Adapting Precomputed Features for Efficient Graph Condensation
Authors: Yuan Li
Jun Hu
Zemin Liu
Bryan Hooi
Jia Chen
Bingsheng He
Venue: ICML 2025 poster (2025)
Keywords: graph neural networks
data condensation
Primary area: deep_learning->graph_neural_networks

Abstract: Graph Neural Networks (GNNs) face significant computational challenges when handling large-scale graphs. To address this, Graph Condensation (GC) methods aim to compress large graphs into smaller, synthetic ones that are more manageable for GNN training. Recently, trajectory matching methods have shown state-of-the-art (SOTA) performance for GC, aligning the model's training behavior on a condensed graph with that on the original graph by guiding the trajectory of model parameters. However, these approaches require repetitive GNN retraining during condensation, making them computationally expensive. To address the efficiency issue, we completely bypass trajectory matching and propose a novel two-stage framework. The first stage, a precomputation stage, performs one-time message passing to extract structural and semantic information from the original graph. The second stage, a diversity-aware adaptation stage, performs class-wise alignment while maximizing the diversity of synthetic features. Remarkably, even with just the precomputation stage, which takes only seconds, our method either matches or surpasses 5 out of 9 baseline results. Extensive experiments show that our approach achieves comparable or better performance while being 96× to 2,455× faster than SOTA methods, making it more practical for large-scale GNN applications. Our code and data are available at https://github.com/Xtra-Computing/GCPA.

Adapting Precomputed Features for Efficient Graph Condensation
Yuan Li Jun Hu' Zemin Liu?
Bryan Hooi Jia Chen
Bingsheng He
Abstract
plex graph structures such as social networks (Fan al., 2019; Zhang et al., 2022) and molecular structures (Guo et al., 2021; Gasteiger et al., 2021). However, real-world large-scale graphs pose significant challenges for training GNNs (Huang et al., 2021; :; Gao et al., 2024). To address this, graph condensation techniques (Jin et al., 2022b;a) pro- pose generating condensed graph from a large original graph, enabling models trained on the condensed graph to be directly applied to the original graph, achieving parable performance on both graphs. GC enhances train- ing efficiency by reducing computational costs on large- scale graphs. Recent studies further demonstrate that GC facilitates efficient GNN training with minor performance loss (Gao et al., 2024; Zhang et al., 2024).
Graph Neural Networks (GNNs) face significant computational challenges when handling large- scale graphs. To address this, Graph Condensa- tion (GC) methods aim to compress large graphs into smaller, synthetic ones that more manage- able for GNN training. Recently, trajectory match- ing methods have shown state-of-the-art (SOTA) performance for GC, aligning the model's train- ing behavior on a condensed graph with that on the original graph by guiding the trajectory of model parameters. However, these approaches require repetitive GNN retraining during conden- sation, making them computationally expensive. To address the efficiency issue, we completely bypass trajectory matching and propose a novel two-stage framework. The first stage, a precom- putation stage, performs one-time message pass- ing extract structural and semantic informa- tion from the original graph. The second stage, a diversity-aware adaptation stage, performs class- wise aligmment while maximizing the diversity of synthetic features. Remarkably, even with just the precomputation stage, which takes only seconds, our method either matches or surpasses 5 out of 9 baseline results. Extensive experiments show that our approach achieves comparable or better performance while being 96x to 2,455 faster than SOTA methods, making it more practical for large-scale GNN applications."
Prior Work. GC methods encompass a variety of ap- proaches, including coreset selection (Welling, 2009; Sener & Savarese, 2018 8), gradient matching (Jin et al., 2022b;a; Yang et al., 2024; Fang et al., 2024), distribution match- ing (Liu et al., 2022; Xiao et al., 2024; Gao et al., 2025), and trajectory matching (Zheng et al., 2024; Zhang et al., 2024). Among them, trajectory matching has emerged as the SOTA method due to its ability to align the train- ing behavior effectively. These methods collect training trajectories from the original graph to guide the training process on the condensed graph, ensuring consistent train- ing behavior between the two graphs. Notably, existing trajectory- methods operate in a Structure-Free (SF) manner, producing condensed graphs without edges, which simplifies the training process. Although these condensed graphs lack edges, they support GNN node classification tasks by adding self-loop edges for each node, ensuring compatibility with GNNs and achieving SOTA.
1. Introduction
Efficiency Issues with SOTA Method. Despite the leading performance, trajectory- -based methods require substantial number of trajectories to achieve optimal results. As illus- trated in Figure 1, trajectory collection involves repeatedly training the model from scratch (e.g., 200 times), taking remarkably long time (e.g., 452 hours on million-node graphs) and dominating the overall condensation time. This inefficiency hinders their applications at scale, e.g., on social networks or e-commerce graphs.
Graph Condensation. Graph learning through GNNs (Kipf & Welling, 2016; Hamilton et al., 2017) has significantly advanced graph data analysis, providing insights into com-
'National University of Singapore *Zhejiang University l GrrbbTax Holdings Pte. Ltd.. Correspondence to: Jun Hu wun.hu@nus.cdu.g>>>
Proceedings of the 4gnd International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
To address the inefficiencies, we propose a novel Graph Condensation framework via Precompute-then-AAaae ap-
Our code and data available https: http://gttt/////// com/ ra- wappptiin//CCi

---

Adapting Precomputed Features for Efficient Graph Condensation
Stage Trajectory Collection (2: 200 re-training rounds)
Stage Trajectory Matching
Ours (SF)
GEOM (SF)
Inilialize GNN
Accuraa (66)
67
67
SFGC
Train 7 epochs
GCDM
66
(> 452 hours for Slage 1 on Ogbn-Producta)
452:8h
65
64
SGDD
72.7h
Time Consumption Each Stage
Ogbn-Products
500,000
1,000,000 1,500,000 2,000,000 Time (s)
Figure 1: Trajectory- methods require repetitive GNN re-training during trajectory collection, which can be highly time-consuming. This stage accounts for the majority of the total runtime in such methods.
Performance vs. condensation time on Ogbn- Products. Our framework significantly reduces condensa- tion time, bypassing the costly trajectory collection stage and outperforming SOTA methods. Here, SF denotes structure-free free graph condensation, where the condensed graphs possess edges.
Figure 2:
proach (GCPA). Our method first employs a precomputa- tion stage and then a diversity-aware adaptation stage, eliminating the need for repetitive re-training with different random initializations. The precomputation stage extracts structural and semantic information from the original graph, achieving competitive performance in a short time. The adaptation stage further refines the precomputed features through class-wise wise alignment and diversity maximization. As a result, we achieve competitive performance (-1.5% to +2.4%) on node classification tasks with substantially faster training time (96x to 2,455x) compared to SOTA (GEOM (Zhang et al., 2024)), as exemplified in Figure 2. We summarize the key contributions of our work as follows:
dimensional features, A E {0, JONNN represents the adja- cency matrix, Y E NxC denotes the ground-truth one-hot node labels among C' classes, and y E IR~ records the la- bels in vector form. Graph condensation aims to generate synthetic graph (or condensed graph, used interchange- ably) corresponding to an existing graph such that a model trained on the synthetic graph is effective when applied to the original graph. Formally, given an original graph T (X, A,YY with N nodes, GC aims to generate a smaller synthetic graph S = A'' Y'} with N' nodes such that a GNN trained on achieves similar performance on as another GNN trained directly on T. In particu- lar, structure- graph condensation emerges as a storage- efficient graph condensation approach where the adjacency matrix is set to identity matrix, A' = so the synthetic graph does not contain structural information.
We propose GCPA, efficient graph condensation frame- work with precomputation and an adaptation stage. Compared to SOTA methods, our framework avoids the costly repetitive re-training during trajectory collection, achieving significant efficiency improvements.
Node classification is prevalent task simplified by graph condensation, involving label assignment based on node features and graph structure. Formally, given a graph & [X, A} ). and a subset of nodes Nz. S N with known la- bels Y I N2 the transductive semi-supervised node classification task involves predicting labels Yu E RC for an unlabeled subset of nodes N2 C N. The optimiza- tion goal can be formulated as bi-level problem:
Our framework is effective. The precomputation stage, which extracts structural and semantic information from the original graph, already matches or surpasses 5 out of 9 baselines. With the adaptation stage, we further enhance performance with class-wise feature alignment and diversity maximization, achieving SOTA performance on 4 out of 7 datasets.
Through extensive experiments on benchmark datasets, we demonstrate that our method achieves better or com- parable accuracy while being up to 2,455 faster than existing methods.
min L(GNN A), Y)
(1)
0s arg arg min L(GNN X A)) A')
where 9 denotes the learnable parameters of an L-layer GNN model, Os represents the optimal GNN parameters learned on the synthetic graph, and L is loss function eval- uating the node classification performance. Existing graph condensation approaches optimize this bi-level problem to learn an optimal synthetic graph S such that trained GNN
2. Preliminaries
Let G = (x, A, denote a graph, where E xd
denotes the node feature matrix with N nodes and d-
2

---

Adapting Precomputed Features for Efficient Graph Condensation
Morge Uniformlly Sampled Nodes
+
P2
Neighbor Aggregation
(Massags
Feature Adaptation Learning via Class Alignment Loss
Structure-based
Semantic-based
Precomputation Stage
Adaptation Stage
Figure 3: Overall pipeline of the proposed GCPA condensation framework. (1) Structure precomputation: Neighbor aggregation is performed to capture structural dependencies. (2) Semantic-based precomputation: Nodes are grouped by semantic relevance and uniformly sampled. (3) Adaptation learning: Synthetic features vl and v2 are pushed away through diversity constraints, while v2 and v3 are pushed away through sampled negative pairs.
with parameters 0s yields optimal performance on T. How- ever, the bi-level optimization problem is computationally intensive, as it involves nested optimization loops. To mit- igate this issue, introduce our framework that directly optimizes synthetic node features for improved efficiency.
ing graph's topology, facilitating the effective propagation of information across nodes. The structure-bant -ased precom- puted features H with K- K-hop neighbor aggregation can be recursively computed as:
for k: = 1,2, ..., ((k) (1 - a: ÁH 1) + +oH (0)
3. GCPA Framework
with Å = D AAD- Ā = A A+I
(2)
The overview of the Graph Condensation framework via a Precompute- teaettetthen-AAAe Ahapt approach (GCPA) is shown in Fig- ure 3. We introduce two stages- precomputation stage and diversity-aware adaptation stage-to to produce structure- free synthetic data. The precomputation stage involves structure-bart based neighbor aggregation and semantic-baag based aggregation on the original graph, achieving competitive per- formance a relatively short time. The adaptation learning stage further refines the precomputed features using a class- wise feature alignment objective improve performance with minor additional costs.
where H = X represents the node feature matrix, K
denotes the number of aggregation hops, and H = (() is the output of the last layer. D the degree matrix of A, where D E, Aij- The coefficient a controls the contribution of raw features each hop. Having processed the structural information, we omit the edges in the follow- up semantic- based precomputation as shown in Figure 3, focusing on processing semantic information.
3.2. Semantic-based Precomputation
To condense set of N aggregated features into N' synthetic node features, we perform semantic-based precomputation by merging uniformly sampled original nodes within each class. Specifically, for each synthetic node v; with class label e (1,2...., C}. we uniformly sample a subset of original nodes in the same class. Then, we compute the semantic-based features by taking the mean of the aggre- gated features of the sampled nodes:
3.1. Structure-based Precomputation
In the context of graph-basee based learning models. neighbor information aggregation refers to the process by which node features are enriched with structural information from neigh- boring nodes. This process allows a node's 's feature vector to incorporate not just its own information but also that of its surrounding neighborhood. Such aggregation is criti- cal for capturing relationships and dependencies in graph- structured data.
X
ΣH, jES,
for 1,2,....
M
(3)
Drawing inspiration from the graph diffusion pro- cess (Gasteiger et al., 2019; Frasca et al., 2020; Hu et al.,
with S, C ITy:
S M, I. = - -
2024b;a), we leverage neighbor structural information to pre- pre-process the original node features. The goal of graph diffusion is to smooth node features based on the underly-
where S; is the set of sampled original nodes for synthetic node i, I. denotes the indices of original nodes belonging to class c, the number of sampled nodes for each synthetic node.
3

