GC4NC: A Benchmark Framework for Graph Condensation on Node Classification with New Insights
Organizations: Emory University · Google Deepmind
Abstract
Graph condensation (GC) is an emerging technique designed to learn a significantly smaller graph that retains the essential information of the original graph. This condensed graph has shown promise in accelerating graph neural networks while preserving performance comparable to those achieved with the original, larger graphs. Additionally, this technique facilitates downstream applications like neural architecture search and deepens our understanding of redundancies in large graphs. Despite the rapid development of GC methods, particularly for node classification, a unified evaluation framework is still lacking to systematically compare different GC methods or clarify key design choices for improving their effectiveness. To bridge these gaps, we introduce GC4NC, a comprehensive framework for evaluating diverse GC methods on node classification across multiple dimensions including performance, efficiency, privacy preservation, denoising ability, NAS effectiveness, and transferability. Our systematic evaluation offers novel insights into how condensed graphs behave and the critical design choices that drive their success. These findings pave the way for future advancements in GC methods, enhancing both performance and expanding their real-world applications. Our code is available at https://github.com/Emory-Melody/GraphSlim/tree/main/benchmark.
Figures & tables
| Dataset | Reduction rate (%) | Coreset | Coarsening | Condensation | Whole | ||||||||||||||
| TM | MM | DM | GM | EM | |||||||||||||||
| Cent-D | Cent-P | Random | Herding | K-Center | Averaging | VNG | GEOM | SFGC | GCSNTK | GCDMX | GCondX ⋆ | GCond | DosCond | MSGC | SGDD | GDEM | |||
| Citeseer | 0.36 | 42.86 | 37.78 | 35.37 | 43.73 | 41.43 | 69.75 | 66.14 | 67.61 | 66.27 | 63.51 | 7 0.65 | 67.79 | 70.05 | 69.41 | 60.24 | 71.87 | 67.88 | 72.6 |
| 0.90 | 58.77 | 52.83 | 50.71 | 59.24 | 51.15 | 69.59 | 66.07 | 70.70 | 70.27 | 62.91 | 7 1.27 | 69.69 | 69.15 | 70.83 | 72.08 | 70.52 | 70.13 | ||
| 1.80 | 62.89 | 63.37 | 62.62 | 66.66 | 59.04 | 69.50 | 65.34 | 73.03 | 7 2.36 | 63.90 | 72.08 | 68.38 | 69.35 | 72.18 | 72.21 | 69.65 | 71.74 | ||
| Cora | 0.50 | 57.79 | 58.44 | 35.14 | 51.68 | 44.64 | 75.94 | 70.40 | 78.14 | 75.11 | 71.58 | 79.21 | 79.74 | 80.17 | 80.65 | 8 0.54 | 80.15 | 54.76 | 81.5 |
| Methods | Cora, r = 2.6% | Citeseer, r = 1.8% | Arxiv, r = 0.5% | |||
| MIA Acc ( ) | Acc ( ) | MIA Acc ( ) | Acc ( ) | MIA Acc ( ) | Acc ( ) | |
| Whole | 74.87 ± 1.16 | 81.50 ± 0.50 | 81.76 ± 1.01 | 72.61 ± 0.27 | 54.26 ± 0.11 | 71.43 ± 0.11 |
| GCond | 72.10 ± 0.96 | 80.54 ± 0.67 | 74.11 ± 0.61 | 69.35 ± 0.82 | 53.04 ± 0.18 | 64.23 ± 0.16 |
