free Self-Supervised Gaussian Embedding of Graphs
Abstract
Graph Contrastive Learning (GCL) has recently emerged as a promising graph self-supervised learning framework for learning discriminative node representations without labels. The widely adopted objective function of GCL benefits from two key properties: alignment and uniformity, which align representations of positive node pairs while uniformly distributing all representations on the hypersphere. The uniformity property plays a critical role in preventing representation collapse and is achieved by pushing apart augmented views of different nodes (negative pairs). As such, existing GCL methods inherently rely on increasing the quantity and quality of negative samples, resulting in heavy computational demands, memory overhead, and potential class collision issues. In this study, we propose a negative-free objective to achieve uniformity, inspired by the fact that points distributed according to a normalized isotropic Gaussian are uniformly spread across the unit hypersphere. Therefore, we can minimize the distance between the distribution of learned representations and the isotropic Gaussian distribution to promote the uniformity of node representations. Our method also distinguishes itself from other approaches by eliminating the need for a parameterized mutual information estimator, an additional projector, asymmetric structures, and, crucially, negative samples. Extensive experiments over seven graph benchmarks demonstrate that our proposal achieves competitive performance with fewer parameters, shorter training times, and lower memory consumption compared to existing GCL methods.
Benchmark Results
Tested on 5 datasets 58 baseline values reported too low
Tables show presented results in global context (best 3 papers + GCN + MLP).
Classic
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | GraphSAGE+CNA | 0.9418
±0.0033 | |
| 2 | GCN+CNA | 0.9366
±0.0048 | |
| 3 | TransformerConv | 0.9350
±0.0058 | |
| 21 | GCN | 0.9004 best reported | best at pub: Understanding Heterophily for Graph Neural Networks (ICML 2024) |
| 39 | SAGE | 0.8911
±0.0012 | |
| 44 | GAT | 0.8897
±0.0014 best reported | best at pub: Learned Low Precision Graph Neural Networks (2020) |
| 130 | BGRL | 0.8730
±0.0010 best reported | best at pub: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) |
| 134 | GRACE | 0.8720
±0.0020 best reported | best at pub: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) |
| 156 | MVGRL | 0.8680
±0.0050 best reported | best at pub: ENGAGE: Explanation Guided Data Augmentation for Graph Representation Learning (2023) best today: Adaptive Negative Scheduling for Graph Contrastive Learning (2026) |
| 177 | DGI | 0.8630
±0.0020 best reported | best at pub: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) |
| 181 | GGD | 0.8620
±0.0020 best reported | best at pub: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) |
| 218 | GMI | 0.8526
±0.0008 best reported | |
| 248 | CCA-SSG | 0.8460
±0.0070 best reported | best at pub: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) Difference within reported uncertainty; not counted as underperformance. |
| 253 | HomoGCL | 0.8450
±0.0050 best reported | best at pub: HomoGCL: Rethinking Homophily in Graph Contrastive Learning (KDD 2023) best today: Rethinking Graph Contrastive Learning through Relative Similarity Preservation (IJCAI 2025) Difference within reported uncertainty; not counted as underperformance. |
| 271 | COSTA | 0.8430
±0.0020 best reported | best at pub: HomoGCL: Rethinking Homophily in Graph Contrastive Learning (KDD 2023) best today: Adaptive Negative Scheduling for Graph Contrastive Learning (2026) |