---

Adapting Precomputed Features for Efficient Graph Condensation
This semantic-based precomputation process effectively condenses the semantic information of multiple nodes within the same class into single synthetic node. Further- more. by maintaining the class distribution in Y through proportional sampling, we fix the synthetic labels Y' to preserve the original class proportions. Consequently, we obtain the precomputed condensed dataset (X', Y"). where X E R.V' and Y' N'C.
With the sampled pairs, we optimize an adaptation loss as follows:
C= - E ((alx
loogo (H,, 23,)
Uniiooi to tyyry log
-(H,, Zi)
(5)
class-wis alignment
3.3. Feature Adaptation Learning
+ 7 B f:y/ log o
-(H; Zi)
Given the limited number of condensed nodes, is crucial that these nodes ideally depict the overall representations of their respective classes (depicted by the background color in Figure 3). Our precomputation stage effectively captures the structural and semantic features of the original graph, setting up a solid foundation for downstream learning. Since the precomputation stage is not directly optimized for the final objective, we further integrate an adaptation learning stage that adjusts the class-wise representations.
intra-class diversity
where (H; Z'z) denotes the inner product between the i-th anchor node's representation H, and the j-th syn- thetic node's ' adapted representation Z'j. t is the index of random negative sample from the synthetic dataset, and o (xx) 1/(1+ exp((xx) is the sigmoid function. The adap- tation module refines the precomputed representations to better align synthetic and original node representations, en- hancing the generalization of the condensed features. We op- timize the model using the A optimizer (Loshchilov & Hutter, 2017), with learning rate v,, weight decay X, and betas (31, 32) as hyperparameters.
To achieve better class-wit wise representation separation, we consider contrastive loss to enhance node features for classification utility (Joshi et al., 2022). For the task of graph condensation, we propose to align the condensed features with the original precomputed features using a class- wise adaptation loss. Specifically, we introduce an adaptation module fndapt : IR xd + N implemented as a Multi- Layer Perceptron (MLP) (Rosenblatt, 1958) with F layers, to better depict the overall representations:
3.4. Complexity Analysis
We analyze the time complexity using GCN as the back- bone for GC. For GCPA, the time complexity is O(KE& Ed + PF N²)))), where E is the number of edges of the original graph and P is the number of epochs. The first term corresponds to precomputation, while the second accounts for adaptation. In contrast, the time complexity of SOTA (GEOM) is O(T + N. + N'd + N2d where T represents the number of repetitive GNN retraining itera- tions for trajectory matching. Compared to GCPA, GEOM includes an additional multiplicative factor T, which can be large (e.g., in the hundreds) and significantly increases GEOM's time complexity. Conversely, GCPA bypasses trajectory matching, eliminating the dependence on T and making it substantially more efficient than GEOM.
Z' ='= 3'= (1 Sadup (X')
(4)
where a hyperparameter controlling the contribution of precomputed representations, Z' represents the adapted synthetic representations. We adopt (Z', Y') as the final condensed dataset after the learning process. Notably, X' is treated as fixed, non non-trainable condition during training, thereby imposing a permanent constraint on Z'. This con- trasts with prior approaches (Jin et al., 2022b; Zheng et al., 2024; Zhang et al., 2024), which use randomly sampled features only for initialization and allow these signals to drift away throughout optimization.
4. Experiments
We further construct the contrastive samples by first sam- pling a sufficient number of anchor nodes H, from the precomputed representations H. These anchors serve as learning targets during the adaptation stage, encouraging the synthetic representations preserve the original feature distributions. The first introduced constraint class-wise alignment, where, for each anchor node in the original graph, we sample a synthetic node belonging to the same class as a positive sample and a set of arbitrary synthetic nodes from different classes as negative samples. Additionally, we introduce an intra-class diversity constraint to encour- age synthetic features within each class to be dissimilar.
In this section, we conduct experiments to validate the ef- fectiveness and efficiency of the proposed framework.
4.1. Experimental Setup
Datasets. Following GCondenser (Liu al., 2024), a com- prehensive graph condensation benchmark, our experiments are conducted on seven benchmark datasets including three smaller networks: CiteSeer, Cora, and PubMed (Kipf & Welling, 2016), and four larger graphs: Ogbn-arxiv, Ogbn- products (Hu et al., 2020). Flickr (Zeng et al., 2020), and Reddit (Hamilton et al., 2017). We use the public data splits
4

---

Adapting Precomputed Features for Efficient Graph Condensation
Table I: Summary of dataset statistics.
Sctting
Dataset
# Train/Val/Tess Nodes
# Nodes
# Edges
# Features
Classes
CiteSeer
120/500/1 1,000
3,327
4,732
3,703
6
Cora
140/500/1 1,000
2,708
5,429
1,433
7
Transductive
PubMed
60/500/1.000
19,717
88,648
500
Ogbn-arxiv
90.941129,799/488.60
169,343
1,166,243
128
40
Ogbn-products
196.6615//99,322222213.000
2,449,029
61 61,859,144
100
47
Flickr
44,625/22.312/22..33
89,250
899,756
500
7
Inductive
Reddit
153,431/23,831/557703
232,965
57,307,946
602
41
for fair comparisons. The dataset statistics and settings are detailed in Table 1. For CiteSeer, Cora, and PubMed datasets, row feature normalization is applied to prepare the data. For Ogbn-arxiv, Flickr, and Reddit datasets, we apply feature standardization. The Ogbn-products dataset retains its feature processing as defined by OGB (Hu et al., 2020).
good condensation results on all datasets. This highlights the non-trivial nature of graph condensation tasks, which necessitate substantial effort.
Two distinct categories of graph condensation methods, including gradient matching and distribution matching, have both shown fair performance on different datasets. Notably, neither category consistently outperforms the other. This variation in performance suggests that multi- ple frameworks might be applicable the task of graph condensation, without universally superior approach.
Baselines. We compare proposed framework the baselines in the following categories: (i) Coreset approach:
K-Center (Sener & Savarese, 2018). (ii) Gradient match- ing approaches: GCond (Jin et al., 2022b), SGDD (Yang et al., 2024), and EXGC (Fang et al., 2024). (iii) Distri- bution matching approaches: GCDM (Liu et al., 2022), SimGC (Xiao et al., 2024), and CGC (Gao et al., 2025). (iv) Trajectory matching approaches: SFGC (Zheng al., 2024)
Recent advancements in trajectory matching, especially the SFGC and GEOM approaches, have demonstrated superior performance on most datasets, affirming the ef- ficacy of trajectory-based methods. Notably, both SFGC and GEOM employ structure-free condensation, indicat- ing that for node classification tasks, providing edges in condensed graphs may not always be necessary.
and GEOM (Zhang et al., 2024). We report results based the following sources, depending on their availability: (a)
official results reported in the original papers of each base- line, (b) reproduced results provided by the GCondenser benchmark (Liu et al., 2024), and (c) results obtained by running the official code.
Our proposed framework achieves superior performance on out 7 datasets, underscoring the effectiveness of our precompute teieee-the-addeeee -dapt approach.
Backbone Models. We use GCN (Kipf & Welling, 2016) and SGC (Wu et al., 2019) as backbone models during condensation and evaluation for fair comparisons. In the crchttacch evaluation, we use more models including MLP (Rosenblatt, 1958), GAT (Veličković et al., 2018), ChebNet (Defferrard et al., 2016), GraphSAGE (Hamilton al., 2017), and APPNP (Gasteiger et al., 2018).
4.3. Efficiency Comparison
We present a comprehensive efficiency comparison of dif- ferent methods using the GCN backbone in Table and the SGC backbone in Table 8 (located in the Appendix). Addi- tionally, Figure illustrates a joint analysis of both accuracy and efficiency. Based on the results, we make the following observations on the efficiency of different approaches:
The details of the evaluation schemes and hyperparameter settings are provided in Appendix A and B, respectively.
4.2. Performance Comparison
As depicted in Figure 4, GEOM, a trajectory-basee method, exhibits leading performance but suffers from poor efficiency. The primary efficiency bottleneck lies in the need for repetitive re-training, which, while effective, leads to severe efficiency issues.
We present the performance of different graph condensation approaches using the GCN backbone in Table 2. Addition- ally, the performance of these approaches with the SGC backbone is shown in Table 7, located in the Appendix. Based on these results, we make the following observations:
As shown in Figure 4, our framework achieves compara- ble or leading performance across datasets. Notably, the framework significantly more efficient than trajectory- based methods, achieving speedups ranging from 96x to 2,455 x compared to these approaches.
The coreset approach, K-Center, which typically employs conventional machine learning techniques, fails to provide
5