| GCondX | 66.83 ± 0.81 | 78.60 ± 0.31 | 71.97 ± 0.58 | 68.38 ± 0.45 | 54.64 ± 0.17 | 59.93 ± 0.54 |
| DosCond | 69.70 ± 0.50 | 81.15 ± 0.50 | 74.33 ± 0.34 | 72.18 ± 0.61 | 54.04 ± 0.79 | 61.06 ± 0.59 |
| SGDD | 70.43 ± 1.63 | 81.04 ± 0.54 | 77.07 ± 4.32 | 69.65 ± 1.68 | 53.29 ± 0.46 | 63.76 ± 0.22 |
| Feature Noise | Structural Noise | Adversarial Structural Noise | |||||
| Dataset | Method | Test Acc. | Perf. Drop | Test Acc. | Perf. Drop | Test Acc. | Perf. Drop |
| Whole | 64.07 | 11.75% | 57.63 | 20.62% | 53.90 | 25.76% | |
| GCond | 64.06 | 7.63% | 65.64 | 5.35% | 66.19 | 4.55% | |
| GCondX | 61.27 | 10.40% | 6 0.42 | 11.65% | 6 0.75 | 11.15% | |
| Citeseer 1.8% | GEOM | 58.77 | 19.53% | 51.41 | 29.60% | 5 7.94 | 20.67% |
| Whole | 74.77 | 8.26% | 72.13 | 11.49% | 66.63 | 18.24% | |
| Random | K-Center | GCondX | SFGC | GEOM | GCond | DosCond | MSGC | Whole | |
| Top 1 (%) | 81.88 | 81.74 | 81.49 | 82.42 | 82.19 | 81.82 | 81.91 | 8 2.40 | 82.51 |
| Acc. Corr. | 0.56 | 0.47 | 0.40 | 0.72 | 0.65 | 0.70 | 0.14 | 0 .71 | - |
| Rank Corr. | 0.64 | 0.60 | 0.57 | 0.71 | 0 .74 | 0.66 | 0.20 | 0.78 | - |
| Graph Property | VNG | GCond | MSGC | SGDD | Avg. | Whole | |
| Density% | Cora | 52.17 | 82.28 | 22.00 | 100.00 | 64.11 | 0.14 |
| (Struc.) | Corr. | -0.81 | 0.07 | 0.55 | 0.13 | -0.02 | - |
| Max Eigenvalue | Cora | 3.73 | 34.90 | 1.69 | 14.09 | 13.60 | 169.01 |
| (Spectra) | Corr. | 0.85 | 0.25 | 0.95 | 0.28 | 0.58 | - |
| DBI | Cora | 3.69 | 1.84 | 0.70 | 4.34 | 2.64 | 9.28 |
| (Label & Feature) | Corr. | 0.81 | 0.93 | 0.94 | 0.97 | 0.91 | - |
Appendix figures & tables20 assets
Supplementary material from the paper’s appendix.
Appendix
| Benchmark Scope | GCondenser [ 27 ] | GC-Bench [ 28 ] | GC4NC |
| Methods | |||
| Coreset & Sparsification | Random, KCenter | Random, KCenter, Herding | Cent-D, Cent-P, Random, KCenter, Herding, TSpanner |
| Coarsening | - | - | Averaging, VNG, Clustering, VN |
| Condensation | |||
| Gradient Matching | GCond, DosCond, SGDD | GCond, DosCond, SGDD | GCond, DosCond, SGDD, MSGC |
| Trajectory Matching | SFGC | SFGC, GEOM | SFGC, GEOM |
| Dataset | #Nodes | #Edges | #Classes | #Features | #Training/Validation/Test |
| Citeseer | 3,327 | 4,732 | 6 | 3,703 | 120/500/1000 |
| Cora | 2,708 | 5,429 | 7 | 1,433 | 140/500/1000 |
| Pubmed | 19,717 | 88,648 | 3 | 500 | 60/500/1000 |
| Arxiv | 169,343 | 1,166,243 | 40 | 128 | 90,941/29,799/48,603 |
| Flickr | 89,250 | 899,756 | 7 | 500 | 44,625/22,312/22,313 |
| 232,965 | 57,307,946 | 210 | 602 | 15,3932/23,699/55,334 |
| Dataset | Labeling Rate | Reduction Rate of Labeled Nodes | Reduction Rate |
| Citeseer | 10% | 0.36% | |
| 3.6% | 25% | 0.9% | |
| 50% | 1.8% | ||
| Cora | 10% | 0.5% | |
| 5.2% | 25% | 1.3% | |
| 50% | 2.6% |
| Dataset | Reduction rate (%) | Node sparsification | Coarsen | Condensation | Whole | |||||||||||||||
