Towards Pre-trained Graph Condensation via Optimal Transport
Authors: Yeyu Yan
Shuai Zheng
Wenjun Hui
Xiangkai Zhu
Dong Chen
Zhenfeng Zhu
Yao Zhao
Kunlun He
Venue: NeurIPS 2025 poster ()
Keywords: Graph Condensation
Data Condensation
Graph neural netwok
Primary area: deep_learning

Abstract: Graph condensation (GC) aims to distill the original graph into a small-scale graph, mitigating redundancy and accelerating GNN training. However, conventional GC approaches heavily rely on rigid GNNs and task-specific supervision. Such a dependency severely restricts their reusability and generalization across various tasks and architectures. In this work, we revisit the goal of ideal GC from the perspective of GNN optimization consistency, and then a generalized GC optimization objective is derived, by which those traditional GC methods can be viewed nicely as special cases of this optimization paradigm. Based on this, \textbf{Pre}-trained \textbf{G}raph \textbf{C}ondensation (\textbf{PreGC}) via optimal transport is proposed to transcend the limitations of task- and architecture-dependent GC methods. Specifically, a hybrid-interval graph diffusion augmentation is presented to suppress the weak generalization ability of the condensed graph on particular architectures by enhancing the uncertainty of node states. Meanwhile, the matching between optimal graph transport plan and representation transport plan is tactfully established to maintain semantic consistencies across source graph and condensed graph spaces, thereby freeing graph condensation from task dependencies. To further facilitate the adaptation of condensed graphs to various downstream tasks, a traceable semantic harmonizer from source nodes to condensed nodes is proposed to bridge semantic associations through the optimized representation transport plan in pre-training. Extensive experiments verify the superiority and versatility of PreGC, demonstrating its task-independent nature and seamless compatibility with arbitrary GNNs.

Towards Pre-trained Graph Condensation via Optimal Transport
Yeyu Yan 1.2, Shuai Zheng Wenjun Huil.2, Xiangkai Zhu Dong Chen
Zhenfeng Zhu Yao Zh Kunlun He
Institute of Information Science, Beijing Jiaotong University, China Visual Intelligence + X International Cooperation Joint Laboratory of the Ministry of Education, China
Jinan University, China *Chinese PLA General Hospital, China yanyeyu-workk oxwail com, zs 19977bjju.. edu...ed..c. cn,
221 12110090@bji tu. edu. en, 180635976300163. com, (hhen20010bjtaa edu. en, zhf zhfzhubbju.. edw.j...euddee...o edu. cn, yzhao@b ju. edu. en, kunlunhe@plagh. org
Abstract
Graph condensation (GC) aims to distill the original graph into a small-scale graph, mitigating redundancy and accelerating GNN training. However, conventional GC approaches heavily rely on rigid GNNs and task-specific supervision. Such a dependency severely restricts their reusability and generalization across various tasks and architectures. In this work, we revisit the goal of ideal GC from the perspective GNN optimization consistency, and then a generalized GC optimiza- tion objective is derived, by which those traditional GC methods can be viewed nicely as special cases of this optimization paradigm. Based on this, Pre-trained Graph Condensation (PreGC) via optimal transport is proposed to transcend the limitations of task- and architecture-dependeer GC methods. Specifically, a hybrid-interval graph diffusion augmentation is presented suppress the weak generalization ability the condensed graph on particular architectures by en- hancing the uncertainty of node states. Meanwhile, the matching between optimal graph transport plan and representation transport plan is tactfully established to maintain semantic consistencies across source graph and condensed graph spaces, thereby freeing graph condensation from task dependencies. To further facilitate the adaptation of condensed graphs various downstream tasks, traceable semantic harmonizer from source nodes to condensed nodes is proposed to bridge semantic associations through the optimized representation transport plan in pre-training. Ex- tensive experiments verify the superiority and versatility of PreGC, demonstrating beskiindependent nature and seamless compatibility with arbitrary GNNs.
Introduction
Graph neural networks (GNNs) have emerged powerful approach to graph data analysis, showing excellent efficacy in a variety of domains, such as recommender systems [5], text analysis [36],53),
social network analysis [52], and so on. However, the dual challenges of exponentially growing graph data volumes (58] and increasingly sophisticated GNN architectures [63] present significant obstacles for GNN training. These challenges become particularly acute in scenarios that require multiple GNN training, such as continual graph learning (2] and federated graph learning [30].
In response to the above issues, graph condensation (GC) synthesize small-scale condensed graph from the real graph. The core objective is to ensure that
25 64 is proposed to optimize and
"Corresponding author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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condensed graphs achieve comparable performance as original graphs when training the same GNN. This approach significantly reduces computational overhead and becomes an up-and-coming solution for efficient graph learning. Depending on matching strategies, GC fall mainly into three types. For example, [25 24, (55] distilled the properties of the real graph by mimicking its gradient changes in
GNN training. [64] 59 replicated the training trajectory on original graphs to synthesize condensed graphs. Other studies [32 [6] generated the graph by aligning the latent space representations of
both graphs and maintaining the distribution consistency. Recently, [16] pointed out that all these methods can be generalized as distribution matching, with the basic pipeline shown in Fig. (a).
Despite these significant efforts, current methods suffer from two critical limita- tions: (i) Architecture-boudd optimiza- tion. Current methods can be regarded as condensing through constraints in the same representation space [16]. There- fore, they have to adopt fixed encoder to map nodes from the graph space to the representation space in condensation. Such an excessive dependence on a spe- eific architecture can degrade the gener- alization of condensed graphs, leading to poor performance on other models. In
(a)
Prodffiedd Lobe'
Conventional Graph Condonsation
You
Archieetaee
Other Saperieed Fart
Original Graph
Condenued Graps
Consennuaion Ahane
Yox Time
(b)
o Semandic Matching
Genralized
Repender Dark
GIA,N) en 5(A
Graph Condensation
Asgesnntaton
GNN
Pank
Rie-tackng (8
Original Groph y
Graph!
addition, the learnable model parameters inherit task bias (46), which causes an- other challenge. (ii) Task-constrained
Figure 1: (a) The conventional GC framework. Due to specific 1 architecture and @ task dependencies. 9 lacks generalization and reusability. (b) Our proposed general- ized GC framework. It solves the above problems by ② and 2, and fine-tuning & in specific scenarios by @
condensation. To simplify optimization [25]. existing methods typically assume that tasks and labels are known, thus the effectiveness of GC depends heavily on the guidance of supervision signals. Unfortunately, this assumption often proves impractical in real-world scenarios. For example, in social networks. multiple tasks may coexist (c.g.. preference classification, income prediction, and relation prediction) [2], and user labels frequently vary owing
to privacy constraints and annotation discrepancies (30] When the task or label changes, conven- tional GC methods require re-condensation to capture new knowledge, significantly diminishing the
reusability of condensed graphs. Although [15] attempted to address label dependency by contrastive learning, its reliance on class similarity hinders its application to other supervised tasks.
To tackle the two critical challenges altogether, we first revisit the GC objective based on the goal of condensed graphs. It aims to achieve comparable performance to the original graph when training the same GNN (i.e. GNN optimization consistency). From this perspective, we derive a generalized GC optimization paradigm without task constraints as shown in Fig. (b) and design a flexible and generalized GC method termed Pre-trained Graph Condensation (PreGC). Inspired by data augmentation [10, graph diffusion augmentation is proposed to perturb node diffusion states via hybrid intervals, thereby enhancing the generalizability of condensed graphs and mitigating their dependence on specific architectures. To decouple the reliance on task labels, we present a novel matching mechanism, named transport plan matching. It maintains consistency in semantic associations between graph space and representation space, thus guaranteeing a unique mapping from each source node to its condensed counterpart. More importantly, transport plans provide explicit feedback on the significance of the source node, alleviating the poor traceability and interpretability issues in conventional condensation methods. After pre- pre-training training, traceable semantic harmonizer via the optimal transport plan is derived, cnabling flexible transfer of task signals from the original graph to the condensed graph for adaptation to diverse tasks.
To the best of our knowledge, PreGC is the first work for generalized graph condensation considering both architecture- and task-agnostic scenarios. Our main contributions are outlined as follows:
Formal theoretical analysis reveals that conventional GC methods are inherently governed by the reconstruction term and the fitting term. Inspired by such an insight, we propose a pre-training framework of graph condensation, generalizing the condensed graph to more scenarios.
2