---

Adapting Precomputed Features for Efficient Graph Condensation
Table 2: Node classification performance comparison using GCN backbone (mean The best and second- results are marked in bold and underlined, respectively. Ours (Pre.) is precomputation-only variant. The Whole column represents the performance obtained by training on the whole dataset.
Dataset
Ratio
K-Cen.
GCond
SGDD
GCDM
SimGC
EXGC
CGC
SFGC
GEOM
Ours (Pre.)
Ours
Whole
0.9%
52.4
70.5
69.5
71.2.ss
73.8
69.2
72.5
71.4
73.01es
72.1 1o2
75.4
CiteSeer
1,8%
64.3+10
70.6+00
70.2+0a
71.9+0
72.2+05
70.1
72.4+02
72.4+04
74.3+0u
72.1+01
74,8+a3
71.4
3.6%
69.1+a.
69.8+14
70.3+L7
72.3+13
71.1+18
70.6+09
72.0+us
70.6=u
73.3+04
72.7+05
74,9
1.3%
64.0a23
79.8
80.1
78.9
80.8=23
$2.0a04
82.7
80.
82.5
80.3
82.1
Cora
2.6%
73.2
80.
80.61ex
79.4
80.91
81.11
82.3
81.7.ms
83.61
80.610s
82.9
81.7
5.2%
16.7
19.3
80.41:
79.9
82.1
82.3
82.5
81.61mx
82.81s7
80.8
82.31cs
0.08%
72.1
67.6+04
76.7+Lu
75.9
74,4+02
77.8+aa
77.6+04
78,4+u
80.1+aa
79.5+13
80.5++4
PubMed
0.15%
76.4ann
74.6208
78.5
77.4
76.0zaa
78.3
77.820s
18.1
79.7
79.7
80.9
79.3
0.3%
78. 2
77.2=a7
78.0
77.6204
76.2
76.2+0.
77.7
78.5
79.5
79.3
81.7
0.05%
47.2
59.2
60.811a
63.31ss
63.61ss
57.610a
64.11s4
65.5
65.5
60.51ov
67.2
Arxiv
0.25%
86.81s
63.21s9
65.811a
59.6184
66.4187
62.31ss
66.41m
68.8 66.
64.6199
67.71oa
71.1
0.5%
60.3++04
64.0+14
66.3+07
62.4+aa
66.8+04
65. 15,0+es
67.2+04
66.8+u4
69.6
65.5+0s
68.1+as
0.025%
55,4
63.7
64.0ao4
66.5aa.
63, 1313
68.02nt
67.1
68.5
64.1 100
69.3
Products
0.05%
57.6+07
67 02.0
65, 65.9
68 4+04
64.8
64.7+14
68, $8,9203
67.9203
69.8 69.8.03
65. 25.9
70.2:44
73.1
0.1%
59.11es
68.01aa
66.1
18.4
67.01s7
66.4100
69.1
10.
71.1
67.7.os
71.5
0.1%
42.0as7
46.5
46.9
44.5
45.3
47.0a.1
46. 46.8
46. 46.62n2
47. Jan
44.4
47.2
Flickr
0.5%
43.2
+7.1
47.1
45.0 45.0tea
45.6 45.6
48.3
47.1
47.0
47.0zuz
45.4
47.1
46.8
1%
44.1
=7.1
#7.1
44.6203
43.8
48.4
47 02nl
47.1
47.3
45.4
47.2
0.05%
46.6+23
88 08.0
90.5+21
88 2412
91.1=
90.2+a.1
90.6e02
89.7=02
91.1=00
90,5+03
90.5+03
Reddit
0.1%
53.0131
89.61a7
91.811
91.81aa
92.0gaa
90.61ne
91.4
90.0gms
91.4gn
91.310a
93.0
94.2
0.2%
58.5
90.
91.6
92.21a1
92.62a1
91.8
91. 01.62go
90.3
91.52ma
91.4201
92.9
Table 3: Efficiency comparison using GCN backbone (total condensation time in seconds).
Dataset
K-Cen.
GCond
SGDD
GCDM
SimGC
EXGC
CGC
SFGC
GEOM
Ours (Pre.)
Ours
CiteSeer (r=1.8%)
7
71
70
57
245
237
32
2,165
10,890
45
Cora (r=2.6%)
5
70
70
240
235
30
2,578
10.144
44
PubMed (r=0.15%)
59
223
291
278
51
8,060
26,432
39
Arxiv (r=0255%)
18
389
759
555
338
126
86,553
104,905
247
Products (+=0.05%)
91
13,554
21,821
11.485
4861
4915
1093
1,509,397
1,912,105
104
2,985
Flickr (r=0.5%)
16
187
1,178
165
425
412
94
96,350
21,061
23
219
Reddit (r-0.1%)
51
2.665
12,126
1563
702
692
182
379.974
128.642
55
505
CiteSeer (r-1.%)
PubMed (r-0.15%
Arxiv (r- 0.25%)
Ours
Ours
GEOM +
Accuracy (96)
GEOM
80.0
GEOM +
74.0
Ours
73.0
78.0
CCGC
SFGC
67.0
X CGO CGOCFGC
SFGC SimGC
76.0
SimGC
66.0
CGC
SFGC
0e0
2e3
5e3
Be3
1e4
0e0
1e4
2e4
0e0
2e4
5e4
Be4
1e5
Products (r=0.)
Flickr (r=0.5%)
Reddit (r=0.1%)
70.0
+ Ours
GEOM
47.0
Ours
GEOM
SFGC
93.0
+ Ours
Accuracy (96)
* CGC
92.0
SimGC
68.0
SFGC
46.5
CGC
GEOM
66.0
46.0
SimGC
SimGC
90.0
SFGC
0e0
5e5
1e6
2e6
2e6
Deo
2e4
5e4 (s)
8e4
1e5
0e0
1e5
2e5
3e5
Time
Time
Figure 4: Evaluation accuracy versus total condensation time using GCN backbone.
6

---

Adapting Precomputed Features for Efficient Graph Condensation
Table 4: Cross-architecture transferability of condensed graphs using GCN backbone.
Dataset
Method
MLP
SGC
GCN
GAT
ChebNet
SAGE
APPNP
Avg.
Std.
GCond
41.8
34.8
39.2
57.4
61.2
47.0
46.8
8.8
CiteSeer
GCDM
62.3
69.6
72.7
58.3
60.2
67.1
71.4
65.9
5.3
0.90%)
SEGC
64.4
64.9
70.4
70.0
69.1
69.5
70.8
68.4
2.5
Ours
66.5
70.9
73.4
73.4
72.8
72.6
72.1
71.7
2.3
GCond
67.7
72.6
79.5
80.7
60.0
78.6
79.0
74.0
7.2
Cora
GCDM
65.3
78.5
80.2
80.1
58.4
77.5
79.3
74.2
8.1
(r
1.30%)
SFGC
68.2
76.2
80.4
79.8
62.1
77.6
81.6
75,1
6.7
Ours
70.5
79.9
81.3
79.1
82.1
78.9
76.2
78.3
3.6
GCond
75.1
55.6
75.0
77.0
74.3
77.2
78.0
73.2
7.3
PubMed
GCDM
73.8
72.9
75.0
73.7
70.5
75.3
76.9
74.0
1.9
(r = 0.08%)
SFGC
73.6
76.8
78.5
76.6
77.2
76.7
78.9
76.9
1.6
Ours
74.2
76.6
76.
76.3
77.3
77.5
76.7
76.4
1.0
GCond
39.2
58.0
$7.0
47.7
36.4
33.5
54.3
46.6
9.5
Arxiv
GCDM
41.6
59.8
60.7
46.5
52.6
55.3
60.3
53.8
6,9
(r= = 0.05%)
SFGC
45.3
62.2
63.3
60.5
50.7
55.4
62.4
57.1
6.4
Ours
46.7
61.6
64.4
63.3
58.4
53.9
59.0
6.2
GCond
36.4
45.7
60.7
48.4
45.2
49.8
60.3
49.5
8.0
Products
GCDM
45.7
60.0
61.2
63.6
66.2
61.6
7.0
0.025%)
SEGC
46.7
61.4
63.4
7.2
Ours
46.3
65.9
67.6
67.8
62.2
62.1
62.5
7.0
GCond
40.8
36.5
40.8
43.0
43.2
44.9
42.0
2.7
Plickr 0.1%)
GCDM
41.7
27.3
37.7
41.5
43.0
43.8
39.4
5.3
SEGC
44.9
38.7
46.2
45.3
43.6
44.9
46.2
44.3
2.4
Ours
44.1
45.1
45.3
45.1
42.4
43.6
45.4
44.4
1.0
GCond
38.7
82.2
79.9
31.2
38.7
41.5
69.8
54.6
20.2
Reddit
GCDM
43.1
87.1
88.1
37.5
55.6
66.2
68.9
63.8
18.3
0.05%)
SEGC
47,5
82,8
87.0
84.4
53.6
71.9
67.5
70.7
14.4
Ours
39.3
91.1
90.9
90.5
61.8
79.1
66.4
74.1
18.1
Table 5: Ablation on precomputation components includ- ing both stuucture-base (Stru.) and semantic-basee (Sem.) aggregation phases. The w/o Both variant initializes the synthetic features using randomly selected original features.
These observations demonstrate that our method not only achieves competitive performance but does so with markedly higher efficiency, addressing one key chal- lenges in scalable graph learning applications.
Dataset
Ours
w/o Stru.
w/o Sem.
w/o Both
4.4. Cross-architecture Transferability
CiteSeer (r=1.80%)
74.8
69.3
67.8102
62.1
Table presents the cross-architecture transferability results of condensed graphs across different models. Our method consistently matches outperforms the top performance across all datasets, underscoring the robustness and general- ization of our framework. The ability to transfer across dif- ferent architectures may be attributed to the similar filtering behavior of popular GNNs, as reported in the literature (Jin et al., 2022b; Zheng et al., 2024). In particular, our frame- work demonstrates outstanding transferability, which may be attributed our direct alignment between original and synthetic features without relying specific GNN models for performance matching.
Cora (r=2.60%)
82.921a
74.4
19.1
72.2
PubMed (5:0.15%)
80.9
77.92ms
78.8104
76. ans
Arxiv (r=0.25%)
67.7
64.22m6
67.3103
63.9
Products (r=0.05%)
70.2+a5
64.6+as
65.7+a2
62.2+a7
Flickr (r=05%%
47.1
46.92m1
47.0aus
46.9
Reddit (r=0.10%)
93.0μm
92.4
92.24m6
92.24M:
Our method is not only efficient than the time- intensive trajectory-based methods but also faster than the majority of other baseline methods on most datasets. These results underscore the superior condensation effi- ciency of our precompute-then thhnaapppt framework.
4.5. Ablation on Precomputation Components
A variant of our method containing only the precompu- tation stage (Pre.), typically taking under 60 seconds to complete, matches or surpasses 5 out of 9 baselines, as de- tailed in Table 2. The results illustrate the capability of the precomputation stage to achieve competitive results a fraction of the time compared to learning-bart baselines.
Table 5 evaluates the impact of structure- based and semantic- based phases of the precomputation stage. The results show that both the structural and semantic components contribute to the performance the framework, particularly on trans- ductive datasets, which reflects the importance of precom-