| Structure-free | Structure-based | |||||||||||||||||||
| Cent-D | Cent-P | Random | Herding | K-Center | Averaging | VN | VNG | GEOM | SFGC | GCSNTK | GCDMX | GCondX ⋆ | GCond | DosCond | MSGC | SGDD | GDEM | |||
| Citeseer | 0.36 | 42.86 2.7 | 37.78 1.3 | 35.37 2.8 | 43.73 1.6 | 41.43 1.4 | 69.75 0.6 | 34.32 5.9 | 66.14 0.3 | 67.61 0.7 | 66.27 0.8 | 63.51 1.9 | 7 0.65 0.5 | 67.79 0.7 | 70.05 2.1 | 69.41 0.8 | 60.24 6.0 | 71.87 0.6 | 67.88 1.8 | 72.6 |
| 0.90 | 58.77 0.5 | 52.83 0.4 | 50.71 0.8 | 59.24 0.4 | 51.15 1.1 | 69.59 0.5 | 40.14 5.3 | 66.07 0.4 | 70.70 0.5 | 70.27 0.7 | 62.91 0.8 | 7 1.27 0.6 | 69.69 0.5 | 69.15 1.2 | 70.83 0.4 | 72.08 0.7 | 70.52 0.6 | 70.13 1.1 | ||
| 1.80 | 62.89 0.4 | 63.37 0.4 | 62.62 0.6 | 66.66 0.5 | 59.04 0.9 | 69.50 0.6 | 41.98 7.0 | 65.34 0.6 | 73.03 0.3 | 7 2.36 0.5 | 63.90 3.4 | 72.08 0.2 | 68.38 0.5 | 69.35 0.8 | 72.18 0.6 | 72.21 0.4 | 69.65 1.7 | 71.74 0.9 | ||
| Cora | 0.50 | 57.79 1.7 | 58.44 1.7 | 35.14 2.5 | 51.68 2.1 | 44.64 4.4 | 75.94 0.7 | 24.62 5.7 | 70.40 0.6 | 78.14 0.5 | 75.11 2.2 | 71.58 0.9 | 79.21 0.4 | 79.74 0.5 | 80.17 0.8 | 80.65 0.6 | 8 0.54 0.3 | 80.15 0.5 | 54.76 4.5 | 81.5 |
| Dataset | Graph 1 | Graph 2 | Graph 3 | Whole | |||
| Score | (%) | Score | (%) | Score | (%) | ||
| Citeseer | 71.93 0.20 | 68.32 | 71.95 0.46 | 68.28 | 71.02 0.37 | 68.15 | 72.6 |
| Cora | 80.43 0.60 | 60.08 | 80.06 0.68 | 60.25 | 80.95 0.26 | 60.48 | 81.5 |
| Pubmed | 79.12 0.14 | 54.32 | 79.15 0.16 | 54.18 | 79.13 0.26 | 54.28 | 78.6 |
| Arxiv | 69.35 0.39 | 17.57 | 69.44 0.41 | 17.55 | 69.30 0.28 | 17.57 | 71.4 |
| Flickr | 47.18 0.13 | 73.05 | 47.19 0.16 | 72.89 | 47.18 0.14 | 72.99 | 47.4 |
| Dataset | Reduc. rate (%) | Coreset & Sparsification | Coarsen | Condensation | Whole | |||||||
| Structure-free | Structure-based | |||||||||||
| Random | K-Center | Averaging | VNG | GEOM | SFGC | GCondX | GCond | DosCond | SGDD | |||
| Citeseer | 0.36 | 37.67 2.45 | 45.11 2.19 | 69.97 0.36 | 64.37 1.29 | 68.90 0.64 | 66.96 1.47 | 68.29 1.30 | 73.63 0.32 | 69.53 0.65 | 71.90 0.24 | 72.6 |
| 0.90 | 47.13 1.32 | 55.09 1.14 | 69.97 0.36 | 69.37 0.62 | 73.20 0.35 | 70.66 0.23 | 69.73 0.46 | 70.93 0.51 | 70.97 0.29 | 70.10 0.73 | ||
| 1.80 | 64.21 0.72 | 62.82 0.78 | 70.01 0.27 | 69.35 0.70 | 74.36 0.30 | 72.37 0.41 | 69.19 0.47 | 70.69 0.47 | 72.73 0.35 | 70.11 0.93 | ||
| Cora | 0.50 | 47.93 0.96 | 49.92 3.06 | 76.55 0.91 | 70.61 0.64 | 79.03 0.61 | 76.80 2.18 | 80.04 0.60 | 80.63 0.48 | 80.43 0.72 | 81.58 0.97 | 81.81 |
| Node reduction rate (%) | Fixed GCN | Validation-selected GCN |
| 0.05 | ||
| 0.10 | ||
| 0.20 |
| Hyperparameter | Values |
| Number of hidden units | {64, 128, 256} |
| Learning rate | {0.01, 0.001} |
| Number of layers | {2, 3, 4} |
| Weight decay | {0, 0.0005, 0.001} |