| 311 | SSGE | 0.8390
±0.0030 | This paper |
| 351 | G-BT | 0.8360 | This paper |
| 370 | RGRL | 0.8350
±0.0070 | This paper |
| 406 | gCooL | 0.8320
±0.0050 best reported | best today: Rethinking Graph Contrastive Learning through Relative Similarity Preservation (IJCAI 2025) Difference within reported uncertainty; not counted as underperformance. |
| 404 | AFGRL | 0.8320
±0.0040 best reported | best at pub: HomoGCL: Rethinking Homophily in Graph Contrastive Learning (KDD 2023) best today: Rethinking Graph Contrastive Learning through Relative Similarity Preservation (IJCAI 2025) |
| 415 | GCA | 0.8310
±0.0030 best reported | best at pub: Graph Contrastive Learning with Implicit Augmentations (2022) |
| 417 | MERIT | 0.8310
±0.0060 | This paper |
| 435 | DeepWalk | 0.8290 best reported | best at pub: Graph Representation Learning: A Survey (2019) |
| 630 | VGAE | 0.8059
±0.0035 best reported | best at pub: Decoupled Self-supervised Learning for Non-Homophilou Graphs (2022) |
| 645 | Node2Vec | 0.8030 best reported | best at pub: Graph Representation Learning: A Survey (2019) |
| 654 | GAE | 0.8011
±0.0022 best reported | best at pub: Graph Masked Autoencoders with Transformers (2022) |
| 724 | MLP | 0.7768 best reported | best at pub: Understanding Heterophily for Graph Neural Networks (ICML 2024) |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | CNA | 0.9575
±0.0058 | |
| 2 | IS-GIB | 0.9390
±0.0187 | |
| 3 | EERM | 0.9112
±0.0145 | |
| 35 | GAT | 0.8080
±0.0026 best reported | best at pub: Feature Expansion for Graph Neural Networks (ICML 2023) |
| 44 | GCN | 0.8033
±0.0088 best reported | best at pub: NTFormer: A Composite Node Tokenized Graph Transformer for Node Classification (2024) |
| 61 | GraphSAGE | 0.7923
±0.0053 | |
| 66 | DGI | 0.7890
±0.0020 best reported | best at pub: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) |
| 73 | GMI | 0.7869
±0.0016 best reported | |
| 87 | MVGRL | 0.7808
±0.0006 best reported | |
| 136 | BGRL | 0.7600
±0.0020 best reported | best at pub: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) |
| 158 | GGD | 0.7550
±0.0010 best reported | best at pub: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) |
| 165 | CCA-SSG | 0.7540
±0.0100 best reported | best at pub: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) |
| 201 | GRACE | 0.7450
±0.0010 best reported | best at pub: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) |
| 226 | SSGE | 0.7410
±0.0030 | This paper |
| 278 | MERIT | 0.7350
±0.0070 | prev best: Multi-View Graph Representation Learning Beyond Homophily (2023) |
| 337 | COSTA | 0.7290
±0.0030 best reported | best at pub: HomoGCL: Rethinking Homophily in Graph Contrastive Learning (KDD 2023) best today: Adaptive Negative Scheduling for Graph Contrastive Learning (2026) |
| 347 | G-BT | 0.7290
±0.0040 | This paper |
| 350 | MLP | 0.7289
±0.0010 best reported | best at pub: On the Equivalence of Graph Convolution and Mixup (TMLR 2023) best today: Gumbel-MPNN: Graph Rewiring with Gumbel-Softmax (2025) |
| 369 | gCooL | 0.7270
±0.0040 best reported | best today: Rethinking Graph Contrastive Learning through Relative Similarity Preservation (IJCAI 2025) |
| 374 | AFGRL | 0.7260
±0.0030 best reported | best at pub: HomoGCL: Rethinking Homophily in Graph Contrastive Learning (KDD 2023) best today: Rethinking Graph Contrastive Learning through Relative Similarity Preservation (IJCAI 2025) |
| 427 | HomoGCL | 0.7230
±0.0070 | This paper |
| 505 | GCA | 0.7150
±0.0030 best reported | best at pub: Topology Reorganized Graph Contrastive Learning with Mitigating Semantic Drift (2024) Difference within reported uncertainty; not counted as underperformance. |