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To empower the condensed graph for prchitecture-agnosee scenarios, hybrid-intens graph diffusion augmentation is presented. which enhances the diversity of node representations by introducing stochasticity into the diffusion process of both the condensed and original graphs.
We propose an optimal transport plan matching to realize semantic alignment without task con- strained. By keeping consistent optimal plan for graph alignment and representation alignment, the uniqueness of semantic associations between source nodes and condensed nodes is guaranteed.
Experiments on four mainstream tasks and nine representative GNN architectures demonstrate the superiority and generalization of PreGC. Further data valuation of the original graph highlights the PreGC's excellent traceability and interpretability.
2
Preliminary
Graph Condensation (GC), Given large-scale original graph G - (A, with node set V (V) N), where A E R NxN denotes the symmetrically normalized adjacency matrix and 25 33 aims to condense small synthetic graph G (A, X) with a condensed node set v (V)- = M) form the real large graph G, where A E R MMM X Mxx, and M < N. At the same time, trained GNNs with the specific task have comparable performance on both & and G, thus accelerating the training of GNNs.
X E IR Nx/ f is a f-dimensional node feature matrix. GC
Graph Optimal Transport. Optimal transport (OT) [39] aims to seek the most cost cost-efficient transport plan π that transforms source distribution μ into target distribution while minimizing the total transport cost (i.e., the optimal transport distance). The elements of æ represent the probability of mass transferring from one location to another. In this work, we primarily concentrate on graph optimal transport 6 43 which extends the OT to compare structured data. Given two graphs
g = (A, X) and 9 (A, X) with the nodes' empirical distribution l E TR-N and E RR,, the
general form of the graph optimal transport distance can be formalized as:
W(G. = min =Ell(a.))
A,A,),,
where 11(H,,) 2) { |-NMM marginals and v/ and and ( denotes the inner product for matrices. C represents the cost function that quantifies the cost of transporting mass from elements in one domain to another.
μ, 1, 2 is set of the joint distributions with
The transport distance quantifies the distribution divergence between condensed and original graphs, offering novel optimization approach for GC. Meanwhile, the transport plan establishes explicit node correspondences, overcoming the limitations of poor traceability in conventional GC methods.
3 Revisiting and Generalizing Graph Condensation
In this section, we revisit the objective of graph condensation and reveal the limitations of existing GC methods, thus motivating the design of PreGC.
Revisiting. For convenience, we follow
33 and start with a vanilla example, which adopts Koorder SGC [48] as the GNN and simplifies the objective of GNNs into the MSE loss: LMSF =
A X YI where Y E R denotes the target variable and W R is the model parameter. For the original graph G, the optimal W arg min ||A A XW
YI can be obtained
w
by MSE loss. In general, the goal of condensation is that the GNN trained on the condensed graph ğ to achieve comparable performance to those trained on G, i.e
min W W|IF min [I(A K X - (
where W = arg min | XW ||? with target variable Y E MCC on G. and t denotes pseudo W
inverse. Subsequently, we further derive the following proposition (Proofs are in Appendix Ap: Proposition 3.1. Suppose that it has an analytical filter g(.) for GNN model, the performance approximation error Ow of condensed graph is jointly bounded by the reconstruction term Oz z and the fitling term Oy.
IW- W| < [M
- (o(L)x) |
IMz(Y) - X (3)
Ow
Oz
Oy
3

---

where (- is any mapping function that aligns (I) and g(L) or Y and Y. L - IN A and L Im A are the Laplacian matrices of 9 and 9.
Proposition 3.1 reveals that the reconstruction term Oz and the fitting term Oy are the two key factors for ideal GC. Unfortunately, current GC methods rely on specific GNNs (i.e. the rigid filter g(-)), resulting in spectrally limited condensed graphs with higher reconstruction errors for Oz. Moreover, the pre-defined task signal in condensation restricts Qy to the specific task. Consequently, the condensed graph can only achieve satisfactory performance with the pre-defined GNNs and task, severely limiting its generalizability and reusability. To overcome these limitations, we focus on proposition 3.1 and relax the above conditionally constrained GC objective into generalized one:
Definition 3.1. Generalized Graph Condensation, Given an original graph G, the objective of ideal graph condensation is to seek condensed graph that simultaneously minimizes the representation- level and the semantic-level discrepancies relative to G:
3" = arg min
Az Z +E
Ay(G, Oy
Oz
where Az (.)))) measures the representation-level discrepancy between representations Z and Z of and G. Dy(:s)) is distance function measuring semantic-level discrepancy, and & serves trade-off parameter balancing the fitting and reconstruction terms.
Definition 3.1 presents general GC optimization paradigm, of which current GC methods can be regarded as variant forms (A (Appendix chows more details). In this formulation, the generalized node representation Z replaces g XL) and the task-specific fitting term Oy is relaxed to semantic aligmment between two graphs. Therefore, by boosting the diversity of the representations Z and Z, and designing unsupervised signals for AY, it is possible to readily establish a pre-trained graph condensation framework. Such framework enhances the generalization of the condensed graph and augments its adaptability across diverse scenarios.
Methodology
Fig. Lil illustrrates the framework of PreGC, which strictly follows the GC objective in Eq. In pre- training phase, graph diffusion augmentation proposed to increase the diversity of representations in Oz to alleviate architectural dependencies. Subsequently, to realize task-agnostic graph condensation,
PreGC adopts an innovative transport plan matching to align semantic associations in Oy. In addition, test-time fine-tuning further achieves dynamic adaptation specific tasks and architectures.
Pre-trained Graph Condensation
Test- test-time Fine-tuning
Graph Space
Representation Space
Tasting
oighaiion Any Aepiniss
Oephodae Aftenment Keyr
Lobel Auslamment dewed AC;
c
Mapzing M.. lpderee
0C Traising
Grape Difiaiion Process
Fine huning Mapping
TtD
Plan Matching Eeties
Ttz.
Task Lobel
8t Offfurion Interval
Transport Cost
Figure 2: Overall pipeline of the proposed PreGC framework. TD and कश denote optimal transport plans for graph alignment and for representation alignment, respectively.
4.1 Diffusion Augmentation via Graph Heat Equation
As analyzed in Section most GC methods adopt K-order SGC 48 or GCN 26 for encoding
graphs to obtain and (42. However, parameters neural networks may inherit task-induced biases, thus distorting the intrinsic representation properties. Although GDEM X and CGC [16] attempt to use parameter- encoders to obtain node representations, they remain architecture-bound resulting in condensed graphs hard to generalize to other GNNs. More critically, under discrete

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propagation, the representation distributions of two graphs evolve at markedly divergent rates due to diameter disparities, inevitably exacerbating the difficulty of GC.
The graph heat equation (GHE) [ as a generalization of the diffusion equation on graph data (graph diffusion), serves as an effective solution to the above problems due to its non-parametric and dynamic continuity properties. Formally, GHE can be defined dZ, LZ,, where Z₁ is the
d:
evolved input representation at time t, and Zo = X. In our proposed PreGC, graph diffusion [1 is used to encode node representations and solved using Explicit Euler method:
Ze+ = Z. S1 = A (,
(5)
where öt is diffusion interval and A (8t) = (1 - 8t)I + 8tA. Finally, the representation (or state) Zr at the terminal time T = K - öt can be formulated as Zr = [A [6 X. Adjusting öt enables smoother propagation rate modulation, mitigating representation instability in discrete propagation.
Graph Diffusion Augmentation. Indeed, deterministic diffusion states capture specific graph spectral responses. To eliminate architectural bias, the condensed graph should maintain state consistency with the original graph for arbitrary diffusion times to inherit the complete spectral properties.
As a sample generation strategy, data augmentation [0 29 aims to enhance the generalizability and robustness of the model through diverse samples. Inspired by this, simple yet efficient diffusion augmentation is proposed to improve the generalizability of the condensed graph by generating diverse diffusion states with hybrid intervals. This process can be formulated as:
2
S(Z,) 1 Z
[A ( x, öt, ~ P, Vot, S
(6)
Amax
where P = ((8mmin 2/Amax ())). T denotes the condensation epoch, and Xmax (L) is the maximum eigenvalue of L. Všt, I 2/ /mxxxxx (L) ensures the stability of the graph diffusion. Similarly, S(Z)) is obtained by keeping the same öt,, as above on G.
It can be seen from Eq. (6) that the stochastic variation of the diffusion interval ôt raises uncertainty of the diffusion state at time T. By aligning the two graph representations Z7 and Zrr, the condensed graph is guided to capture the diffusion trajectories of the original graph across varying states. In addition, we can derive the following proposition (Proofs are in Appendix (A):
Proposition 4.1. When enough diffusion times {ōl, YT=1 are sampled from the distribution P over the interval I 160min 2/ 3/mmax (L)), the spectral response function is satisfied by sampling coverage over the entire spectral range:
lim Pr(max sup min (01( - +,(66)) < = Vi
v,
T +00
ōtel
where (8())) Kött. is the spectral response function for the i-th eigenvalue A
Proposition 4.1 establishes a fundamental probabilistic guarantee for the completeness of spectral coverage in graph diffusion augmentation. This strategy breaks through the fixed-architecture constraints conventional GC methods and effectively preserves the original spectral properties.
4.2 Optimal Transport Plan Matching
For the generalized GC objective in Eq. 44) the distance function Ay(;)) )) is challenging to define without any task signal. However, semantic information should be intrinsic property of the graph, rather than being influenced by the downstream task. In other words, the semantics of the same node should remain unchanged in both the graph and the representation spaces. This insight inspired us to ensure the consistency of semantic associations between G and & by defining two similarity metrics in graph and representation spaces, respectively. In fact, it is nontrivial to directly compare these two metrics, since they belong to two different spaces and are two disjoint objects by nature.
Fortunately, optimal transport theory provides an elegant framework for this purpose. Therefore, we present an optimal transport plan matching, which aims to find the consistent optimal plan for graph alignment and representation alignment. For the graph space, natural idea is to consider both node features and structure information in the transport plan. In PreGC, the fused Gromov-Wassersin 43 is adopted to obtain the optimal transport plan Ti E R NMM,
T (G. 9)
arg min (1 X + (1 7)J J (A, (TD.D)
(8)
()),)))))))
5