---

Adapting Precomputed Features for Efficient Graph Condensation
Table 6: Impact of diversity constraint.
Ogbn-arxiv (0.25%)
Flickr (0.5%)
68
Accuracy (%)
47.0
Dataset
Y - 0
7 = 0.01
7 = 0.1
? =
66
CiteSeer (r=1.00%)
72.1
73.5+04
74.8=03
74,2+aa
46.5
Method
Coru (r=2.60%)
81,4+02
82.9
80.3=04
79.7+aa
Ours
PubMed (r=0.15%)
79.9
80.2+aa
80,9
79.3+a4
64
46.0
SOTA
Arxiv (r=0.25%)
64.6204
65.81ns
67.7
66. 66.1aax
100
200
300
0
200
400
Products (f=0.05%)
65.9
66.3ana
70.2
68.4
Epoch
Epoch
Flickr (r=0.5%)
45.4
47.1
46.8
25.9
Reddit (r=0.10%)
91.3 2012
92.1
93.0=:..
92.7aa4
Figure 5: Effect of adaptation learning over different num-
bers of adaptation epochs.
putation transductive datasets where the complete graph structure is available. We also observe that removing struc- tural components typically results in a larger performance drop compared to the removal of semantic components, This indicates the critical role of structure- capturing representative features the original graph. In conclusion, the structural and semantic components are both pivotal to achieving optimal performance in our framework, but their impact varies with the nature of the datasets.
Citeseer PubMed
Apxiv Flickr
Citeseer PubMed
Andiv Fllckr
80
80
70
70
aggregation in
60
60
50
0
2
1
10
50
100
(a) Precomputation Hops
(b) Semantic Aggr. Size M
4.6. Impact of Diversity Constraint
80
80
Table 6 evaluates the impact of the diversity constraint model performance across different datasets. The hyperpa- rameter yy controls diversity, with higher values promoting greater diversity in the synthetic features. We observe that while the constraint generally enhances performance, exces- sively high values y can negatively affect the results.
70
70
60
60
50
0
0.25
0.5
0.75
10
50
(c) Residual Coefficient /
(d) Negative Samples S
4.7. Impact of Adaptation Learning
Figure 6: Impact of hyperparameters on accuracy (%)
We demonstrate the impact of the adaptation learning stage in Figure 5. On the presented large datasets, the precompu- tation stage (Epoch 0) produces condensed representations with sub-optimal performance. The adaptation learning fur- ther improves the representations by aligning them with the original node representations, achieving SOTA performance after sufficient training epochs.
suggests that the adaptation learning phase is relatively in- sensitive to these hyperparameters, reducing the need for exhaustive tuning and enhancing the method's applicability across datasets and deployment settings.
5. Related Work
4.8. Extended Analysis of Hyperparameters
Graph Condensation (GC) (Jin et al., 2022b;a; Yang et al., 2024), derived from dataset distillation (Wang et al., 2018; Zhao et al., 2021). aims to produce smaller version of a graph while retaining information from the original. Its optimization goal is for GNNs trained on the condensed graph to perform similarly to those trained on the original. GC methods can be categorized into two classes: structured GC, which generates both node features and graph structure; and structure-free GC, which only focuses on synthesizing node features but does not generate explicit graph structure.
As shown in Figure 6a, increasing the number of structure- based precomputation hops K generally improves accuracy across datasets. A similar trend is observed in Figure 6b for the semantic aggregation size M. These positive cor- relations highlight the importance of both structure- and semantic-based aggregation modules during precomputa- tion. Notably, performance gains begin saturate beyond K-2 and 50 indicating diminishing returns once the representations are sufficiently aggregated.
In contrast, Figures 6c and 6d show that varying the residual coefficient 3 and the number of negative samples yields less significant changes in final accuracy. This robustness
Structured Graph Condensation synthesizes graph struc- ture using a neural network that generates links between nodes based on their features. GCond (Jin et al., 2022b) uses
8

---

Adapting Precomputed Features for Efficient Graph Condensation
gradient matching loss between the original and condensed graphs, but its nested optimization loop limits efficiency. GCDM (Liu et al., 2022) generates smaller graphs with distribution similar to the original graph, using a distribution matching loss measured by maximum mean discrepancy. SGDD (Yang et al., 2024) incorporates the original graph structure through optimal transport. SimGC (Xiao et al., 2024) aligns condensed and original graphs using pre- trained SGC model without external parameters for efficient graph condensation. EXGC (Fang et al., 2024) employs mean-field variational approximation and gradient-based ex- plainability to efficiently condense large graphs. CGC (Gao et al., 2025) introduces a training- free graph condensation framework that transforms class-level distribution matching into a clustering- -based class partition problem.
achieving SOTA results with further adaptation stage. Our framework demonstrates that precomputation is promis- ing solution for efficient graph condensation, and it is also flexible, as it can be further enhanced through adaptation learning. In the future, we plan to explore more precompu- tation techniques for graph condensation.
A Acknowledgements
This research is supported by the National Research Founda- tion, Singapore and Infocomm Media Development Author- ity under its Trust Tech Funding Initiative. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore and Infocomm Media Development Authority.
Structure-Free Graph Condensation assumes that struc- tural information can be embedded directly into the syn- thetic node features, bypassing the need to generate graph structure. GCondX (Jin et al., 2022b), a variant of GCond, focuses solely on feature learning via gradient match- ing without the inner loop. SFGC (Zheng et al., 2024) matches training trajectories with expert guidance, and GEOM (Zhang et al., 2024) adjusts the matching range for different node difficulties. While these condensed graphs do not possess edges, they enable GNN- node classifi- cation by introducing self-loops for each node, simplifying the training process and achieving SOTA performance.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential socictal consequences of our work, none which we feel must be specifically highlighted here.
References
Cai, C., Wang. D., and Wang. Y. Graph coarsening with neu- ral networks. In 9th International conference on Learning Representations, 2021.
Graph Coarsening methods (Cai al., 2021; Loukas & Vandergheynst. 2018; Huang et al., 2021; Deng et al., 2019) reduce graph size by clustering original nodes into super- nodes, preserving the overall structural or spectral properties while enabling more efficient downstream processing
Caliñski, T. and Harabasz, J. A dendrite method for clus- ter analysis. Communications in Statistics-thes and Methods, 3(1):1-27, 1974.
Coreset methods (Sener & Savarese, 2018; Welling, 2009; Wolf, 2011) select subset of the original nodes and retain the induced edges. K-Center (Sener Savarese, 2018) trains a GCN on the original graph to generate embeddings, from which k-nearest nodes are sampled to form subgraph.
Davies, D. L.. and Bouldin, D. W. A cluster separation measure. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-1(2)2244-227, 1979.
Defferrard, M., Bresson, X., and Vandergheynst, P. Con- volutional neural networks on graphs with fast localized spectral filtering. Advances in neural information pro- cessing systems, 29, 2016.
Different from existing methods, our framework employs a streamlined precomputation and adaptation process that extracts and aligns features efficiently, avoiding the costly re-training in SOTA methods.
Deng, C., Zhao, Z., Wang, Y.. Zhang, Z., and Feng, Z. Graphzoom: A multi-level spectral approach for accurate and scalable graph embedding. In International Confer- ence on Learning Representations, 2019.
6. Conclusion
In this paper, we propose framework, GCPA, for effi- cient graph condensation. The method consists only of a precomputation stage that performs one-time message pass- ing and a diversity-aware adaptation stage. Compared to SOTA methods, our framework avoids costly repetitive re- training, achieving up 2,455x faster training speed. Our framework is also effective, surpassing or matching a ma- jority of baselines with just the precomputation stage and
Fan, W., Ma, Y., Li, Q., He, Y., Zhao, E.,, Tang, J., and Yin, D. Graph neural networks for social recommendation. In The world wide web conference, pp. 417-426, 2019.
Fang, J., Li, X., Sui, Y., Gao, Y., Zhang, G., Wang, K., Wang, X., and He, X. Exgc: Bridging efficiency and explainability in graph condensation. In Proceedings of the ACM Web Conference 2024, pp. 721-732, 2024.

---

Adapting Precomputed Features for Efficient Graph Condensation
Frasca, F., Rossi, E., Eynard, D., Chamberlain, B., Bron- stein, M., and Monti, F. Sign: Scalable inception graph neural networks. arXiv preprint arXiv:2004. 2020.
Jin, W., Tang, X., Jiang, H., Li, Z.., Zhang, D., Tang, J., and Yin, B. Condensing graphs via one-step gradient matching. In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 720-730, 2022a.
Gao, X., Chen, T., Zhang, W., Li, Y., Sun, X., and Yin, H. Graph condensation for open-world graph learning. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 851-862, 2024.
Jin, W., Zhao, L.,, Zhang, S., Liu, Y., Tang, J., and Shah, N. Graph condensation for graph neural networks. In International Conference on Learning Representations, 2022b.
Gao, X., Ye, G., Chen, T., Zhang, W., Yu, J., and Yin, H. Re- thinking and accelerating graph condensation: A training- free approach with class partition. In Proceedings of the ACM on Web Conference 2025, pp. 4359 4373, 2025.
Joshi, C. K., Liu, F., Xun, X., Lin, J., and Foo, C. S. On representation knowledge distillation for graph neural networks. IEEE transactions on neural networks and learning systems, 2022.
Gasteiger, J., Bojchevski, A., and Günnemann, S. Predict then propagate: Graph neural networks meet personal- ized pagerank. In International Conference Learning Representations, 2018
Kipf, T. N. and Welling, M. Semi-supervised classifica- tion with graph convolutional networks. International Conference on Learning Representations, 2016.
Gasteiger, J., Weißenberger, S., and Günnemann, S. Diffu- sion improves graph learning. Advances in neural infor- mation processing systems, 32, 2019.
Liu, M., Li, S., Chen, X., and Song. Graph condensation via receptive field distribution matching. arXiv preprint arXiv:2206.1 2022.
Gasteiger, J., Becker, F., and Günnemann, S. Gemnet: Uni- versal directional graph neural networks for molecules. Advances in Neural Information Processing Systems, 34: 6790-6802, 2021.
Liu, Y., Qiu, R., and Huang, Geondenser: Benchmarking graph condensation. arXiv preprint arXiv:2405. 142466 2024.
Loshchilov, I. and Hutter, F. Decoupled weight decay reg- ularization. In International Conference Learning Representations, 2017.
Guo, Z., Zhang, C., Yu, W., Herr, J., Wiest, O., Jiang, M., and Chawla, N. V. Few-shot graph learning for molecular property prediction. In Proceedings of the web conference 2021, pp. 2559-2567, 2021.
Loukas, A. and Vandergheynst, P. Spectrally approximat- ing large graphs with smaller graphs. In International conference on machine learning, pp. 3237-3246. PMLR, 2018.
Hamilton, W. Ying, Z., and Leskovec, J. Inductive repre- sentation learning on large graphs. Advances in neural information processing systems, 30, 2017.
Hu, J., Hooi, B., and He, B. Efficient heterogeneous graph learning via random projection. IEEE Trans. Knowl. Data Eng., 36(12):8093 0108099-810,,,, 2024a.
Rosenblatt, F. The perceptron: probabilistic model for information storage and organization in the brain. Psy- chological review, 65(6))386, 1958.
Hu, Hooi, B., Qian, S., Fang, Q., and Xu, C. MGDCF: distance learning via markov graph diffusion for neural collaborative filtering. IEEE Trans. Knowl. Data Eng., 36(7):3281-3296, 2024b.
Rousseeuw, P. J. Silhouettes: graphical aid to the inter- pretation and validation of cluster analysis. Journal of computational and applied mathematics, 20:53-65, 1987.
Sener, O. and Savarese, S. Active learning for convolutional neural networks: A core-set approach. In International Conference on Learning Representations, 2018.
Hu, W., Fey, M., Zitnik, M., Dong, Y., Ren, H., Liu, B., Catasta, M., and Leskovec, Open graph benchmark: Datasets for machine learning on graphs. Advances in neural information processing systems, 33:22118-22111 2020.
Veličković, P., Cucurull, G., Casanova, A., Romero, A.. Liò, P., and Bengio. Graph attention networks. In International Conference on Learning Representations, 2018.
Huang, Z., Zhang, S., Xi, C., Liu, T., and Zhou, M. Scal- ing up graph neural networks via graph coarsening. In Proceedings of the 27th ACM SIGKDD conference on knowledge discovery data mining, pp. 2021.
Wang. T., Zhu, J.-Y. Torralba, A., and Efros, A. A. Dataset distillation. arXiv preprint arXiv: 1.111111.10900 10959, 2018.
10