| 512 | RGRL | 0.7150
±0.0090 | This paper |
| 625 | VGAE | 0.6990
±0.0057 best reported | best at pub: Decoupled Self-supervised Learning for Non-Homophilou Graphs (2022) |
| 692 | DeepWalk | 0.6785
±0.0079 best reported | best at pub: Decoupled Self-supervised Learning for Non-Homophilou Graphs (2022) |
| 739 | GAE | 0.6639
±0.0024 best reported | best at pub: Decoupled Self-supervised Learning for Non-Homophilou Graphs (2022) Difference within reported uncertainty; not counted as underperformance. |
| 766 | Node2Vec | 0.6552
±0.0018 best reported |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | LGTL | 0.9518
±0.0011 | |
| 2 | MixHop (LO) | 0.9513
±0.0038 | |
| 3 | GraphSAGE (LO) | 0.9511
±0.0036 | |
| 18 | GCN | 0.9296
±0.0015 best reported | best at pub: Node Identifiers: Compact, Discrete Representations for Efficient Graph Learning (ICLR 2024) |
| 20 | GAT | 0.9233
±0.0018 best reported | best at pub: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) |
| 116 | SAGE | 0.8881
±0.0015 | |
| 136 | CCA-SSG | 0.8840
±0.0060 best reported | best at pub: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) |
| 144 | BGRL | 0.8830
±0.0010 best reported | best at pub: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) |
| 194 | GRACE | 0.8730
±0.0010 best reported | best at pub: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) |
| 195 | GMI | 0.8726
±0.0016 best reported | |
| 196 | MVGRL | 0.8725
±0.0012 best reported | |
| 204 | DGI | 0.8708
±0.0017 best reported | |
| 208 | COSTA | 0.8698
±0.0033 best reported | |
| 210 | MLP | 0.8690
±0.0074 best reported | best at pub: On the Equivalence of Graph Convolution and Mixup (TMLR 2023) |
| 223 | DeepWalk | 0.8652
±0.0012 best reported | |
| 251 | Node2Vec | 0.8520
±0.0002 best reported | |
| 255 | GCA | 0.8512
±0.0023 best reported | best at pub: Graph Masked Autoencoders with Transformers (2022) |
| 256 | MERIT | 0.8505
±0.0010 best reported | best at pub: Multi-View Graph Representation Learning Beyond Homophily (2023) |
| 275 | GGD | 0.8420
±0.0010 best reported | best at pub: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) |
| 299 | VGAE | 0.8307
±0.0029 best reported | best at pub: Decoupled Self-supervised Learning for Non-Homophilou Graphs (2022) |
| 304 | GAE | 0.8294
±0.0031 best reported | best at pub: Multi-View Graph Representation Learning Beyond Homophily (2023) |
| 341 | SSGE | 0.8160
±0.0010 | This paper |
| 365 | HomoGCL | 0.8110
±0.0030 | This paper |
| 384 | AFGRL | 0.8080
±0.0060 best reported | best at pub: HomoGCL: Rethinking Homophily in Graph Contrastive Learning (KDD 2023) best today: Rethinking Graph Contrastive Learning through Relative Similarity Preservation (IJCAI 2025) Difference within reported uncertainty; not counted as underperformance. |
| 407 | gCooL | 0.8050
±0.0040 best reported | best today: Rethinking Graph Contrastive Learning through Relative Similarity Preservation (IJCAI 2025) Difference within reported uncertainty; not counted as underperformance. |
| 414 | G-BT | 0.8040
±0.0010 | This paper |
| 477 | RGRL | 0.7980
±0.0030 | This paper |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | GCN | 0.9302
±0.0011 best reported | best at pub: Augmentation-Free Graph Contrastive Learning with Performance Guarantee (2022) |
| 2 | RSE + RGCN (70b) | 0.8852
±0.0019 | |
| 3 | LEMP+T | 0.8825 | |
| 9 | SAGE | 0.8771 | |
| 31 | GAT | 0.8533 best reported | |
| 89 | RGRL | 0.8029
±0.0072 best reported | best at pub: Relational Self-Supervised Learning on Graphs (2022) |
| 94 | GRACE | 0.8014