---

where @ denotes the tensor-man matsor multiplication, and y represents the coefficient used to control the importance of structure and features. (XX, 1), |X - Xjا12 ll and [ A) = |Ass A are cost metric functions that measure feature discrepancy and structure discrepancy, respectively. For the node presentation, we leverage Wasserstein distance to obtain the optimal plan T2 E R NxM. M.
1 Ziy) = arg min (K(Z) =IElI(a.))
(9)
Intuitively, the optimal transport plan characterizes the transferring probability from source nodes to condensed nodes, and semantic association consistency in different spaces is preserved by minimizing the divergence between two transport plans. The transport plan matching loss is defined as follows:
Lplan (T 2 Z7 Z7
(10)
Without loss of generality, .) set to the Frobenius norm in this paper.
4.3 Node Significance Evaluation
Unlike conventional GC works without explicittly considering node contributions for condensation, the optimal resresentation transport plan n² here not only delineates the semantic associations between both graphs, but also inversely reflects the potential significance of the source nodes. Specifically, let's first define mask matrix Tmaak E R xM to filter out the most influencial nodes from source graph 9 on each condensed node i.e. we have (Tmask) )i.j = ( when (Tz k.j is among the top-H of the j-th column, otherwise (Tmask)i, vector s RR of the source graph can be obtained by:
Then, the node significance score
S Tmaak 1M
(1))
where 1s E RM denotes the all-ones vector. Essentially, the significance explicitly reveals the contributions of different source nodes for the condensed graph. Noticeably, the evaluation of node significance based on are two-folds. On the one hand, it provides more transparent interpretability for graph condensation, and on the other hand, also helps to guide node labeling effectively in an active learning way 20 Section 5.2 further confirms the validity of the node significance evaluation.
4.4 Condensed Graph Optimization
Pre-training. Benefiting from Eq. (9), the reconstruction term Oz can be achieved by diffusion alignment between Zip and Zip. Therefore, according Eq. (4) the total loss be denoted as:
(12) 39 After obtaining the condensed graph 9. the transport plan nz which reflects the transfer probabilities between the source nodes from & and the condensed nodes, is also retained. To ensure the uniqueness of the semantic mapping, we discretize Tzz to obtain the semantic assignment matrix M E R XM;
Liotal Lonat &Cptan
Loost Az (Z7, ZF)
where ( denotes the Wasserstein distance
M.j I(j arg max (T))
(13)
When the condensed graph & needs to be trained on downstream tasks, the target variable Ý E MxC can be obtained from the original label by the traceable semantic harmonizer Mz(-)::
= Mz (r) D M, Yor
(14)
where Y,, E R. Ner Ner xC is a training label with Ner labelled nodes, and Ω = (i.. in.,... } is the set of their row indices. Ma E N., x denotes the aabbassignment matrix defined by (Mą)k.) = M Dg = diag(M,). 1N..) MMM, and diag(-) is used to create diagonal matrix.
iN..
Test Time Fine-tuning. To further unlock the potential of condensed graphs while improving the performance of condensed graphs different tasks and GNN architectures, draw the idea of fine-tuning pre-trained models 3 to propose PreGCn, a strategy for achieving finer semantic alignment by fine-tuning the assignment matrix. Specifically, given a decay rate [0, 1]. the assignment matrix Mi is updated every Tup epochs according to predefined tasks and GNN:
M + eM + (1 - -e)Mup
(15)

---

where Ma E RNNM is obtained from #5(Y, Y) similar as Eq. ( Y and Y are the predicted outputs by the specific GNN encoded from G and g. respectively.
Through the above optimization framework, we empower GC with pre-training and fine-tuning capabilities. The overall pipeline of PreGC are summarized in Appendix
5 Experiments
5.1 Experimental Settings
Datasets. To comprehensively evaluate the condensation performance of PreGC, five graph datasets (Cora, Citeseer, Pubmed (26), OGB-Arxiv [60], and H&M (2) are utilized in experiments. Different tasks are tested to validate the generalization of PreGC, including node classification (NC), node clustering (NClu), link prediction (L.P), and node regression (NR). Specificll,, OGB-Arxiv includes two NC tasks: predicting paper topics (T) and publication years (Y). H&M includes an NC task: predicting product categories (C), and an NR task: predicting product prices (P). Two condensation ratios (r M are considered. More details are provided in Appendix C.1
N
Baselines & Implementations. We compare the proposed PreGC with six representative graph condensation methods (GCDM [3), GCond 25,, SFGC [64], SGDD [55) GDEM [33], and CGC [T6). Following [25] 42 the condensed graph is synthesized according to the default settings the above methods, and a certain GNN (default is SGC [48) is trained on this graph and evaluated on
the test set of the original graph. Moreover, other eight representative GNNs (GCN (6), APPNP
277. k-GNN 388 GAT 44. GraphSAGG [9, SSGC [65]. BernNet (33 and GPRGNN S are
selected to evaluate the effectiveness of PreGC. The details of baselines, task settings, condensation
settings, and parameter settings are provided in Appendix C.2 and C.3
Table 1: Performance comparison different tasks (We report test accuracy (%) on NC, NMI (%) on NClu, and AUC (%) on LP tasks. Bold entry is the best result, underline marks the runner-up).
GCDM (2022)
GCond (2022)
SPGC (2023)
SGDD (2023)
GDEM (2024)
CGC (2025)
PretiC (Our)
Dataset
Task
Ratio
Whole
1.3%
61.511.2
79.011.
75.400.
79.220.6
8088004
79.060.8
81.1003
SC
79.060.6
79.241.4
81.1808
80.700.1
$1.6019
80.800.8
1.3%
36.940.7
58.141.6
56.0220
59.0:1.2
57.2+0
58.2+1.2
$4.6+10
NC-
2.6%
45.34005
57.6099
57,5+0.7
58.7:0.5
57.060.5
50 Am0.5
50.660.5
19.3&1.5
Cora
1.3%
56.869.77
56.4330
NIA
57.083.
5.0.
75.362.2
1P
2.0%
57.241.8
62.241.6
NIA
604a22
39.7225
46.514.1
79.243.2
81.9+4.8
1.3%
19.164.7
26.8+10.1
NIA
14.356.
29.543.8
8.960.7
33.6+110
LP-xChh
2.6%
13.1
32.016.0
NIA
13443.3
35.2x1.4
10.3
34.922.1
42.343.3
0.9%
56,721.2
68.211.0
67.120.9
68.921.6
70,620.3
20.5+0.4
70.620.5
NC
1.8%
$2.12009
67.9.0.7
65.121.5
69.721.7
70.5:044
70.620.4
70.820.8
24.24
0.9%
25.3.0.7
41 41.012.7
42As18
43.0a133
42.8.0.7
40.6
43.8a1.4
NC-
1.8%
20.842.5
38.1+0.8
42.9H0.9
43.5H0.7
45.2e0.4
4..7+188
43.960.6
42.420.5
Citeserr
0.9%
61.513.8
61.512.0
NIA
60.022.
65.8437
NIA
66.51.66
LP
1.8%
57.565.3
62.843.4
NIA
56942.7
64.363
58.5239
69.323.3
80.844.9
0.9%
7,240.9
20.343.3
NIA
10.1+122
210+2.5
NIA
32.7e0.7
LP-
1.88
14.743.0
N/A
9.2414
16943.3
7.24144
33.111.2
33A43388
0.08%
71.4116
72,940.6
77.40.22
78.4004
77.8.0.1
78.7.0.3
79.940.4
NC
0.15%
60.840.8
71.6114
78.120.4
78.6005
76.820.6
75.420.5
$1.020.5
78.860..
0.08%
24.8+2.8
21.1+5.5
35.36016
35.680
35.5el.11
37,05616
35.6e1188
NC-
0.15%
19.410.5
18.9.0.7
36 9..02
36.1.0.3
34.150.4
38 38.000.4
38.360.7
32.1
Pabmed
72.142.9
22.943.8
NIA
72.663.1
43.52166
82.912.0
1P
0.15%
68.911.2
73.743.5
NIA
58.4332
56.1a2.3
84.842.0
86.312.4
95.0+1.0
14.1+3.0
6.264.4
NIA
3060.4
1/66022
N/A
22.5el0.7
1P-
0.15%
13.740.4
4.1+1.8
N/A
12.0+1.8
20.1+4.2
23,5+0.3
24.0=1.0
30.743.3
5.2 Experimental Results
Performance of PreGC on Various Tasks. We first demonstrate the generalizability of the condensed graph obtained by PreGC as shown in Table I Clearly, PreGC achieves optimal performance in most cases. In particular, PreGC consistently outperforms baselines on LP, demonstrating that the condensed graph of PreGC can well capture the topological relationships in the original structure. SFGC is unable to perform LP task (i.c. "N/A") due to its emphasis on structure-fre free during
7

