GC4NC: A Benchmark Framework for Graph Condensation on Node Classification with New Insights

Shengbo Gong; Juntong Ni; Noveen Sachdeva; Carl Yang; Wei Jin

cs.LGJun 24, 2024

GC4NC: A Benchmark Framework for Graph Condensation on Node Classification with New Insights

Authors: Shengbo Gong, Juntong Ni, Noveen Sachdeva, Carl Yang, Wei Jin

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
		Coreset					Coarsening		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

Table 1: Node reduction performance: test accuracy (%,

\uparrow

), or anomaly-class average precision (AP; %,

\uparrow

) for Yelp . Means are shown; Yelp uncertainties are in Table 9 . Yelp uses a matched twelve-configuration GCN search selected by validation AP (Appendix A.6.1 ). The best and second-best available results, excluding Whole, are bold and u nderlined; ties share a rank. Blue and red denote structure-free and structure-based condensation.

Methods	Cora, r = 2.6%		Citeseer, r = 1.8%		Arxiv, r = 0.5%
Methods	MIA Acc ( $\downarrow$ )	Acc ( $\uparrow$ )	MIA Acc ( $\downarrow$ )	Acc ( $\uparrow$ )	MIA Acc ( $\downarrow$ )	Acc ( $\uparrow$ )
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

Table 2: Privacy preservation evaluation. "MIA Acc" measures how well an attacker can infer whether a node is in the training or test set. We also report node classification accuracy ("Acc"), aiming to emphasize the balance between model performance and privacy preservation.

		Feature Noise		Structural Noise		Adversarial Structural Noise
Dataset	Method	Test Acc. $\uparrow$	Perf. Drop $\downarrow$	Test Acc. $\uparrow$	Perf. Drop $\downarrow$	Test Acc. $\uparrow$	Perf. Drop $\downarrow$
	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%

Table 3: Denoising ability evaluation. "Perf. Drop" shows the relative loss of accuracy compared to the original results before corruption. The best results are in bold and results that outperform whole dataset training are u nderlined. Structure-free and Structure-based methods are colored blue and red .

	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	-

Table 4: NAS evaluation. Best result is in bold . Runner-up is u nderlined. Worst is colored red .

Figure 4: Condensed graph performance evaluated by different GNNs. The relative accuracy refers to the accuracy preserved compared to training on the whole dataset.

Figure 5: Test accuracy for different methods with different initialization.

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	-

Table 5: Graph properties of condensed graphs on Cora from different structure-based GC methods. The " Corr. " row shows the correlation of certain property between the condensed graph and the whole graph across five datasets. The "Whole" column of the " Corr. " row displays the average correlation value of the four methods.

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 $\downarrow$
Gradient Matching	GCond, DosCond, SGDD	GCond, DosCond, SGDD	GCond, DosCond, SGDD, MSGC
Trajectory Matching	SFGC	SFGC, GEOM	SFGC, GEOM

Appendix

Table 6: Comparison between our GC4NC and two concurrent works. "OOM" means if the benchmark explore when the GC methods report out-of-memory error. In "Impact of Initialization".

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
Reddit	232,965	57,307,946	210	602	15,3932/23,699/55,334

Appendix

Table 7: Datasets Statistics

Dataset	Labeling Rate	Reduction Rate of Labeled Nodes	Reduction Rate $r$
Citeseer		10%	0.36%
	3.6%	25%	0.9%
		50%	1.8%
Cora		10%	0.5%
	5.2%	25%	1.3%
		50%	2.6%

Appendix

Table 8: Explanation of Reduction Rate under transductive and inductive settings

Dataset

Reduction rate (%)