±0.0048 best reported | best at pub: Large-Scale Representation Learning on Graphs via Bootstrapping (ICLR 2021) best today: Adaptive Negative Scheduling for Graph Contrastive Learning (2026) |
| 100 | BGRL | 0.8000
±0.0010 best reported | best at pub: Learning Graph Augmentations to Learn Graph Representations (2022) best today: PyG-SSL: A Graph Self-Supervised Learning Toolkit (2024) |
| 128 | SSGE | 0.7919
±0.0057 best reported | best today: Self-Supervised Graph Learning via Spectral Bootstrapping and Laplacian-Based Augmentations (2025) Difference within reported uncertainty; not counted as underperformance. |
| 129 | COSTA | 0.7912
±0.0002 best reported | best at pub: Graph Soft-Contrastive Learning via Neighborhood Ranking (2022) best today: Adaptive Negative Scheduling for Graph Contrastive Learning (2026) |
| 130 | CCA-SSG | 0.7908
±0.0053 best reported | best at pub: Rethinking and Simplifying Bootstrapped Graph Latents (2023) best today: Bootstrap Latents of Nodes and Neighbors for Graph Self-Supervised Learning (2024) |
| 143 | AFGRL | 0.7890
±0.0060 best reported | best at pub: Augmentation-Free Self-Supervised Learning on Graphs (AAAI 2021) best today: PyG-SSL: A Graph Self-Supervised Learning Toolkit (2024) |
| 145 | HomoGCL | 0.7884
±0.0047 best reported | best today: Rethinking Graph Contrastive Learning through Relative Similarity Preservation (IJCAI 2025) Difference within reported uncertainty; not counted as underperformance. |
| 147 | gCooL | 0.7874
±0.0004 | This paper |
| 151 | DGI | 0.7865
±0.0690 best reported | best at pub: Bootstrap Latents of Nodes and Neighbors for Graph Self-Supervised Learning (2024) best today: Quantizing Text-attributed Graphs for Semantic-Structural Integration (KDD 2025) Difference within reported uncertainty; not counted as underperformance. |
| 153 | GGD | 0.7862
±0.0047 | This paper |
| 162 | GCA | 0.7840
±0.0013 best reported | best at pub: Learning Graph Augmentations to Learn Graph Representations (2022) best today: Augmentation-Free Graph Contrastive Learning with Performance Guarantee (2022) Difference within reported uncertainty; not counted as underperformance. |
| 164 | MERIT | 0.7835
±0.0005 | This paper |
| 181 | GAE | 0.7787
±0.0053 | prev best: Augmentation-Free Graph Contrastive Learning with Performance Guarantee (2022) |
| 180 | VGAE | 0.7787
±0.0053 | prev best: Barlow Graph Auto-Encoder for Unsupervised Network Embedding (2021) |
| 190 | MVGRL | 0.7757
±0.0046 | This paper |
| 198 | DeepWalk | 0.7721 best reported | best at pub: Barlow Graph Auto-Encoder for Unsupervised Network Embedding (2021) |
| 211 | G-BT | 0.7683
±0.0073 | This paper |
| 234 | GMI | 0.7490
±0.0010 best reported | best at pub: Learning Graph Augmentations to Learn Graph Representations (2022) Difference within reported uncertainty; not counted as underperformance. |
| 237 | MLP | 0.7469
±0.0085 best reported | best at pub: Fully-inductive Node Classification on Arbitrary Graphs (ICLR 2024) best today: View Space: Learning Representation across Arbitrary Graphs (ICML 2025) |
| 258 | Node2Vec | 0.7180
±0.0010 best reported | best at pub: Learning Graph Augmentations to Learn Graph Representations (2022) Difference within reported uncertainty; not counted as underperformance. |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | DoG | 0.9690
±0.0030 | |
| 2 | GWN-fa | 0.9667
±0.0026 | |
| 3 | GWN-sym | 0.9666
±0.0026 | |
| 6 | GraphSAGE | 0.9593
±0.0025 | |
| 12 | GCN | 0.9514
±0.0025 best reported | best at pub: Unifying Graph Convolutional Neural Networks and Label Propagation (2020) best today: Graph Wave Networks (WWW 2025) |
| 21 | GAT | 0.9437