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condensation, limiting its task reusability. In addition, Jsoos the generalization of PreGC under different supervision tasks and the flexibility task migration. Owing to the limitations that need to initialize condensed labels by the original class labels, existing methods can't be directly applied to node regression (i.e. "N/A"). By virtue of its unsupervised matching strategy, PreGC becomes the first GC method capable of adapting to NR task. Regardless of the tasks, PreGC consistently achieved optimal performance. In addition, the fine-tuning strategy PreGCr achieves 1.46% gain compared to PreGC on H&M dataset.
Table 2: Performance comparison to baselines on different supervised tasks ("Y-T"" means conden- sation on the "Y" task and testing on the "T" task. We report test accuracy (%) on NC and R² (%) on NR tasks. Imp. indicates the performance improvement over the best baseline).
OGB-Arriv (r 1.25%)
H&M (r 2.5%)
T-I
Y-T
Y-Y
T-Y
C-+C
P-C
P-P
C-P
Whole
0.34
55.37 0:21
77.07 : 0.26
50
0.64
GCDM
13.08+1.04
46.48+2.40
44.20H0.84
46 16.17=1.25
57. 17.15+0..0
N/A
N/A
26.08+00.7
GCand
56.7450.65
41.38+1 .5
49 49.25el0.07
42.52=2:11
64,57=11.53
NIA
NIA
15.29e1.14
SPGC
50 69.64:0.2
50.15e1.55
44
4.81=1.04
SGDD
68+0.82
27.80+2.09
41.70+2.25
42.30=1.55
61.85+0.30
N/A
18.85+2.13
GDEM
53.0349771
40.01+0087
49. 49.17H41.32
47 576000066
52.61+3005
NIA
NIA
6.8942.99
CGC
$8.11
53.32e1 61
49.50 00.5000.72
48.5360.34
64,35e0.33
N/A
NIA
15.95e1.11
PreGC
60:55:0:43
59.6240.47
52,54+0 30
$1.40+0.36
69.06e0).25
68.56-0220
34 39-9940.
33.17+1 415
Imp.
0.911
6:301
2,847
2.96
4.497
7,097
PreGCa 60.8120.40
60.8620.4
52,47:0.30
$2,41:0255
69,98:0.24
28.98
34.7550.74
34.6320.69
Imp.
7,541
2.979
3,881
5:417
8.551
Generalizability of PreGC across Different GNNs, Ideal condensed graphs should perform well in different GNNs, and to this end, each synthetic graph evaluated by nine GNNs, and the results are shown in Table 3 It is evident that the PreGC exhibits the highest average accuracy, suggesting that the condensed graph extracted by PreGC can consistently benefit various GNNs. Furthermore, only SFGC and PreGC have average accuracy exceeding that of training on the raw graph. However, SFGC can lead to limited application scenarios due to its lack of explicit structure.
Table 3: Node classification performance across different GNNs on Pubmed (r = 0. 0.15%). Highlight marks the lossless results and Avg. is the average results of GNNs.
SGC
GCN
APPNP
k-GNN
GAT
SAGE
SSGC
Bern.
Avg.
Whole
78.941.2
78:2400.
79.240.4
77.040.8
77.240.4
77,000.
79.040.3
77,440.5
79,840.3
78.1+1.0
GCDM
60.840.8
$6.811.1
70.010.9
72.310.5
76.220.8
53,310.8
69.810.4
68.211.3
71.120.7
66.517.7
GOoed
7166116
73.1+1.7
73.040.2
65.9+244
21.742.9
69.41155
73.3+0.4
72.6+2.2
72.4+0.8
71.4+2.4
SPGC
78.1+0.4
78.940.7
79.740.
78.140.7
77.540.
75.948.5
78.740.3
78.4404
79.8+0.2
78.3+1.1
78640.5
76.740.7
76.740.7
73,413.0
77,410.7
GDEM
76.8+0.6
77.3+0.3
77.7+0.4
78.240.5
26.7+0.7
76.4+1.3
27.5+1.1
76.940.4
27.0+0.3
77.240.6
CGC
78.420.5
79.510.3
78.840.4
75.210.9
78.920.5
76.4.0.7
79.040.4
77.310.4
78.540.6
78.1
PreGC
81,000.5
80.360.4
80.960.3
79.180.5
77.8.1.0
77.961.0
81,360.4
80.200.
80.010.5
79.861.3
Performance Gap between Condensed Graph and Original Graph. According to 16
49 33 the
quality of condensed graphs can be quantified through class-level features on NC task. Therefore, we present the labeled reconstruction error LRE x Σk A A with the class la- bel to evaluate condensed graphs. This formulation systematically captures multiscale receptive fields through the expectation over propagation steps to evaluate the distribution discrep- ancy. We set K = and the case l: 0 is also taken into account, which reflects the difference between the condensed and original features. As shown in Fig. 3 com- pared with two representative GC methods 16 33). the LRE of PreGC is consistently lower than that of the other two methods on different data and tasks. Since PreGC does not use any task labels in condensation, this is more challenging than baselines and also demonstrates the effectiveness of traceable semantic harmonizer.
40sk mi.33%
Figure 3: The LRE of different GC methods.
8

---

Data Valuation based on Node Significance. To confirm the validity of the significance evaluation in Section 4.3] we rescreened the same number of nodes as the original training set based on Eq. (II) and retrained them on the original graph. Fig. 4 (b) shows visualization of the distribution of nodes in the rescreened training set (Dark nodes denote the training set, and distinct colors denote different classes). Compared to the default training set provided by [26) (Fig. (a)), the rescreened training set demonstrates greater coverage and distributional dispersion. We further measure distribution differences with the average nearest neighbor distance d |.-1 Liev,. min i 2,112 (larger d
indicates wider coverage of the selected labels, and 2-order diffusion is considered. i.e., Z = A
by default), and the results also confirmed the above viewpoint. Fig. [ compares the performance of SGC trained with different training sets. Leveraging the rescreened nodes as supervision yields higher classification accuracy and markedly improves the training efficiency of the source graph.
(N) Deliult (á 1.66)
(b) PreGC-2.6%% 1.79)
(c) Performance on Cora
Figure 4: Performance of the training set selected via node significance on Cora.
Table 4: The prerformance of GC mehtods with different trainig ratio on OGB-Arxiv.
Training Ratio
0.15
0.3
0.45
0.6
0.75
Count
Whole
66.71 43
69.5 00.53++.11 II
70.44 70.44.0.51
71 11.6990022
71.72=0.57
GCDM
35.98=1.31
36.32+1.08
36.0920.88
36.9420.60
38.3520.67
GCond
56.1720.30
$6.95a0.31
56.83:0.49
57.4720.36
58.0020.34
34
x5
SFGC
59.09=0.42
59.8540.33
60.04:0.53
60.4620.51
60,59:0.48
x5
SGDD
58.01a0.77
58.7910.40
59.0320.44
59 59.5420.53
60.24400.6
X5
GDEM
52.23+0.32
54.20+0.57
54.81=0.44 .4
54 14.9800.48
55.41=0.31
x5
CGC
58.39=0.35
59 04+0.49
58.92=0.72
59 53+0.36
60.05+0.29
x5
PreGC
60.53:0.32
61.96:00.6
61.72:00.3
63.372a049
63.58+0.82
x1
Flexibility of PreGC. The existing condensation methods limited to concentrating under certain proportions of labels. When new labels are added in the original graph, these methods must re- condensation to capture this new knowledge in the condensed graph, which undoubtedly increases the condensation cost and limits its flexibility. PreGC can quickly transfer new labels to the condensed graph through the traceable semantic harmonizer without re-condensation. As shown in Table 4 we rescaled the training set ratio of the original graph and recorded the number of re-condensation (Count). With the structure and features of the condensed graph fixed, the condensed graph by PreGC shows similar patterns of change as the original graph as the training ratio increases. This label allocation strategy significantly improves the reusability of PreGC. Notably, existing baselines require repeated re-condensation to incorporate new label knowledge (x5), whereas PreGC achieves this with a single pass of semantic assignment matrix M.
A Alltion Study. To validate the efficacy of crucial optimization strategies, we con- duct a comprehensive analysis on H&M dataset through ablation studies. The ex- perimental results are shown in Table 5 wherein "PreGCwl Aug is the PreGC that removes the graph diffusion augmentation and "PreGCwio Cooa is the adoption of the existing strategy, i.e. predefined node labels before GC. We observe that both graph diffusion augmentation and Linat contribute significantly to the improvement of the GC optimization. Notably, "PreGCwin Leo.t induces label dependency in condensation, rendering the condensed graph via category prove inadequate for achieving superior performance on price prediction. Furthermore, the combination of these two modules increases the stability of the condensed graph.
Table Experimental results for the ablation study.
2.5%
5.0%
H&M
Category
Price
Category
Price
PreGe
19.3
34.4a0.9
71.144.3
36.9&1.1
PreGCCwt Awy
66.640.7
32.2+0.9
67.9+0.4
33.3+1.4
PreGCxxl Loost
67.5+0.5
29.1
68.9+0.8
31.7+1.5