---

Adapting Precomputed Features for Efficient Graph Condensation
Welling, M. Herding dynamical weights to learn. In Pro- ceedings of the 26th annual international conference machine learning, pp. 1121-1128, 2009.
Wolf, G. W. Facility location: concepts, models, algorithms and studies, series: Contributions management sci- ence. International Journal of Geographical Information Science, 25(2):331-333 2011.
Wu, F., Souza, A., Zhang, T., Fifty, C.,, Yu, T., and Wein- berger, K. Simplifying graph convolutional networks. In International conference on machine learning, pp. 6861- 6871. PMLR, 2019.
Xiao, Z.,, Wang, Y., Liu, S., Wang, H., Song, M., and Zheng, Simple graph condensation. Joint European Confer- ence on Machine Learning and Knowledge Discovery in Databases, pp. 53 13-71.30 Springer, 2024.
Yang, B., Wang, K., Sun, Q.. Ji, C.,, Fu, X., Tang, H., You, Y., and Li, Does graph distillation see like vision dataset counterpart? Advances in Neural Information Processing Systems, 36, 2024.
Zeng. H., Zhou, H., Srivastava, A., Kannan, R., and Prasanna, V. GraphSAINT: Graph sampling based in- ductive learning method. In International Conference on Learning Representations, 2020.
Zhang, Y., Gao, S., Pei, J., and Huang, H. Improving social network embedding via new second-order continuous graph neural networks. In Proceedings of the 28th ACM SIGKDD conference knowledge discovery and data mining, pp. 2515-2523, 2022.
Zhang, Y., Zhang. T., Wang, K., Guo, Z.,, Liang, Y., Bres- son, X., Jin, W., and You, Y. Navigating complexity: Toward lossless graph condensation via expanding win- dow matching. In International Conference Machine Learning, pp. 60379 603955. PMLR, 2024.
Zhao, B., Mopuri, K. R., and Bilen, H. Dataset condensation with gradient matching. In International Conference Learning Representations, 2021.
Zheng, X., Zhang, M., Chen, C., Nguyen, Q. V. H., Zhu, X., and Pan, S. Structure-free graph condensation: From large-scale graphs condensed graph-free data. Ad- vances in Neural Information Processing Systems, 36, 2024.

---

Adapting Precomputed Features for Efficient Graph Condensation
A. Evaluation Scheme
Following GCondenser (Liu et al., 2024), we evaluate all methods using three different condensation ratios (r) for each dataset. Specifically, the condensation ratio r defined as the fraction of condensed nodes to the total number of original nodes N. where < r 1. In the transductive setting, N denotes the total node count in the entire large-scale graph, whereas in the inductive setting, N refers to the node count within the training sub-gra sub-graph of the complete large-scale graph. The evaluation has two phases: (i) the condensation phase: synthesizes the condensed graph from the original graph, and (ii) the evaluation phase: the GNN is trained on the condensed graph, and the performance is evaluated on the original test nodes. We repeat the experiments five times and report the average node classification accuracy with standard deviation. The experiments are conducted on a single NVIDIA H100 GPU (80GB).
B. Hyperparameter Settings
For our method, tune the structure-basse precomputation hops K E: (1, 4}. damping factor 0.20, 0.5, 0.75), residual coefficient 3 E {0, 0.25, 0.5, 0.75), diversity coefficient E (0, 0.001, 0.01, 0.01,0.1,1) 0.1, semantic- based aggregation size M E [1, 10, 50, 100), number of negative samples S E (1,5, 10, 50), number of adaptation layers {1, 2,3. and hidden dimension of the adaptation module {128, 256, 512]. We tune all hyperparameters on the validation set. We adopt the default settings of AdamW, including learning rate v) = 0.001, 31 = 0.9, 82 = 0.999, and = 0.01 for weight decay.
C. Performance and Efficiency using SGC Backbone
Figure 7 and 8 present node classification performance and efficiency comparisons using the SGC backbone, respectively.
Table 7: Node classification performance comparison using SGC backbone (std) The best and second- results are marked in bold and underlined, respectively. Ours (Pre.) is the precomputation-only variant. The Whole column represents the performance obtained by training the whole dataset.
Dataset
Ratio
K-Cen.
GCond
SGDD
GCDM
SFGC
GEOM
Ours
Whole
0.9%
52.7
71.9gu
71.1
66.0
65.2
60.1
72.3
CiteSeer
1.8%
66.8
71.0
69.9
66.7
67. 07.0=as
65, 25,2
72.7=a3
70.3
3.6%
68. 100
72.5=12
70.8
69.1+12
68. 88.8+02
67.7+03
72.7=96
1.3%
63.82n0
80.6
62.4gss
77.0
73.8
69.2
80.9
Cora
2.6%
70.3
81.0=u2
80,8
78.9
77.
69.6+15
81.5+aa
79.2
5.2%
77.1
80.9.uu
81.4
77.9
79.2
37.3
81.9:sa
0,08%
70.5
75.9=07
76.4s0p
73.3+12
73.9+05
73.8+03
76.6=95
PubMed
0.15%
75.8=a0
75.2=00
78.0=a3
74.7+06
75.8+02
77.4+04
76.9+0m
76.9
0.3%
750
75.7
76.1an1
76.5
75.8
75.8
76.8
0.05%
51.8
65.5
64.5
60.8
16.1
62.01sa
67.2
Arxiv
0.25%
58.2
66.5_us
62.7 Lay
66.71ss
62.8 1
67.3
68.8
0.5%
60.3
67.2=a1
66.9
62.4
66.4
63.6e03
67.3+
0.025%
48.6=us
64.0=a2
64.9
$7.7+02
62.2+aa
61.1
65.0:sa
Products
0.05%
$2.2
64.0
62.3
58.2493
62.2
62.4
65.1
64.7
0.1%
55.4
64.4
64.3 jus
60.8
61.9
63.1 13s1
65.0
0.1%
14.5
43.7_ms
43.61sx
40.3
45.3
33.6164
45.61.83
Flickr
0.5%
36.
42.2=a2
41.6+16
40.8+01
45.7+04
37.4+02
46.5+*22
44.2
1%
41.1-us 36.5
43.2+
42.7+04
46.1 1
38. 38.1+42
46.8
0.05%
$4.0za1
89,7=no
90.5
90.34a8
90.9
59.4+15
91.5
Reddit
0.1%
78.6
91.8_nz
91.9
88.1 1zx
92.61o2
81.7
92.61m
93,2
0.2%
83.8
12.1
86.3
91.7
92.6
86.7
92.7
D. Visualization
Figure 7 displays visualization results between SFGC condensed features and ours. Our condensed graphs exhibit clear clustering patterns on all presented datasets with minimal inter-class mixing, in contrast to the SFGC graphs, which show less distinct class separation. The comparison is more evident on larger datasets such as Ogbn-arxiv and Flickr, where SFGC fails to produce clear clustering patterns. To quantify these clustering patterns, we follow previous work (Zhang et al., 2024)
12

---

Adapting Precomputed Features for Efficient Graph Condensation
Table 8: Efficiency comparison using SGC backbone (total condensation time in seconds).
Dataset
K K-Center
GCond
SGDD
GCDM
SFGC
GROM
Ours (Pre.)
Ours
CiteSeer (r=18%%
5
42
51
47
1,652
6,920
6
26
Cora (r:2.6%)
4
42
48
47
2,011
6,031
4
21
PubMed (r=0.15%)
4
34
204
42
7,555
22,201
5
40
Arxiv (r=0.25%)
6
283
1,485
242
78,586
84,356
20
71
Products (r=0.05%)
44
2,011
2,007
1,545
1,357,845
1,687,718
104
586
Flickr (r=0.5%)
177
300
258
99,254
19,202
23
Reddit (r=0.1%)
508
9,203
505
360,327
100,354
55
91
SFGC
Ours
SFGC
Ours
SC:
2.22
1.03
3:45
CH: 5.30
CH: 26.55
CH: 1.97
CHi 23.64
+
*
(a) CiteSeer (r=1.8%) 1.8%)
(b) Cora (r=2.6%)
SFGC
Ours
SFGC
Durs
SC: 9..44 D8: 2:50
Sc: 0.26
Sc 20.8 DB: 9.06
sc: 0.09
D8: 1.27
(c) Ogbn-arxiv (r=0.25%)
(d) Flickr (r=0.5%)
Figure 7: The t-SNE visualization of synthetic node features using GCN backbone. The node classes are represented by colors. The clustering metrics including Silhouette Coefficient (SCT), Davies-Bouldin Index (DBI), and Calinski-Harabass Index (CH1) are reported for each plot. The arrows and | denote that higher value indicates better clustering pattern for SC and CH, while lower value indicates better clustering for DB.
to utilize clustering metrics including the Silhouette Coefficient (Rousseeuw, 1987), the Davies-Bouldi Index (Davies & Bouldin, 1979), and the Calinski- Index (Caliñski & Harabasz, 1974), all of which indicate that our condensed graphs demonstrate better clustering patterns. The visualization results show that our framework effectively optimizes the condensed features, forming robust representations to preserve the original graph's 's classification capabilities.
E. Structure-free Features via Precomputation
During the precomputation stage, transform the raw features to structure-free features via precomputation. We show that when using SGC as the backbone GNN, the precomputed features coupled with an identity adjacency matrix are equivalent to the raw features coupled with the original graph structure. We start by defining the SGC network on the original graph:
SGC(X, A; = AD-AA x
(6
where X the raw node features.
+ I represents the adjacency matrix with self-loops, D denotes the degree matrix of A, Ki is the number of propagation layers, and @ the weight matrix.
13