±0.0028 best reported | best at pub: Unifying Graph Convolutional Neural Networks and Label Propagation (2020) best today: Graph Wave Networks (WWW 2025) |
| 43 | DGI | 0.9350
±0.0030 best reported | best at pub: HAT-GAE: Self-Supervised Graph Auto-encoders with Hierarchical Adaptive Masking and Trainable Corruption (2023) best today: PyG-SSL: A Graph Self-Supervised Learning Toolkit (2024) |
| 45 | SSGE | 0.9346
±0.0045 | This paper |
| 48 | CCA-SSG | 0.9337
±0.0020 best reported | best at pub: MA-GCL: Model Augmentation Tricks for Graph Contrastive Learning (AAAI 2022) |
| 51 | GCA | 0.9332
±0.0012 best reported | best at pub: Beyond Smoothing: Unsupervised Graph Representation Learning with Edge Heterophily Discriminating (AAAI 2022) |
| 52 | gCooL | 0.9332
±0.0002 best reported | best today: Rethinking Graph Contrastive Learning through Relative Similarity Preservation (IJCAI 2025) |
| 53 | BGRL | 0.9331
±0.0013 best reported | best at pub: Spectral Augmentation for Self-Supervised Learning on Graphs (ICLR 2022) best today: Adaptive Negative Scheduling for Graph Contrastive Learning (2026) |
| 56 | AFGRL | 0.9330
±0.0020 best reported | best at pub: Rethinking and Simplifying Bootstrapped Graph Latents (2023) best today: PyG-SSL: A Graph Self-Supervised Learning Toolkit (2024) Difference within reported uncertainty; not counted as underperformance. |
| 69 | GRACE | 0.9301
±0.0003 best reported | best at pub: Topology Reorganized Graph Contrastive Learning with Mitigating Semantic Drift (2024) best today: A Fused Gromov-Wasserstein Approach to Subgraph Contrastive Learning (TMLR 2025) |
| 76 | COSTA | 0.9295
±0.0012 best reported | best at pub: Rethinking and Simplifying Bootstrapped Graph Latents (2023) best today: Adaptive Negative Scheduling for Graph Contrastive Learning (2026) Difference within reported uncertainty; not counted as underperformance. |
| 77 | G-BT | 0.9295
±0.0017 best reported | best at pub: Smoothed Graph Contrastive Learning via Seamless Proximity Integration (2024) best today: Rethinking Graph Contrastive Learning through Relative Similarity Preservation (IJCAI 2025) Difference within reported uncertainty; not counted as underperformance. |
| 82 | RGRL | 0.9294
±0.0013 | This paper |
| 88 | MLP | 0.9280
±0.0020 best reported | best at pub: Unifying Graph Convolutional Neural Networks and Label Propagation (2020) best today: Revisiting the Necessity of Graph Learning and Common Graph Benchmarks (2024) |
| 94 | HomoGCL | 0.9228
±0.0003 | best today: Rethinking Graph Contrastive Learning through Relative Similarity Preservation (IJCAI 2025) |
| 98 | GGD | 0.9270
±0.0040 best reported | best today: Diffusion on Graph: Augmentation of Graph Structure for Node Classification (TMLR 2025) |
| 102 | MERIT | 0.9251
±0.0014 | prev best: Eliciting Structural and Semantic Global Knowledge in Unsupervised Graph Contrastive Learning (AAAI 2022) Improvement within reported uncertainty; not highlighted. |
| 117 | VGAE | 0.9211
±0.0009 | This paper |
| 116 | MVGRL | 0.9211
±0.0012 | This paper |
| 152 | GMI | 0.9108
±0.0056 best reported | best at pub: Topology Reorganized Graph Contrastive Learning with Mitigating Semantic Drift (2024) |
| 163 | GAE | 0.9005
±0.0035 best reported | best at pub: HAT-GAE: Self-Supervised Graph Auto-encoders with Hierarchical Adaptive Masking and Trainable Corruption (2023) Difference within reported uncertainty; not counted as underperformance. |
| 184 | Node2Vec | 0.8510 best reported | best at pub: Learning Graph Augmentations to Learn Graph Representations (2022) Difference within reported uncertainty; not counted as underperformance. |
| 188 | DeepWalk | 0.8461
±0.0022 | This paper |