---

More experimental analysis, including generalizability of PGC across different GNNs, comparison of LRE with different GC methods, data valuation, and sensitivity analysis are detailed Appendix [ In addition, we also discuss the related work and limitations in Appendix
6 Conclusions
In this work, we revisit the limitations of current GC methods and formulate generalized GC objec- tive, Following this paradigm, we further propose a pre-trained graph condensation framework for distilling task-agnostic and architecture-flext condensed graphs on specific GNNs, a bybriddinterre graph diffusion augmentation is proposed to en- hance generalization by increasing node state uncertainty. Subsequently, plan matching mechanism derived to eliminate task constraints, which relies on matching optimal transport plans for graph alignment and representation alignment preserve semantic association consistency. In addition, supervised fine-tuning further refines the condensed task signals and improves the performance the condensed graph on specific tasks. The experimental results consistently demonstrate the efficacy and reusability of PreGC, showcasing its generalization on various GNNs and tasks.
condensed graph. To avoid excessive dependence of
Acknowledgments and Disclosure of Funding
This work was supported in part by the National Key Research and Development Program of China (2024YFF0505704), National Natural Science Foundation of China (62476022, 62506029), Beijing Natural Science Foundation (L251043 4254085), and China Postdoctoral Science Foundation (0055M77189)). The authors would also like thank the anonymous reviewers for their constructive comments and valuable suggestions on this paper.
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Appendix
This the appendix of our work: "Towards Pre-trained Graph Condensation via Optimal Transport". In this appendix, we provide more details of the proposed PreGC in terms of theoretical proofs, dataset statisties, method analysis, experimental settings with some additional results, and related works.
A Proof of Propositions
Proposition 3.1. Suppose that it has an analytical filter g(:) for GNN model, the performance approximation error Ow of condensed graph is jointly bounded by the reconstruction term Oz and the fitting term Oy.
|W-
[MZ(Y)ll (M (((L))))(((((()))))))
(o(L)X)))
Ow
Oz
Oy
(16)
where Mz(-)) ) is any mapping function that aligns g g( X and g(L) or ý and Y. L IN - and L I A are the Laplacian matrices of G and 9.
Proof. Given GNN with analytical filtering function g()), have (())XW Y the original graph and XW) Y on the condensed graph G. respectively. Due M2(() can align the first dimensions, we also have Mz M=(g(L)X)W = -Mz(Y) for G. Therefore, the expression can be denoted as:
IW- W-WWFp (XX)
(|M2((L)))) (U
(z (G(L))) (17)
[[(M2((L)) ( + (
& (M ( - ( || (M - - (|)
For the trajectory matching methods 664 59 they integrated the above objectives into the process of graph condensation, that is:
arg min| W- W|r=
arg min { (Yll ( - - - [
(18) According to [16]. different matching strategies can be summarized into the same optimization objective. Thus, Proposion 3.1 shows that by minimizing the reconstruction term Oz and the fitting term Oy, the performance derived from can be approximated that g.
This completes the proof of the proposion.
Proposition 4.1. When enough diffusion times (8t-) are sampled from the distribution over the interval I = jōtmin 2/ Xmax (L)), the spectral response function is satisfied by sampling coverage over the entire spectral range:
lim Pr(max sup min (0,(5t) - +,( < <)) 1, v4 T-x
(19)
where (8t) =
- Kött. is the spectral response function for the i-th eigenvalue X
Proof. Given a graph g = (A, X), ) the diffusion state with diffusion intervals 8l, at time K8t, can be denoted as:
- Kôt. Lx Kott T
(20)
For the i-th Laplacian eigenvalue X; we write the corresponding spectral response (b(61) e KX with = 1, N. Here, begin with the following the lemma to better support Proposition 4.1:
22

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Lemma 1. For any two different diffusion intervals 5t,, 8tg E I, the difference in spectral response satisfied:
| +,(6t)) KAe Kozd: lôt1 -ôt2l
(21)
where v, lies between öt1 and öt2.
Proof. According to the mean value theorem, for the function (,(5t,) = e -K there exist d/s [minn(tt, (tz)), max(ot1. öt2)] such that:
(,t))) 2,())) (0))) - 8t1)
(22)
(((1)) KAie" KViA,
Taking absolute values on both sides of the equation:
(6tt)) ()(6t)) le Kót1 c -K Kdse Koid, .881 8t2l
(23)
This completes the proof of the lemma.
From Lemma 1, each o; is Lipschitz I with constant:
L sup ((t) KλGE K KAj StET
(24)
Let I, = max; I = KANe AN where An = Amax (L). According to the lemma, for any
öt I, if there exists sampled point öt, such that jôt öt, < L. then for all i, the spectral response functions satisfy:
(0t())) (,tt)) Kond. 1ôt - öt, S Li 1ôt öt S Li
y L
< 4p
(25)
To ensure this condition, we divide the interval I = [ôtmin 2/ Amax (L)) into m
L
Amn (L) 66mmm)))))) sub-intervals, each of length [/] 2/ Ama (L)-5 < G. Assuming the diffusion
times (ōl, (ō6-)-=1 sampled from distribution P with positive density over 7, for each subinterval, the probability that at least one sampling point falls into it is:
1 (1 Prain)
(26)
where Pmin > is the minimum probability of sampling a point in any subinterval. When T + oc. we have p + 1. The probability that all m subintervals have at least one sampling point is pd and pm - when T - 20.
When all subintervals contain at least one sampling point, for any 8t E I, there exists a sampling point öt, within the same subinterval or in an adjacent subinterval, with |öt öt, < f. This ensures that:
max sup min (61) (,,66)) << x stel
(27)
Therefore:
lim Pr(max sup min (5t) +:(bt=)) < x0) 11 44 > 0
(28)
T
Stêf
That is, when T is sufficiently large, the probability that the sampling points form yo-net covering T
tends to 1, ensuring coverage of the entire function family (0:(8t) = e Kōt :i = 2, N).
This completes the proof of the proposition.
B Implementation Details of PreGC
B.1 Algorithm
The detailed algorithm of PreGC is shown in Algorithm | In detail, the topology generator GEN(O) and the condensed features X are initialized before condensation. To accelerate the synthesis of the condensed graph, the clustering centers of K-means are adopted as the initialization of the condensed features. Here, GEN(O) can be any strategy used to generate graph structure, and
23