---

Adapting Precomputed Features for Efficient Graph Condensation
Then, revisit the feature precomputation introduced in Equation when a 0:
X' = D -D-AAAA 2
k x,
where X' denotes the precomputed features, which is the result of applying the same transformation as in the SGC but isolated from the learning weights θ
As result, SGC with precomputed features and identity adjacency matrix becomes:
SGC((',I:)) = (D (D= x'e X'e (D- ( XO,
(8)
Therefore, draw the equivalence between SGC computation the original graph and the cructure-free precomputed features:
SGC( = SGC(C,
The equivalence shows that although the original features and condensed features are differently distributed, they perform equivalently when coupled with their corresponding graph structure using the SGC backbone. Drawing inspiration from this equivalence under the SGC backbone, our framework focuses on initializing and refining features in the precomputed feature space, enabling effective training of message- paassiggg GNNs on the structure- free condensed graphs
14


Reviews:
[{"review_id": "WnalJptsAt", "reviewer": "Reviewer_Z5gs", "summary": "This paper proposes an efficient graph condensation method composed of aggregation and contrastive learning stages. Extensive experiments indicate that this approach achieves performance comparable to state-of-the-art condensation methods, while significantly improving computational efficiency.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the examination of our work and the thoughtful comments provided. Kindly find our responses to the raised comments and questions below.\n\n**Q1: The performance on Citeseer is unexpectedly high, surpassing even the most advanced GNNs on the original Citeseer graph.**\n\nThank you for your observation. We note that on Citeseer, both baseline methods and our method have shown improved results compared to GCN trained on the original graph. This enhancement is likely due to the condensation process reducing noise in the data. The reproducible code is available [here](https://anonymous.4open.science/r/Precompute-Adapt-Graph-Condensation-6A76/). We note that 75.4 remains within a reasonable range considering recent GNNs achieving 77.5 on Citeseer [a].\n\n|Orig.GCN|GCDM|SFGC|GEOM|GCPA|\n|---|---|---|---|---|\n|71.4|72.3|72.4|74.3|75.4|\n\n**Q2: GCPA shows higher variability on Pubmed and Products. Addressing sampling randomness or imposing additional constraints within the contrastive learning process could help.**\n\nThank you for your valuable feedback. To address the instability, we apply constraints including AdamW [b] and L2 regularization to the adaptation model, which constrains the impact of random samples. The updated results are presented below. We will update these changes in the revised manuscript.\n\n| Dataset | Ratio | GCPA (Previous) | GCPA (with AdamW and L2 reg.) |\n|---|---|---|---|\n|PubMed|0.08\\%|80.2±1.9|80.5±0.4|\n||0.15\\%|80.5±0.8|80.9±0.3|\n||0.3\\%|81.6±2.4|81.7±0.4|\n|Products|0.025\\%|69.3±0.2|69.3±0.2|\n||0.05\\%|69.9±0.7|70.2±0.5|\n||0.1\\%|71.3±0.7|71.5±0.4|\n\n**Q3: Some important baselines focused explicitly on efficient graph condensation methods are omitted, notably references [1] and [2].**\n\nThank you for your feedback. We kindly note that the references are missing, so we follow **Reviewer rs1P Q1** to compare with SimGC [c], EXGC [d], and CGC [e]. We set the **smallest condensation ratios** and present **accuracy** with **total running time**. Considering the inconsistent time measurements (e.g., CGC reports only the condensation time), we uniformly run evaluation to measure total running time. Our method consistently outperforms the baselines, while the efficient baselines underperform GEOM. We will revise the manuscript to incorporate these methods. (For fair comparison, we adopt the new baselines’ finer hyperparameter search and update GCPA results accordingly, which slightly differ from those in the paper.)\n\n|Dataset|SimGC|EXGC|CGC|GEOM|GCPA|\n|---|---|---|---|---|---|\n|Citeseer|73.8 (245s)|69.2 (237s)|72.5 (32s)|73.0 (6,920s)|**75.4** (45s)|\n|Cora|80.8 (240s)|82.0 (235s)|82.7 (30s)|82.5 (6,031s)|**82.9** (44s)|\n|Arxiv|63.6 (362s)|57.6 (338s)|64.1 (126s)|65.5 (84,356s)|**67.2** (247s)|\n|Products|63.3 (4,861s)|62.1 (4,915s)|68.0 (1,093s)|68.5 (1,687,718s)|**69.3** (2,985s)|\n|Flickr|45.3 (425s)|47.0 (412s)|46.8 (94s)|47.1 (19,202s)|**47.2** (219s)|\n|Reddit|91.1 (702s)|90.2 (692s)|90.6 (182s)|91.1 (100,354s)|**91.3** (505s)|\n|AvgDiff|**-2.6**|**-4.2**|**-1.4**|**-0.9**|-|\n\n**Q4: In Line 168, the claim regarding the \"non-learning process leading to sub-optimal representations\" may not always hold true.**\n\nWe acknowledge the misleading claim and revise the paragraph:\n\nOur precomputation stage effectively captures the structural and semantic features of the original graph. Since the precomputation stage is not directly optimized for the final objective, we further integrate an adaptation learning stage that adjusts the class-wise representations.\n\n**Q5: Unclear term \"SF\".**\n\nThank you for pointing this out. In Figure 2, \"SF\" stands for \"structure-free\", indicating that the condensed graphs possess no edges.\n\n**Q6: Inconsistent term \"anchor\".**\n\nThank you for pointing out the inconsistency. To clarify, anchors refer to the sampled features $H_i \\in H$, where $H$ denotes the precomputed features. The anchors serve as learning targets during the adaptation stage, preserving the original feature distributions.\n\n**Q7: Figure 3’s caption is not organized by different modules.**\n\nWe appreciate your suggestion and revise the caption:\n\nOverview of GCPA framework. (1) *Structure-based precomputation:* Neighborhood aggregation is performed to capture structural dependencies. (2) *Semantic-based precomputation:* Nodes are grouped by semantic relevance using uniformly sampled representations. (3) *Adaptation learning:* Synthetic features v1 and v2 are pushed away through diversity constraints, while v2 and v3 are pushed away through sampled negative pairs.\n\n[a] From cluster assumption to graph convolution: Graph-based semi-supervised learning revisited\n\n[b] Decoupled Weight Decay Regularization\n\n[c] Simple graph condensation, ECML-PKDD 2024\n\n[d] Exgc: Bridging efficiency and explainability in graph condensation, WWW 2024\n\n[e] Rethinking and accelerating graph condensation: A training-free approach with class partition, WWW 2025", "claims_and_evidence": "Clearly stated and well-supported.", "methods_and_evaluation_criteria": "Adequately described and appropriate.", "theoretical_claims": "Not applicable.", "experimental_designs_or_analyses": "Requires additional analysis regarding the stability of the proposed method.", "relation_to_broader_scientific_literature": "Clearly connects with existing literature on Graph Condensation and Contrastive Learning.", "essential_references_not_discussed": "No critical omissions identified.", "other_strengths_and_weaknesses": "## Strengths\n\n- The proposed method is novel, and the results are robust.\n- Significant improvement in efficiency, indicating strong potential for practical applications.\n\n---\n\n## Weaknesses\n1. The performance of the GCN on the Citeseer dataset is unexpectedly high (75.4 with a condensed graph versus 71.4 with the original graph), surpassing even the most advanced GNNs on the original Citeseer graph. This anomaly could potentially be attributed to a bug or other underlying issues. Further analysis and discussion are required to clarify this discrepancy.\n2. GCPA shows higher variability compared to baselines on datasets like Pubmed and Products, as evidenced by higher standard errors in Table 2. This may stem from naive uniform sampling and inherent instability in contrastive learning. Addressing sampling randomness or imposing additional constraints within the contrastive learning process could help.\n3. Some important baselines focused explicitly on efficient graph condensation methods are omitted, notably references [1] and [2]. Including these baselines would strengthen comparative validity. This causes the speed up not convincing, as the proposed method only compare with slowest method though it’s SOTA. It’d better to compare the proposed method with both SOTA and most efficient graph condensation method with acceptable performance.\n4. In Line 168, the claim regarding the \"non-learning process leading to sub-optimal representations\" may not always hold true. For example, some training-free GNNs (e.g., reference [3]) can achieve performance on par with trainingable GNNs. Thus, it is suggested to add more disucssions and clarificaitons on this aspect.", "other_comments_or_suggestions": "1. **Minor Ambiguities:** Figure 2 contains an unclear abbreviation \"(SF).\" Clarification is needed.\n 2. **Presentation and Writing Consistency:**\n    - The term \"anchor\" appears inconsistently (mentioned only twice in paper and not illustrated explicitly in Figure 3).\n    - Figure 3’s caption is not organized by different modules, making it difficult to follow.", "questions_for_authors": "Please see above", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4fPGejJ0OL", "reviewer": "Reviewer_sWb8", "summary": "This paper introduces Graph Condensation via a Precompute-then-Adapt Approach (GCPA), an efficient method for condensing large-scale graphs to accelerate Graph Neural Network (GNN) training. The proposed framework is more computationally efficient than trajectory matching methods and instead consists of two stages: (1) a precomputation stage, which extracts structural and semantic information using a single pass of message passing, and (2) an adaptation stage, which refines the synthetic features using class-wise feature alignment and diversity maximization. The method is evaluated across multiple benchmark datasets, showing up to 2,455× speedup while achieving performance comparable to or better than state-of-the-art (SOTA) graph condensation methods.\n\n## update after rebuttal\nAs I noted during the rebuttal, I am still concerned about the hyperparameters, and the core contribution driving performance seems relatively straightforward. I will therefore maintain my score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the examination of our work and the thoughtful comments provided. Kindly find our responses to the raised comments and questions below.\n\n**Q1: Code cannot be opened.**\n\nWe apologize for the inconvenience. It appears there was a temporary issue. We have refreshed the repository and it is now accessible.\n\n**Q2: The authors do not discuss the effect of many hyperparameters.**\n\nThank you for pointing this out. In the extended experiments, we tune the key hyperparameters below on the validation set and analyze their impact:\n\n- Semantic aggregation size $M$\n- Negative sample size $S$\n- Damping factor $\\alpha$\n- Precomputation hops $K$\n- Residual coefficient $\\beta$\n\nThe results below show that the model is generally robust to these settings. Some hyperparameters ($M$, $S$) benefit from larger values, while others ($\\alpha$) have stable optimal choices. A few others ($K$, $\\beta$) require more careful tuning. We will clarify these findings in the revised version.\n\n|$M$|Arxiv|Flickr|\n|---|---|---|\n|1|66.7|46.8|\n|10|66.5|46.6|\n|50|66.9|46.9|\n|100|67.7|47.1|\n\n|$S$|Arxiv|Flickr|\n|---|---|---|\n|1|67.0|46.7|\n|5|67.7|47.0|\n|10|67.7|47.0|\n|50|67.7|47.1|\n\n|$\\alpha$|Arxiv|Flickr|\n|---|---|---|\n|0|67.0|45.3|\n|0.25|67.7|47.1|\n|0.5|66.5|47.0|\n|0.75|65.6|47.0|\n\n|$K$|Arxiv|Flickr|\n|---|---|---|\n|0|64.2|46.9|\n|1|65.0|47.0|\n|2|67.7|47.0|\n|3|63.9|47.1|\n|4|63.8|47.0|\n\n|$\\beta$|Arxiv|Flickr|\n|---|---|---|\n|0|66.9|46.4|\n|0.25|67.2|47.1|\n|0.5|67.1|46.4|\n|0.75|67.7|46.3|\n\n**Q3: Does it truly make sense to coarsen the graph using only non-learnable message passing, with the learning component restricted to (the updated) node features?