Node sparsification

Coarsen

Condensation

Whole

Structure-free

Structure-based

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

\pm

2.7

37.78

\pm

1.3

35.37

\pm

2.8

43.73

\pm

1.6

41.43

\pm

1.4

69.75

\pm

0.6

34.32

\pm

5.9

66.14

\pm

0.3

67.61

\pm

0.7

66.27

\pm

0.8

63.51

\pm

1.9

7 0.65

\pm

0.5

67.79

\pm

0.7

70.05

\pm

2.1

69.41

\pm

0.8

60.24

\pm

6.0

71.87

\pm

0.6

67.88

\pm

1.8

72.6

0.90

58.77

\pm

0.5

52.83

\pm

0.4

50.71

\pm

0.8

59.24

\pm

0.4

51.15

\pm

1.1

69.59

\pm

0.5

40.14

\pm

5.3

66.07

\pm

0.4

70.70

\pm

0.5

70.27

\pm

0.7

62.91

\pm

0.8

7 1.27

\pm

0.6

69.69

\pm

0.5

69.15

\pm

1.2

70.83

\pm

0.4

72.08

\pm

0.7

70.52

\pm

0.6

70.13

\pm

1.1

1.80

62.89

\pm

0.4

63.37

\pm

0.4

62.62

\pm

0.6

66.66

\pm

0.5

59.04

\pm

0.9

69.50

\pm

0.6

41.98

\pm

7.0

65.34

\pm

0.6

73.03

\pm

0.3

7 2.36

\pm

0.5

63.90

\pm

3.4

72.08

\pm

0.2

68.38

\pm

0.5

69.35

\pm

0.8

72.18

\pm

0.6

72.21

\pm

0.4

69.65

\pm

1.7

71.74

\pm

0.9

Cora

0.50

57.79

\pm

1.7

58.44

\pm

1.7

35.14

\pm

2.5

51.68

\pm

2.1

44.64

\pm

4.4

75.94

\pm

0.7

24.62

\pm

5.7

70.40

\pm

0.6

78.14

\pm

0.5

75.11

\pm

2.2

71.58

\pm

0.9

79.21

\pm

0.4

79.74

\pm

0.5

80.17

\pm

0.8

80.65

\pm

0.6

8 0.54

\pm

0.3

80.15

\pm

0.5

54.76

\pm

4.5

81.5

Appendix

Table 9: Complete node reduction results: test accuracy (%,

\uparrow

), or anomaly-class AP (%,

\uparrow

) for Yelp . Yelp reports mean

\pm

population standard deviation after the matched validation-AP search in Appendix A.6.1 ; Random, Herding, and K-Center pool thirty fits from three saved reduction seeds, and other methods use ten fits.

\star

denotes the original GCondX dispatch to DosCondX with GCondX settings. Other datasets retain their reported error bars; GDEM

\dagger

marks additional fixed-GCN measurements (ten evaluations). The best and second-best available results, excluding Whole, are bold and u nderlined; ties share a rank. Edge sparsification is separate in Table 10 .

Dataset	Graph 1		Graph 2		Graph 3		Whole
Dataset	Score	$r_{E}$ (%)	Score	$r_{E}$ (%)	Score	$r_{E}$ (%)	Whole
Citeseer	71.93 $\pm$ 0.20	68.32	71.95 $\pm$ 0.46	68.28	71.02 $\pm$ 0.37	68.15	72.6
Cora	80.43 $\pm$ 0.60	60.08	80.06 $\pm$ 0.68	60.25	80.95 $\pm$ 0.26	60.48	81.5
Pubmed	79.12 $\pm$ 0.14	54.32	79.15 $\pm$ 0.16	54.18	79.13 $\pm$ 0.26	54.28	78.6
Arxiv	69.35 $\pm$ 0.39	17.57	69.44 $\pm$ 0.41	17.55	69.30 $\pm$ 0.28	17.57	71.4
Flickr	47.18 $\pm$ 0.13	73.05	47.19 $\pm$ 0.16	72.89	47.18 $\pm$ 0.14	72.99	47.4

Appendix

Table 10: Reproduced TSpanner performance with complete edge-retention counts. Graphs 1–3 are independently generated sparsified graphs at the same distance threshold

t=4

. Each score is the mean

\pm

standard deviation of ten GCN evaluations on that fixed graph. Scores are test accuracy (%), or anomaly-class AP (%) for Yelp . Yelp uses the validation-AP search in Appendix A.6.1 , including its reproduced Whole baseline; other Whole entries retain the original published accuracy.

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

\pm

2.45

45.11

\pm

2.19

69.97

\pm

0.36

64.37

\pm

1.29

68.90

\pm

0.64

66.96

\pm

1.47

68.29

\pm

1.30

73.63

\pm

0.32

69.53

\pm

0.65

71.90

\pm

0.24

72.6

0.90

47.13

\pm

1.32

55.09

\pm

1.14

69.97

\pm

0.36

69.37

\pm

0.62

73.20

\pm

0.35

70.66

\pm

0.23

69.73

\pm

0.46

70.93

\pm

0.51

70.97

\pm

0.29

70.10

\pm

0.73

1.80

64.21

\pm

0.72

62.82

\pm

0.78

70.01

\pm

0.27

69.35

\pm

0.70

74.36

\pm

0.30

72.37

\pm

0.41

69.19

\pm

0.47

70.69

\pm

0.47

72.73

\pm

0.35

70.11

\pm

0.93

Cora

0.50

47.93

\pm

0.96

49.92

\pm

3.06

76.55

\pm

0.91

70.61

\pm

0.64

79.03

\pm

0.61

76.80

\pm

2.18

80.04

\pm

0.60

80.63

\pm

0.48

80.43

\pm

0.72

81.58

\pm

0.97

81.81

Appendix

Table 11: Experiment results under hyperparameter searching . The search space is shown in Table 13 . The best results, excluding the whole graph training results, are marked in bold .