---

with the symmetric normalization function Sym(-) for convenience. we adopt (
))
Then, node representations 7 and Zip are generated via the graph diffusion with stochastic states. By calculating the optimal transport of the two graphs in graph space and representation space, one can obtaín the transport plans TD and =2 respectively, and the transport cost Lenst. Plan matching loss can be obtained from the difference between metrics TD and T'z. Finally, the PreGC minimizes the loss Liotal by iteratively optimizing @ and X. Our work signifcantly differs from other GC methods, PreGC truly enables pre-training and provides transparent interpretability of the condensation.
Algorithm 1 Pre-trained Graph Condensation (PreGC)
Require: Real graph G = (A, X), learning rates ?h, and f22
Ensure: small-scale condensed graph (A, X) with the mapping matrix M.
1: Condensation Phase */
2: Initialize and X.
3: for 0, 1........ T do
4: 1 Graph Diffusion Augmentation #/
5: Compute A GEN(O);; then 9 = (A, X)
6: Randomly generate diffusion interval 5t,
7:
Graph diffusion on G and & to genenrate Zi and 2% according to Eq. 6 respectively
8:
1e Transport Plan Matching */
9:
Calculate graph transport plan Th according to Eq. 8
10:
Calculate representaion transport plan TTz and transport cost Ccost according to Eq.
Compute plan matching loss Cptan according Eq. 1T
12:
o Condensed Graph Updating #1
13:
Compute total loss Liotal Loost + &Cplan
14:
if (7(T + T2)
then
15:
Update θ - 7n1 VeCioual
16: else
17: Update X + - - V Scotal
18: end if
19:
end for
20:
Compute according to Eq.
À GEN(O)
21: return A, X. and M
B.2 Details of Optimal Transport Solution
Since Eq. (9) can be regarded special form of Eq. (8) (i.e., 7 1), our discussion primarily focuses on solving for TD in Eq. (8).
Eq. (8) decomposes into two terms: the former is Wasserstein distance that operates solely node features. This term can be efficiently approximated using the Sinkhorn- Knopp algorithm [[].
which iteratively converges the optimal transport plan x². Specifically, the Sinkhorn-Knopp algorithm introduces entropy regularizer and alternates between Sinkhorn projections, (())
exp(-X(X, X) and (+++1) = Fi (F)((()))) where represents the number of iterations and X denotes the weigth of regularization. The projections (T) = π 2 (11 T) and F2(TT) 2 (T111)
denote the column normalization and row normalization, respectively, where @ is element-wise division. They ensure that the rows and columns satisfy the given marginal distribution constraints.
In the limit, this alternating projection will converge to minimizer lim 7(1) = =
The latter term corresponds to Gromov- Wasserstein (GW) distance, which can be formulated as nonconvex quadratic programming problem. We use semi-relaxed Gromov-Wasserssee Wasserstein divergence 45] and optimize with the conditional gradient solver. The gradient of with respect to the GW term is V 2J 2A(AA, A) & The conditional gradient algorithm consists in solving linearization (P, v at each iteration, where P E TR XMM the extreme-poin solution that arises from the linearized sub- aob-problem solved. It can be solved by gradient descent with a direction (0) followed by line search for the optimal step.
In the optimization process, we utilize the third-party library functions GeomLoss and Python Optimal Transport (POT) to solve Eq. (8) and Eq. (9), respectively.
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C Experimental Details
C.1 Dataset Details
In this work, five graph datasets are adopted to evaluate the effectiveness of our condensation method. and their data statistics are shown in [ab Among them, three citation networks (Cora, Citeseer, and Pubmed) [26) are used for node classification (NC), graph clustering (GC), and link prediction (LP) tasks. OGB-Arxiv [60] is used to test the performance of GNNs on topic classification and publication year prediction tasks. H&M [ is subset that is obtained from a co-purchase network of products from the H&M company. The nodes are products whose features consist of the product's graphical appearance, perceived color, etc., and they are connected edges if they are frequently purchased by the same user. H&M has two semi-supervised tasks: predict the categories of products (node classification) and predict their average prices (node regression).
Table DI: Details of dataset statistics. NC, NClu, LP, and NR denotes node classification, node clustering, link prediction, and node regression tasks respectively.
Dataset
Nodes
Edges
Features
Task
Classes
Traing Validation Test
Cora 2,708
5,429
1,433
NC NClu LP
7
140/ 1,000
Citeseer
3,327
4,732
3,703
NC/ / NClu LLP
/00/500/
500 1,000
Pubmed
19,717
44,338
500
NC NClu LP
3
60 500 1,000
NC-Topic (T)
40
4,585 2,517 7,065
OGB-Arxiv
14,167
33,520
128
NC- Year (Y)
5
7,125 / 3,492 3,550
NC-Category (C)
21
7,837 / 33,908 3,763
H&M
15,508
920,284
191
NR-Price (P)
7,724 3,843 3,941
C.2
Baselines & GNNS
Baselines. In this work, six baselines have been selected for comparison.
GCDM B2]:
An efficient GC method that generates condensed graphs based on distribution matching by optimizing the maximum mean discrepancy between class representations.
A representation GC method that utilizes the gradient matching to align the model gradient from both graphs.
GCond 25]
SFGC 64
The first structure-free GC method that synthesizes condensed features by trajectory matching while preserving the topological information of the original graph.
generalized graph condensation method with enhanced graph structure modeling by topological mapping.
SGDD 55
GDEM 33 A GC method for topology enhancement by eigenbasis matching.
CGC 16
An efficient GC method with graph generation that introduces the training-free strategy
in distribution matching.
To illustrate the distinctions among various approaches and emphasize our contributions, present the relationship between six representative graph condensation methods and the generalized graph condensation paradigm, as shown in Table [2
GNN architectures. To evaluate and compare the generalization of PreGC and other graph conden- sation methods, we test the node classification performance with nine different GNN architectures: SGC [4]. GCN [26] APPNP (27]. k-GNN [38], GAT 44], GraphSAGE (SAGE) [9) SSGC [65], BernNet (Ber.) 23 and GPRGNN (GPR.) g. Note that the focus of our work is on verifying the generalization of condensed graphs rather than the effectiveness of certain GNN architecture. Therefore, we harmonize the GNN parameters for all datasets to ensure the faimess of the exper- iments. Concretely, the smoothing factor APPNP, SSG, and GPR. is uniformly set to 1, the numer of attention heads in GAT is set to 4. For GNNs 27 65 23 g that can capture long-range dependencies, we set the number of propagation layers K by default and K otherwise. For other hyperparameters,
the combination of parameters provided by PyG by default.
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Table D2: Comparison of different graph condensation methods. For convenience, SGC À K A is uniformly set to GNN encoder in condensation, and A (hr) (1 - ōt)1 + StA.
Condensation Phase
Testing Phase
Label Alignment
Graph
Optimization Objective
Representaion
Graph Interpretability
Task Adaptation
Condensation
Z
GCDM
Distribution
(Ardv22))
Before
Matching
Ox
AXIAXXI ........ 11
NIA
Classification
Gradient
(CCLR22)
Refore
Matching
Ow (wio (3y)
A
N/A
Classification
SPGC
Trajectoyy
(Ncu
Refore
Matching
Ow (wio Oyy
1
N/A
Classification
SGDD
Gradient
(NcudPS22)
Before
Matching
Ow (wfo Oy)
A*X
Topology
Classification
GDEM
Eigenbasis
(CML244)
Before
Matching
Oz
X LX
Topology
Classification
CGC
Distribution Matching
(WWW25)
Before
Oz
NIA
Classification
PreGC
Transport Plan
After
Malching
Oz. Oy
[A
Semantics
Any Tasks
C.3 Experimental Setup
Task setting. Unless otherwise noted, we use the default training/validaationnessis ratios to evaluate the performance of the condensed graphs for supervised tasks. Specifically, for Whole the training set of the original graph is used for training. For graph condensation methods, we train on the condensed graph, and validation and testing are always evaluated on the validation and test sets of the original graph. For link prediction task, we follow [42 and randomly select 10% ratio of edges as validation and test sets, respectively. In node clustering task, the K-means algorithm is adopted to the learned embeddings of all nodes (For example, "NC-NClu" is denoted as the embedding obtained after training on node classification as input for clustering) and adopts normalized mutual information (NMI) to assess the quality of the clustering results. We report test accuracy (%) on NC, NMI (%) on NClu, AUC (%) on LP and R² (%) on NR tasks. The average results and standard deviations of performing different random seeds on each task are reported.
Condensation setting. For condensation methods. the hyperparameters are set according to the original paper if available. Otherwise, we follow the hyperparameter combinations GC-Bench [4). Except for the OGB-Arxiv dataset which uses the normalized features provided by [60], all other graph data are condensed using the original features. For our proposed PreGC, we set the importance coefficient = 0.5 in Eq. (8) by default, 8/ = 0.1 for all datasets.
Evaluation setting. To verify the effectiveness of various graph condensation strategies, all graphs, except H&M adopt one-layer GNN encoder, are uniformly configured with a two-layer SGC as the GNN encoder and a single-layer MLP the predictor by default to adapt different downstream tasks. For all GNNs, the hidden unit is set to 64, with a learning rate of 0.01, weight decay of 0,0005 and the Adam optimizer employed.
Experiment environment. The experiments are conducted on the machine with Intel(R) Xeon(R) Silver 4314 CPU @ 2.40GHz, and NVIDIA GeForee RTX 4090 with 24GB memory and CUDA 12.0. The operating system is Ubuntu 20.04.6 with 384GB memory.
D More Experimental Results
D.1 Performance of PreGC on OGB-Arxiv and H&M datasets.
In this section, we show the experimental results at other condensation ratios, as shown in Table B1 For transfer tasks such as "Y-'T", the condensed graph is first generated on the source task "Y". The testing phase involves: (1) computing the transport plan between original and condensed graphs, then (2) transferring the labels of task "T" from the original graph to the condensed graph for evaluation.
26