**\n\nThank you for the thoughtful comment. We agree that relying on non-learnable message passing can introduce limitations and may not generalize to all datasets. However, in many real-world datasets—such as those used in graph condensation benchmarks—non-learnable message passing combined with MLPs (e.g., SIGN [a]) have shown competitive or even SOTA performance. A likely reason is that these benchmark datasets exhibit strong homophily or stable neighborhood patterns, allowing fixed message passing to capture key structural information effectively.\n\nOur method follows a similar principle—non-learnable message passing combined with MLP adaptation—and achieves strong results, suggesting that this design can be effective in practice.\n\n**Q4: The margin between the best result of the proposed method and the strongest baseline remains relatively small in almost all cases.**\n\nThank you for the observation. Our method is primarily designed for efficiency rather than solely maximizing performance. Given that current SOTA methods can be impractical (e.g., requiring up to 452 hours), our goal is to offer a more efficient alternative while aiming to match, not necessarily surpass, SOTA performance. Notably, our approach achieves significant speedup and even delivers leading results, which we find both promising and encouraging.\n\n**Q5: The condensed graph has no edges, so why would it make sense to apply GNN on it?**\n\nYour are correct that the condensed graph has no edges. This setup was first investigated in SFGC [b], where only self-loops are available for GNN message passing. Although counterintuitive, these methods—including ours—have shown superior performance. A potential reason is that the model learns to encode structural information into the condensed features for GNN to learn effectively, even without edges. We follow this established setting and believe it’s an interesting direction for further investigation.\n\n**Q6: Does GCN backbone play a role in the condensation process of the proposed method?**\n\nYou are correct that GCPA does not rely on GCN during condensation. We appreciate your observation and will clarify this in revised version:\n\n- Condensation: All methods except GCPA use GCN to guide learning.\n- Evaluation: All methods including GCPA use GCN for evaluation.\n\n**Q7: Why adaptation has only a minor impact in some cases? Intuitively, I would have expected adaptation to play a more significant role than precomputation.**\n\nThank you for the observation. The impact of adaptation indeed varies in a large range, affected by the quality of the precomputed features, which in turn is influenced by factors like data noises. We illustrate this by adding noise below. When the data is clean, precomputed features are already strong, so adaptation yields modest improvements. When we add noise and degrade the precomputed features, the adaptation becomes much more beneficial.\n\n|$\\sigma$ (Gaussian noise)|PubMed (Precomp.)|PubMed (Adapt)|Improve|\n|---|---|---|---|\n|0|79.7|80.5|+0.8|\n|0.01|75.7|77.4|+1.7|\n|0.05|57.4|61.4|+4.0|\n|0.1|49.8|57.1|+7.3|\n|1|42.7|54.4|+11.7|\n\n[a] SIGN: Scalable Inception Graph Neural Networks\n\n[b] Structure-free Graph Condensation: From Large-scale Graphs to Condensed Graph-free Data", "claims_and_evidence": "The authors make strong claims regarding the efficiency of their approach. This claim is backed up well in the paper, e.g., in Figures 2, 4.", "methods_and_evaluation_criteria": "The authors’ evaluation protocol follows standard practice and aligns with previous research.", "theoretical_claims": "The authors don’t provide any theoretical claims.", "experimental_designs_or_analyses": "The authors’ experimental design follows standard practice and aligns with previous research.", "relation_to_broader_scientific_literature": "The authors include a \"Related Work\" section that effectively discusses relevant topics.", "essential_references_not_discussed": "No, I didn't identify any essential references that were missing.", "other_strengths_and_weaknesses": "**Strengths**:\n\n1. The proposed technique noticeably reduces the time required for the graph condensation process and effectively addresses the issue of repeated training consumption in large-scale graphs for existing structure-free methods, which seems reasonable to me.\n\n2. The proposed technique performs well on benchmark datasets with various condensation ratios, even during initial precomputation. \n\n**Weaknesses**:\n1. The method introduces a large number of hyperparameters beyond the standard ones used for training (e.g., hidden dimensions, number of layers, etc.), including:\n* $K$ – the number of precomputation hops\n* $\\alpha$ – the damping factor\n* $\\beta$ – the residual coefficient\n* $\\gamma$ – the diversity coefficient\n* $M$ – the semantic-based aggregation size\n* $S$ – the number of negative samples\n\nHowever, apart from $\\gamma$, the authors do not discuss or demonstrate the effect of these hyperparameters on the method’s performance. This is particularly concerning given the large number of hyperparameters involved.\n\n2. The precomputation stage is mostly without learning, with the exception of Equation 4, where $f_{\\text{adapt}}$ operates only on the (updated) node features. I understand how this could be advantageous, but it also introduces certain limitations. Does it truly make sense to coarsen the graph using only non-learnable message passing, with the learning component restricted to (the updated) node features? I appreciate the authors' acknowledgment of this issue and their attempt to address it in the discussion (lines 172–186) via Equation 4. However, this approach still seems somewhat problematic to me.\n\n3. The results, as shown in Table 2 for example, generally surpass the baselines. However, the margin between the best result of the proposed method and the strongest baseline remains relatively small in almost all cases, which makes the method less compelling.", "other_comments_or_suggestions": "See questions below.", "questions_for_authors": "1. If I understand correctly, the condensed graph has no edges. So why would it make sense to apply a GNN to train on the condensed graph?\n\n2. In Table 3, does the GCN backbone play a role in the condensation process of the proposed method, or is it only relevant to the baselines? If I understand correctly, the condensation is determined by Equations 2, 3, and 4, none of which involve a GNN.\n\n3. In Table 2, I notice that in some cases, the results after the precomputation stage are very similar to those after both stages (precomputation and adaptation). Could you provide some insight into why the adaptation stage appears to have only a minor impact in these cases? Intuitively, I would have expected adaptation to play a more significant role than precomputation.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1anO35rnOP", "reviewer": "Reviewer_nZN9", "summary": "This paper propose the GCPA method, which not only bring the unbelievable efficiency into the graph condensation process but also gains considerable results, for example, for the Ogbn-products dataset, the conventional trajectory method calls for 452 hours in collecting the trajectories, but the GCPA only cost much less time, while even achieving the SOTA results.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the examination of our work and the thoughtful comments provided. Kindly find our responses to the raised comments and questions below.\n\n**Q1: Can you ablate the adaptation stage to the other expensive matching processes? The precomputation stage seems like a normal trick.**\n\nThank you for the suggestion. We would like to clarify that our framework—precomputation followed by adaptation—is fundamentally different from existing methods, which typically follow an initialize-then-learn paradigm. The key difference lies in where **message passing**—the operation that captures graph structural information—is performed, as shown below:\n\nExisting methods (e.g., CTRL, GEOM):\n\n- Initialization: random sampling, clustering, etc.\n- Learning: GNN training with repetitive **message passing** (the main source of computational cost)\n\nOurs:\n\n- Precomputation: one-time **message passing**\n- Adaptation: MLP training (without message passing)\n\nOur framework replaces the costly, repeated message passing (in the learning stage) in existing methods with a one-time message passing step (in the precomputation stage), achieving both efficiency and strong performance. This is not just a simple preprocessing trick, and we believe it constitutes a meaningful contribution.\n\nAs for replacing adaptation with existing matching processes, such changes may lead to redundant message passing (since precomputation already involves message passing, and existing matching processes involve repeated message passing during learning) and defeat the purpose of our efficiency gains. Nonetheless, we believe it deserves future exploration.\n\n**Q2: I cannot fully understand the setting of Appendix D.**\n\nThank you for the question. In Appendix D, we highlight the effect of precomputation by comparing two settings:\n\n- GCPA (Ours): \n    1. Structurally precompute features $H$\n    2. Semantically precompute condensed features $\\hat{X}'$\n    3. Adapt condensed features $Z'=f_{adapt}(\\hat{X}')$ with precomputed features $H$\n- GCPA without Precomputation (Ours with Random Initialization): \n    1. Randomly initialize condensed features $X_{rand}'$, \n    2. Adapt condensed features $Z'=f_{adapt}(X_{rand}')$ with original features $X$\n\nTable 9 shows that removing precomputation leads to loss of structural information, and hence significantly worse performance. We conclude that precomputation is not merely an initialization step—it plays a critical role by embedding structural information and guiding adaptation learning. This ablation confirms that precomputation meaningfully contributes to performance, beyond what adaptation alone can achieve.\n\n**Q3: A deep explanation of the effectiveness of your methods? Clear motivation why you use such a precomputation stage and contrastive learning technique? Not that convincing, just a combination of the data process to me, better inspiration is expected.**\n\nThe two stages–precomputation and adaptation–are motivated by the need to maintain strong structural and semantic guidance from the precomputed features. In many existing methods, precomputed features serve only as **temporary constraints**—initializing the learnable condensed features $Z'$—which can then vanish from the original signal as training progresses. In contrast, our approach employs a **permanent constraint**, where the precomputed features $\\hat{X}'$ continuously guide the adaptation: $Z' = f_{adapt}(\\hat{X}')$, ensuring their influence remains intact throughout training.\n\nWe illustrate this distinction using a variant of GCPA, replacing $f_{adapt}$ with learnable condensed features $Z'$. The performance drop indicate that our permanent constraint is effective in preserving critical information.