| Dropout rate | {0, 0.5} |
| Dataset | Density % | Max Eigenvalue | DBI AGG | Homophily |
| Cora | 14.82 | 1.57 | 1.09 | 0.33 |
| Whole | 0.14 | 169.01 | 4.67 | 0.81 |
| Citeseer | 11.86 | 1.51 | 1.46 | 0.33 |
| Whole | 0.08 | 100.04 | 8.49 | 0.74 |
| Pubmed | 6.90 | 0.02 | 1.36 | 1.00 |
| Whole | 0.02 | 172.16 | 5.01 | 0.80 |
| Random | K-Center | GCondX | SFGC | GEOM | GCond | DosCond | MSGC | |
| Recall@5 | 0.0 | 0.0 | 0.2 | 0.0 | 0.2 | 0.0 | 0.0 | 0.0 |
| Recall@10 | 0.0 | 0.0 | 0.2 | 0.1 | 0.2 | 0.2 | 0.1 | 0.1 |
| Recall@20 | 0.4 | 0.35 | 0.5 | 0.45 | 0.4 | 0.45 | 0.25 | 0.4 |
| Architecture | Search Space |
| Number of propagation | {2, 4, 6, 8, 10} |
| Residual coefficient | {0.1, 0.2} |
| Hidden dimension | {16, 32, 64, 128, 256, 512} |
| Activation function | {Sigmoid, Tanh, ReLU, Linear, Softplus, LeakyReLU, ReLU6, ELU} |
| Graph property | Dataset and | VNG | GCond | MSGC | SGDD | Avg. | Whole |
| Density% | Citeseer 1.8% | 36.95 | 84.58 | 22.50 | 100.00 | 61.01 | 0.08 |
| (Structure) | Cora 2.6% | 52.17 | 82.28 | 22.00 | 100.00 | 64.11 | 0.14 |
| Arxiv 0.5% | 100.00 | 75.40 | 8.17 | 99.91 | 70.87 | 0.01 | |
| Flickr 1% | 100.00 | 100.00 | 3.44 | 99.96 | 75.85 | 0.01 | |
| Reddit 0.1% | 100.00 | 2.67 | 32.07 | 74.85 | 52.39 | 0.05 | |
| Corr. | -0.81 | 0.07 | 0.55 | 0.13 | -0.01 | - |
| Datasets | VNG | GCond | MSGC | SGDD | GCondX | GEOM | Avg. | Whole |
| Citeseer 1.8% | 4.14 | 1.40 | 1.98 | 3.47 | 2.90 | 2.55 | 2.74 | 12.07 |
| Cora 2.6% | 3.69 | 1.84 | 0.70 | 4.34 | 2.18 | 3.16 | 2.65 | 9.28 |
| Arxiv 0.5% | 2.27 | 2.62 | 2.49 | 2.80 | 5.52 | 4.37 | 3.35 | 7.12 |
| Flickr 1% | 5.60 | 7.14 | 7.33 | 13.57 | 22.93 | 6.04 | 10.43 | 31.02 |
| Reddit 0.1% | 1.51 | 2.16 | 1.49 | 1.53 | 0.57 | 2.96 | 1.70 | 9.59 |
| Corr. | 0.81 | 0.93 | 0.94 | 0.97 | 0.95 | 0.78 | 0.90 | - |
| VNG | GCond | MSGC | SGDD | Correlation | |
| Avg ACC | 63.34 | 69.40 | 69.02 | 68.95 | - |
| Density (%) | -0.81 | 0.07 | 0.55 | 0.13 | 0.91 |
| Max Eigen | 0.85 | 0.25 | 0.95 | 0.28 | -0.51 |
| DBI | 0.81 | 0.93 | 0.94 | 0.97 | 0.96 |
| DBI-AGG | 0.99 | 0.93 | 0.95 | 0.89 | -0.80 |
| Homophily | -0.83 | -0.68 | -0.46 | -0.80 | 0.54 |
| Feature Noise | Structural Noise | Adversarial Structural Noise | |||||
| Dataset | Method | Test Acc. | Perf. Drop | Test Acc. | Perf. Drop | Test Acc. | Perf. Drop |
| Whole | 64.07 | 11.75% | 57.63 | 20.62% | 53.90 | 25.76% | |
| Random | 56.91 | 9.11% | 6 1.56 | 1.69% | 5 9.42 | 5.12% | |
| KCenter | 52.80 | 10.57% | 55.41 | 6.15% | 5 5.07 | 6.73% | |
| GCond | 64.06 | 7.63% | 65.64 | 5.35% | 66.19 | 4.55% | |
| GCondX | 61.27 | 10.40% | 6 0.42 | 11.65% | 6 0.75 | 11.15% | |
Graph Neural Networks
Graph Neural Networks learn from graph-structured data by combining information across connected nodes, edges, and larger structures.
See all 752 papers