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In particular, existing condensation methods are incompatible with node regression tasks, making condensed graphs unavailable when the source task is "P". Not surprisingly, PreGC consistently outperforms existing GC methods. In particular, the fine-tuning PreGC,i.e. PreGCn, achieves more significant gains on the transfer task. This suggests that the semantic distribution will significantly different for different tasks. However, existing methods can only be adapted to single categorization task, resulting limited application of their condensed graphs.
Table El: Performance comparison to baselines on different supervised tasks.
OGB Arai (r 2.8%)
H&M (r 5.0%)
T--T
Y--T
Y--Y
T--Y
C-2C
P--C
P-4P
C-
Whole
0:24
55.37 1 0:21
77.07 # 0.26
50.50 0.04
0.04
GCDM
35.4923.01
M1.991.59
45 15.6920:95
34.66g3.24
59.6911 18
NAA
NIA
19.23x1.87
Geund
58.1910663
19.9211.66
46.8831 1.05
47.18a1006
51,91a2.71
NiA
NIA
15.35a1.45
59.7250.26
47.9320.57
59.3142.28
SGDD
60.1610.33
$1.8:10,00
(9.8920.53
64.50=1.61
62.64.0.51
NIA
NIA
17.7543.19
CDEM
56.6910.70
34.1910.03
49 49.9110.63
48.5 48.54.0.62
$1.7120.89
NIA
NIA
[1.73a1772
CGC
59.7120.42
36.78s1.25
50.7960.29
49.7130.54
66.00:0.29
29.7940.64
PreGC
62 6660.2
01.940:49
52,9620.35
52.05.0422
21,0640.26
69.88.041
36.9411.05
25.2540.87
Imp.
5:11
4.46T
6.441
PwGCC
62.90=0.49
62,8500.48
53.1140.17
53,20H0.21 3.497
71.54e01.00
71.44:0322
37.0760.96
37.01a0994
Imp.
2,441
6,071
2.329
4.947
7:221
D.2 Generalization Ability of PreGC across Different GNNs.
This section presents additional generalization results of PreGC across different GNNs, as detailed in Overall, the condensed graphs synthesized via PreGC consistently outperform other condensation methods across diverse datasets, demonstrating the effectiveness of graph diffusion augmentation. While the most competitive baseline CGC surpasses PreGC on certain GNNs, its performance remains limited to specific architectures and shows inferior results on some spatial GNNs (e.g., k-GNN and GraphSAGE).
Tables F2 F4
Table E2: Node classification performance across different GNNs on Cora (r 2.6%) Highlight marks the lossless results and Avg. is the average results of GNNs).
SGC
GCN
GNN
SSGC
Avg.
80.8:0.8
81.0:0.6
81.910.8
77.7+1.0
81,810.8
81.310.6
81.5e0.3
81.240.8
80.7x1.4
69.2+0.4
67.4+0.9
74.3+0.3
44.8+2.3
68.240.8
73.940.7
70.4+1.2
66.4+2.5
64.6+11.1
OCoed
79.010.6
79.110.5
78.740.8
63.243.1
73.3u1.6
66.711.5
78.511-1
74.413.0
72.011.6
73.915.8
SPOC
79.2+1.4
80.0+0.5
80.340.5
78.240.7
79.9+0.7
77.5+0.6
80.740.7
80.0+0.7
79.5+0.4
79.5e1.0
81.1+0.8
79.8+0.77
81.241.4
79.240.6
79.5+2.4
78.8+1.4
78.7+3.0
GDEM
80.810.4
80 R10.4
81.110.9
78.510.7
80.3,0.4
79.110.4
81,240.3
80.910.5
80.610.4
80/410.9
CGC
81.8:0.6
81. 140.9
81.6+0.8
40.4+7.0
81.040.9
66.4+1.2
81.140.3
81.74166
81:440.6
75.2+14.0
PreGC
81.610.9
81,84:1.1
82,350.4
78.861.0
80.011.0
79,850.6
82,151.0
$1.120.5
82.340.6
81.1133
Table E3: Node classification performance across different GNNs Citeseer (r 0.9%).
SAGE
SSGC
Avg.
Whole
69.8177
GCDM
54.721.2
53.120.9
36.421.6
33.721.2
56.121.2
41.741.2
55.911.5
51.920.7
49.211.9
50.517.9
GCoed
68:9+1.0
70.14007
60.141.3
52.0+3.4
49,445.
62.741.1
60.742.0
$9.944.0
48.8+5.8
59.247.8
SPGC
67.140.9
65.7=24
65.44166
65.740.7
6848199
63.252.2
66.6+2.
SGDD
68.9+1.6
7..6110
69.0:1.9
62.144.2
69.9+1.2
7.3.3.11
GDEM
70.640.3
69.2+0.8
70.140.
68. 68.6+0.7
69 20.260.7
68.440.7
69.5411.
69.240.8
69,54014
69.441.7
CGC
20.5e01.
72.45115
71.5ell.7
63.3+2.0
N04408
63.0e1.5
72.2al1.8
72.2sl1.5
64.9+1.0
68.944.1
PreC
71,0s1.0
69.8e11.7
D.3 Performance Gap between Condensed Graph and Original Graph
Figs. El and E2 show the labeled reconstruction eтror of different condensation methods. It can be clearly observed that PreGC consistently and significantly outperforms existing graph condensation methods, particularly on the CiteSeer and PubMed datasets. This result not only confirms the effectiveness of PreGC preserving the critical properties of original graphs during condensation, but also reveals its robustness architecture dependencies.
27

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Table E4: Node classification performance across different GNNs on OGB OGB-Arxiv-topic (r = 2.5%).
GCN
APPNP
1:4INN
SAGE
SSGC
GPP..
Whole
64,420.2
63.820.5
65.420.5
64.220.5
63.810.3
61.880.2
64.740.4
63,056.2
64.520.8
63.921.0
GCoodd
58.5400.5
58.7411.
58.140.9
27.842.6
39.8+4.6
29.242.2
60.158.22
52.5&144
46.821.5
47.9+1299
SPOC
59.720.3
58.821.1
59.920.8
57.066.2
59.520.5
50.121.3
00.420.5
57.521.9
58.320.8
57.943.1
SGDD
60.540.3
54.710.9
56.510.9
44.112.8
31.112.0
11.311.9
60.310.4
33.943.8
37.913.1
16.7111.5
55.341.3
55,2+1.0
CGC
39.720.4
58.720.5
61.840.8
31,721.1
00.520.5
37,61199 19
00.120.2
57.121.0
53.92088
53.5211,0
PreGC
62.9e10.5
02.5ell.7
64.9e0..
62.9
61.4+0.6
$9.5+10.5
64.2e0.7
59,841.5
63.2+l0.5
62,4+1.8
Table E5: Node classification performance across different GNNs on OGB-Arxiv-year (r 2.5%).
SGC
GCN
APPNP
i:CINN
GAT
SACIE
SSGC
Bern.
CPRR.
Whols
55.42112
57.75004
55.9H114
56.8ell.5
56.1ei..5
54.560.5
55.5e60.4
53.940.
54.2
55.5&1.3
GCDM
45.711.0
51,020.2
50,920.22
43.721.1
30.210.4
41.842.1
49.920.7
47.9963.4
50.5+0.4
SHGC
50.1896
52,020.5
51.340.3
S1A4014
31.941.5
45.41.22
31,24004
51.3+0.1
51.640.1
51.7+2.11
SGDD
29.921.
50.960.2
50.720.1
51.120.
51.120.0
42.420.11
50.520.4
43.023.7
48.5&1.3
48.72:355
49.04016
51.140.3
43.0+3.1
51.120.4
39.1+10.8
50,950-11
PrGC
53.0e11.4
53.360.3
53.5e0.1
52.740.3
53.040.2
52.1=0.4
53.5e0.4
51.1+1.0
51.940.1
52.740.8
144
Figure El: The LRE of different GC methods with the smaller condensation ratio.
Figure E2: The LRE of different GC methods with the larger condensation ratio.
28

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D.4 Data Valuation based on Node Significance
This section evaluates the node significance of the original graph, i.e., the ability to explicitly represent the importance of the source node through the transport plan. We rescreened the number of nodes as the original training set based on Eq. (IT). For example, in the Cora dataset, the default number of training nodes 140 (Default), and select the top-140 raw nodes based on node significance s and label them as the training nodes. The training nodes selected through the transport plan prove more valuable than the default training set, yielding significant improvements in the performance of GNN, shown in Figs. E3 and E4 This matching method not only enhances the effectiveness data annotation, but also elucidates the relative influence different nodes during the condensation process, thereby improving the interpretability of graph condensation.
(a) Default (d 2.88)
(b) PreGC-0.9% (d 3.03)
(6) Performance on Citeseor
Figure E3: Performance of training set selected with transport plan on Citeseer.
10C
(s) Dd= 0.122)
(b) ProGC-0.09% 0.174)
(0) Performance Pobmed
Figure E4: Performance of training selected with transport plan Pubmed.
D.5
Sensitivity Analysis
Our investigation further examines the sensitivity of several crucial hyperparameters: the loss balancing coefficient& & in condensation, the iteration interval Tap and the decay rate e in supervised fine-tuning, illustrated in Fig. ES The sensitivity to the balance coefficient varies across different datasets. Generally, the best results are achieved when & = 1 or 2. Fig. ES E5 illustrates the impact of supervised fine-tuning the H&M dataset (The dark blue part represents the condensed graph without supervised fine-tuning), where a trade-off between the values of Tap and is required to ensure optimal performance of PreGC.
71
Parmelyy 1950
70
02
3.2
10
Dedicari
(n) Perfomsance of PreGC with different
(b) Performance of ProOC, with differont
Label distribution H&A
Figure E5: The impact different parameter combinations on PreGC.
29