\n\n||Arxiv|Flickr|\n|---|---|---|\n|GCPA-Variant (Temporary constraint with learnable $Z'$)|66.9|46.2|\n|GCPA (Permanent constraint with learnable $f_{adapt}$ and fixed $\\hat{X}'$)|**67.7**|**47.1**|\n\nBesides, we provide theoretical insights on the effectiveness of our method from the perspective of mutual information. We kindly refer to **Reviewer rs1P Q3** for this discussion.", "claims_and_evidence": "As far as I can see, this paper mainly claims that they achieve better performance with 96x to 2,455x faster than the SOTA methods. \nFor the evidence, they provide the detailed experiments in Tab. 2, and provide the time comparison in Fig.4.", "methods_and_evaluation_criteria": "This is what I concern most, because if we only see the individual part of the precomputation stage, each of them seems like a normal step that was widely used in the previous literature. For example, in the CTRL[A], the authors discuss the different ways of initial sets selection. Therefore, from my understanding, the authors just change the expensive matching process to a simple contrastive learning process. \n\tThen, my questions are listed as follow:\nIf you can ablate the adaptation stage to the other expensive matching processes? Because the precomputation stage seems like a normal trick to me? (I see the experiments on the appendix D, maybe it is associated with my question? But I cannot fully understand them.)\nCould you give a deep explanation of the effectiveness of your methods? The current one is more like a technique report, I didn’t see the clear motivation why you use such a precomputation stage and contrastive learning technique? \nI am not fully convinced by the story, but I appreciate the simple idea and impressive experiment results. I’ll give the borderline accept for now, and decide later depending on the answer in the rebuttal period.\n\n[A] CTRL: GRAPH CONDENSATION VIA CRAFTING RATIONAL TRAJECTORY MATCHING. Zhang et.al. Arxiv 2023.", "theoretical_claims": "I didn’t see any proof.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "It is a big innovation since the authors significantly reduce the condensing time but hold considerable results.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths:\n1. Good results, comprehensive comparsion on addtional PubMed, Products datasets.\n\nWeakness:\n1. Not that convincing, just a combination of the data process to me, better inspiration is expected.", "other_comments_or_suggestions": "No.", "questions_for_authors": "See above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "No."}}, {"review_id": "eu1R7PlyLJ", "reviewer": "Reviewer_rs1P", "summary": "To address the efficiency issue in graph condensation (GC), this paper proposes GCPA, a two-stage framework comprising precomputation and diversity-aware adaptation. The precomputation stage aggregates structural and semantic information for competitive performance, while the adaptation stage refines features via class-wise alignment with minimal cost. Experiments on seven benchmarks confirm its superior efficiency.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the examination of our work and the thoughtful comments provided. Kindly find our responses to the raised comments and questions below.\n\n**Q1: Some recently proposed efficient GC are not discussed, including SimGC[1], EXGC[2], and CGC[3].**\n\nWe thank the reviewer for highlighting the recent efficient GC methods. We set the **smallest condensation ratios** and present **accuracy** with **total running time**. Considering the inconsistent time measurements (e.g., CGC reports only the condensation time), we uniformly run evaluation to measure total running time. Our method consistently outperforms the baselines, while the efficient baselines underperform GEOM. We will revise the manuscript to incorporate these methods. (For fair comparison, we adopt the new baselines’ finer hyperparameter search and update GCPA results accordingly, which slightly differ from those in the paper.)\n\n|Dataset|SimGC|EXGC|CGC|GEOM|GCPA|\n|---|---|---|---|---|---|\n|Citeseer|73.8 (245s)|69.2 (237s)|72.5 (32s)|73.0 (6,920s)|**75.4** (45s)|\n|Cora|80.8 (240s)|82.0 (235s)|82.7 (30s)|82.5 (6,031s)|**82.9** (44s)|\n|Arxiv|63.6 (362s)|57.6 (338s)|64.1 (126s)|65.5 (84,356s)|**67.2** (247s)|\n|Products|63.3 (4,861s)|62.1 (4,915s)|68.0 (1,093s)|68.5 (1,687,718s)|**69.3** (2,985s)|\n|Flickr|45.3 (425s)|47.0 (412s)|46.8 (94s)|47.1 (19,202s)|**47.2** (219s)|\n|Reddit|91.1 (702s)|90.2 (692s)|90.6 (182s)|91.1 (100,354s)|**91.3** (505s)|\n|AvgDiff|**-2.6**|**-4.2**|**-1.4**|**-0.9**|-|s\n\n**Q2: The novelty of the paper is limited. It seems that the structure-precomputation only uses the graph diffusion operators which is widely used to perform message passing, and semantic precomputation is a simple average operation.**\n\nOur work is motivated by the need to maintain strong structural and semantic guidance from the precomputed features. In many existing methods, precomputed features serve only as **temporary constraints**—initializing the learnable condensed features $Z'$—which can then vanish from the original signal as training progresses. In contrast, our approach employs a **permanent constraint**, where the precomputed features $\\hat{X}'$ continuously guide the adaptation: $Z' = f_{adapt}(\\hat{X}')$, ensuring their influence remains intact throughout training.\n\nWe illustrate this distinction using a variant of GCPA, replacing $f_{adapt}$ with learnable condensed features $Z'$. The performance drop indicate that our permanent constraint is effective in preserving critical information.\n\n||Arxiv|Flickr|\n|---|---|---|\n|GCPA-Variant (Temporary constraint with learnable $Z'$)|66.9|46.2|\n|GCPA (Permanent constraint with learnable $f_{adapt}$ and fixed $\\hat{X}'$)|**67.7**|**47.1**|\n\n**Q3: A lack of theoratical analysis on why the method is effective. The proposed method appears to be very simple. Could the author provide a theoratical analysis on how the proposed GCPA can achieve comparative or even superior data utility (i.e., classification performance) than the previous GC methods?**\n\nWe appreciate the reviewer’s question and provide theoretical insight below.\n\n**[Under SGC model, graph condensation reduces to feature set condensation]**\n\nIn Appendix G, we establish that, for the Simple Graph Convolution (SGC) model, the node embeddings can be precomputed as $X'=(\\tilde{D}^{-0.5}\\,\\tilde{A}\\,\\tilde{D}^{-0.5})^KX$. Hence, replacing $A$ by an identity adjacency $I$ while using $X'$ is equivalent to using $A$ with the original features $X$. Thus, condensing a graph under SGC is essentially condensing the precomputed features $X'$.  \n\n**[Contrastive loss increases mutual information]**\n\nOur contrastive loss uses logistic cross-entropy to distinguish positive (same-class) pairs from negative (randomly sampled) pairs. This is known to maximize a lower bound on $\\mathrm{JS}(p^+\\|p^-)$, where $p^+$ and $p^-$ are the positive and negative pair distributions. As $\\mathrm{JS}(p^+\\|p^-)$ increases, $X'$ and the condensed features $Z'$ become more class-discriminative, thereby increasing their mutual information on class-relevant signals. Combined with the SGC equivalence, it follows that condensing $X'$ under this contrastive loss effectively preserves and amplifies the class-relevant information in the original graph features $X$.\n\nTo support the claim on the effectiveness of our method, we present KSG mutual information (MI) [a] with accuracies below. The results show a clear increase in mutual information during both stages, along with improved accuracies. This trend highlights the importance of class separation in the adaptation process, thereby contributing to the strong performance of GCPA by helping to preserve class boundaries.\n\n|Stage|Arxiv-MI|Arxiv-Acc|Flickr-MI|Flickr-Acc|\n|---|---|---|---|---|\n|Precomputation|0.016|64.6|0.011|45.4|\n|Adaptation at epoch 10|0.044|65.3|0.016|45.9|\n|Adaptation at epoch 50|0.397|66.8|0.148|46.3|\n|Adaptation at epoch 100|0.567|67.2|0.347|46.8|\n\n[a] Estimating mutual information", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "There is no theoretical claims and proofs in this paper.", "experimental_designs_or_analyses": "I have reviewed all aspects of the experiment section, including the experimental setup, performance evaluation, efficiency evaluation, transferability evaluation, ablation study, and parameter analysis. Overall, the experimental design and analyses are reasonable. However, some important baselines are missing.", "relation_to_broader_scientific_literature": "The contributions of this paper are related to previous studies on graph neural networks (GNN) [1], and data condensation (DC)[2].\n\n[1] Wu, Z., Pan, S., Chen, F., Long, G., Zhang, C., & Philip, S. Y.  A comprehensive survey on graph neural networks. TNNLS 2020.\n\n[2] Cui, J., Wang, R., Si, S., & Hsieh, C. J. (2022). Dc-bench: Dataset condensation NeurIPS2022", "essential_references_not_discussed": "The key contribution of the paper is efficient GC. However, some recent proposed method are not discussed, including SimGC[1], EXGC[2], and CGC[3].\n\n[1] Xiao, Z., Wang, Y., Liu, S., Wang, H., Song, M., & Zheng, T. Simple graph condensation. In ECML-PKDD 2024\n\n[2] Fang, J., Li, X., Sui, Y., Gao, Y., Zhang, G., Wang, K., ... & He, X. (2024, May). Exgc: Bridging efficiency and explainability in graph condensation. In WWW2024\n\n[3] Gao, X., Ye, G., Chen, T., Zhang, W., Yu, J., & Yin, H. (2024). Rethinking and accelerating graph condensation: A training-free approach with class partition. arXiv preprint arXiv:2405.13707.", "other_strengths_and_weaknesses": "Strength:\n1. The paper is generally well-written.\n2. The efficiency improvement in GC appears to be significant.\n\nWeakness:\n1. The novelty of the paper is poor. \n2. A lack of theoratical analysis on why the method is effective.\n3. Some important baselines are missing.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1. The novelty of the paper is limited. It seems that the structure-precomputation only uses the graph diffusion operators which is widely used to perform message passing, and semantic precomputation is a simple average operation. Could the author further clarify the novelty of this paper.\n\n2. The proposed method appears to be very simple. Could the author provide a theoratical analysis on how the proposed GCPA can achieve comparative or even superior data utility (i.e., classification performance) than the previous GC methods?\n\n3. Why the author do not compare GCPA with recently proposed efficient GC methods, including SimGC[1], EXGC[2], and CGC[3].\n\n[1] Xiao, Z., Wang, Y., Liu, S., Wang, H., Song, M., & Zheng, T. Simple graph condensation. In ECML-PKDD 2024\n\n[2] Fang, J., Li, X., Sui, Y., Gao, Y., Zhang, G., Wang, K., ... & He, X. (2024, May). Exgc: Bridging efficiency and explainability in graph condensation. In WWW2024\n\n[3] Gao, X., Ye, G., Chen, T., Zhang, W., Yu, J., & Yin, H. (2024). Rethinking and accelerating graph condensation: A training-free approach with class partition. arXiv preprint arXiv:2405.13707.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: After careful consideration of the reviews and subsequent discussions, I recommend accepting your paper for ICML 2025, though I note this is a borderline case. The reviewers unanimously acknowledged the impressive efficiency gains and competitive performance on several benchmarks, which represent a significant practical contribution to the field. However, concerns were raised about the justification of your approach, its novelty, and the sensitivity to numerous hyperparameters. For the camera-ready version, I strongly encourage you to incorporate the insights from your rebuttal into the main paper, expand your hyperparameter analysis, and include comparisons with recent efficient graph condensation methods. Your work addresses an important problem in graph neural networks with a solution that makes graph condensation substantially more practical, and these improvements would strengthen its impact.
Decision: Accept (poster)