Node reduction rate (%)	Fixed GCN	Validation-selected GCN
0.05	$46.08\pm 0.00$	$48.14\pm 2.59$
0.10	$46.08\pm 0.00$	$47.89\pm 2.08$
0.20	$46.08\pm 0.00$	$47.47\pm 1.44$

Appendix

Table 12: Supplementary macro-F1 diagnostic for GDEM on Yelp before and after downstream GCN search. Test macro-F1 (%) is reported as mean

\pm

standard deviation over ten evaluation seeds (1–10). Both columns evaluate the same saved condensed graphs. This separate search selects by validation macro-F1; the primary AP results in Tables 1 and 9 select by validation AP.

Panel of Figure 6 — Figure 6: Performance vs. Total Time and Epoch Time on Arxiv (left two) and Reddit (right two).

Figure 7: Hyperparameters Sensitivity Analysis on Condensed Graphs .

Figure 8: Hyperparameters Sensitivity Analysis on Whole Graphs .

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}

Appendix

Table 13: Hyperparameter Search Space for Sensitivity Analysis

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

Appendix

Table 14: Property preservation check for GDEM, a method explicitly preserve the graph property.

Figure 9: Performance of condensed graphs evaluated by different GNNs.

Figure 10: Condensed graph performance evaluated using different models including SGFormer on Cora .

	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

Appendix

Table 15: Recall@

K

of top-

K

architectures evaluated on Cora with

r=0.5

Architecture	Search Space
Number of propagation $K$	{2, 4, 6, 8, 10}
Residual coefficient $\alpha$	{0.1, 0.2}
Hidden dimension	{16, 32, 64, 128, 256, 512}
Activation function	{Sigmoid, Tanh, ReLU, Linear, Softplus, LeakyReLU, ReLU6, ELU}

Appendix

Table 16: Architecture search space for APPNP.

Graph property	Dataset and $r$	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	-

Appendix

Table 17: Graph properties in condensed graphs from different structure-based GC methods. The " Corr. " row shows the correlation of certain properties between the condensed graph and the whole graph across five datasets.

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	-

Appendix

Table 18: DBI in condensed graphs from both structure-based and structure-free GC methods, continued from Table 17 .

	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

Appendix

Table 19: Correlation between condensed graph properties and model performance.

		Feature Noise		Structural Noise		Adversarial Structural Noise
Dataset	Method	Test Acc. $\uparrow$	Perf. Drop $\downarrow$	Test Acc. $\uparrow$	Perf. Drop $\downarrow$	Test Acc. $\uparrow$	Perf. Drop $\downarrow$
	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%

Appendix

Table 20: Denoising effects of selected methods. "Perf. Drop" shows the relative loss of accuracy compared to the original results of each method before being corrupted. The best results are in bold and results that outperform whole dataset training are u nderlined. Structure-free and structure-based methods are colored as blue and red .

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Graph Neural Networks learn from graph-structured data by combining information across connected nodes, edges, and larger structures.

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An Efficient and Scalable Graph Condensation with Structure-Preserving

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Graph condensation (GC) is pivotal for enabling Graph Neural Networks (GNNs) deployment in resource-constrained scenarios by compressing large-scale graphs into compact synthetic counterparts. Existing GC methods commonly suffer from computational inefficiency due to coupled optimization as well as encountering poor generalization across GNN architectures. To address these challenges, this study proposes an Efficient and Scalable Graph Condensation with Structure-Preserving (SP-ESGC), which possesses a decoupled design that separates node condensation from graph structure generation. Specifically, it first employs heat kernel feature propagation to generate node representation via spectral graph theory-inspired diffusion. Further, a novel hybrid clustering strategy is designed to extracts discriminative intra-class centroids from the node representation. Finally, a pre-trained edge predictor infers transferable structural patterns from the original graph, ensuring accurate synthetic graph generation. Extensive experiments on real-world graph datasets demonstrate that the proposed SP-ESGC implementes a precise GC with significantly high computational efficiency. Moreover, SP-ESGC also generalizes well across diverse GNN architectures.

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