---

D.6 Ablation Study
In this section, we add ablation experiments of PreGC on the OGB-Arxiv dataset, with the results presented in Table Consistent with the observations on the H&M dataset, both key modules play a significant role in graph condensation. Notably, the removal of graph diffusion augmentation leads to increased performance fluctuations, which demonstrates that graph diffusion augmentation can enhance the generalization capability of the condensed graph.
Table E6: Experimental results for the ablation study on OGB-Arxiv.
1=1.5%%
r=2.5%
OGB Arxiv
Topic
Year
Topic
Year
PreGC
60.55+0.43
52.34+0.30
62.66+0.47
52.96=0033
PreGCowo Aug
59.21=0.75
51.10#0.66
59.93+0.68
51.06+0.79
PreGCwM who Ecost
58.84+0.51
49.4240.58
59.99+0.42
50.55+0.49
D.7 Computational Cost
In this section, we conduct additional experiments to compare the computational efficiency of PreGC with existing baselines on the Cora dataset. It covers peak memory usage and runtime across the pre-processing, condensation. and fine-tuning stages (CGC is excluded as it is training- ffree method). The results are shown in Table E7 denotes the result of a single execution. Note that in practical implementations, multiple preprocessing often be necessary depending on specific settings.). It can be observed that PreGC achieves significantly faster condensation time compared to most GC methods. Even when fine-tuning is required, it only increases few additional time costs. Although GDEM spends the least time during the condensation process, it requires extra preprocessing time for eigenvalue decomposition. Overall, both PreGC and GDEM exhibit substantially lower training times than other GC methods. The underlying reason is that existing GC methods employ nested bi-level optimization: outer loop updates the GNN parameters, while an inner loop optimizes the condensed graph. In contrast, PreGC leverages graph diffusion, a parameter-fres message passing mechanism, eliminating the conecconumingg outer loop. This not only reduces condensation time but also prevents the condensed graph from overfitting to architecture-speciff parameters. Furthermore, when downstream tasks or label distributions change, existing baselines must re-condense the graph to capture this updated knowledge. In contrast, PreGC is task- and label-agnostic, thus requiring only a single condensation and can be reused multiple times.
Table E7: The computational cost on Cora dataset (r = 1.3%).
Memory (MB)
Pre-processing (S)
Condensation (S)
Fine-tuning (S)
Total (S)
GCDM
1,276
1542.38
1542.38
GCond
1,280
2,304.41
2,304.41
SPGC
14.01*
1,937.65
SGDD
1,644
2,364.18
2,364.18
GDEM
2,688
4.56*
7.48
12:04
PreGC
1,714
22.36
0.81
23.17
E Discussion
E.1
Related Work
Graph Reduction. Unlike model quantization 47 and knowledge distillation [37. graph reduction
22 is a data-centric approach. It aims to reduce the size of graph by decreasing the number of nodes and edges, allowing efficient and effective GNN training. includes graph sparsification [
64
graph coarsening [41, 28], and graph condensation 25 55]. Graph sparsification aims to reduce edge density while preserving key structural or spectral properties the original graph. Graph coarsening aggregates nodes or edges form smaller super-node or super-edge, often maintaining hierarchical relationships. Their core principle to maintain specific large-scale graph properties (e.g., spectra and principal cigenvalues) in smaller graphs that have been thinned coarsened, essentially still
30

---

modifying the original graph. In contrast, graph condensation directly optimizes and synthesizes a small-scale condensed graph, which breaks the information constraints imposed by the original graph and promotes consistent GNN test performance.
Graph Condensation 25 a.k.a., graph distillation [3]]] aims condense large-scale graph
data into a small-scale synthetic graph, reducing garph redundancy and accelerating GNN training. Due to its superior ability in graph reduction, it has found a wide range of applications, including
accelerating inference [4] graph continuous learning 34
and federated graph learning MT
Early GC methods inspired by data condensation [61 proposed matching strategies such as gradient matching [25] [3), trajectory matching 64 59 distribution matching [32] 16 [4]] etc.,
focusing more on learning the consistency of the representation space and proposing related variant of structure-free. SGDD 5 was the first to propose Laplacian energy distributions to metricize the relationships among structures. GDEM [R] analyzed the upper limit of the existing GC performance is limited by the GNN architecture and improves this problem by eigenbasis matching. In addition, the parameters different GNNs also bring implicit bias that would be inherited by the condensed graph. [16] and [17] proposed efficient graph condensation without parameters to avoid this problem. [54] synthesized a condensed soft label using multiple self-supervised tasks as auxiliary information to enhance the fitness of condensed graphs. Despite the great efforts made by the above methods, they are still limited to a particular task (e.g., node classification). Recently, SGDC [46] proposed self-supervised graph-level condensation, which is free from task dependency by contrast learning. [1] extended it to self- coffssupereieed node-level condensation, improving the generalization ability in condensed graphs on unsupervised tasks. However, the essence of the above methods is still based on class partitioning, which cannot be applied to other supervised tasks (e.g., node regression), hindering the reusability of condensed graphs.
To our best knowledge, we are the first attempt study generalized graph condensation. PreGC has the advantages of being GNN-parameter--free and task-independent, allowing the condensed graph to be flexibly adapted to arbitrary GNN architectures and diverse downstream tasks.
Applications of Graph Condensation. In addition to the aforementioned research, graph conden- sation has been further extended to other domains [40] 4 For example, Jin et al. 24] were the first to generalize graph condensation from node-level to graph-level tasks and significantly reduced condensation cost using one-step gradient matching. Xu et al. (50) reformulated the paradigm of graph-level condensation via kernel ridge regression while preserving the performance of the original graph data. Gupta al. (18) and Wang et al. further extended graph condensation to model-agnostic and self-supervised graph-level settings, rapidly advancing the field. Moreover, owing to the training efficiency of condensed graphs, Liu et al. [35] 33) integrated graph condensation with continual learning, both mitigating catastrophic forgetting and accelerating model training. Yan al. [51] and Zhang et al. (57) applied graph condensation to federated graph learning, enabling distributed graph learning under privacy constraints by sharing knowledge through condensed graphs.
E.2 Limitation
Although the proposed PreGC is highly generalizable and reusable, eliminating the need to repeat the condensation, it still suffers from a shortcoming. PreGC inevitably introduces additional complexity to the condensation process due the computational constraints of optimal transport. Potential improvements in computational efficiency can be explored through graph partitioning techniques or linear optimal transport approximations. However, since the main contribution of this work is to propose a generalized graph condensation paradigm and pre-trained condensation framework, we defer the study of improving computational efficiency to future research.
E.3 Broader Impacts
Although graph condensation is pivotal for scaling graph neural networks to large-scale graph data, it also entail fairesss and privacy risks. Due the condensation process selectively preserves or amplifies structural patterns, it may inadvertently over-represent majority groups while suppress- ing minority substructures. Moreover, the re-synthesis of node features complicates established anonymization guarantees, as condensed features may be re-identified via topological cues. Accord- ingly, the condensation procedure should be considered to mitigate unfairness induced by structural bias while simultancously preventing the leakage of sensitive attribute information.
31


Reviews:
[{"review_id": "w7E4kLQl8Y", "reviewer": "Reviewer_NPxj", "summary": "This submission proposes PreGC, a novel graph condensation framework that distills large-scale graphs into compact synthetic graphs via optimal transport and graph diffusion augmentation. PreGC tries to eliminate task and architecture dependencies in conventional graph condensation methods by (1) using hybrid-interval graph diffusion to enhance generalization across GNN architectures and (2) employing optimal transport plan matching to preserve semantic consistency without task-specific labels. Experiments partially validate PreGC's superiority in cross-task/cross-architecture settings and its interpretability through node significance evaluation.", "questions": "Please mainly answer the concerns about the submission listed in C1 and C2.", "limitations": "N/A", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\nS1 Clearly identifies limitations of existing graph condensation methods (task/architecture dependency) and establishes a generalized GC optimization paradigm. Theoretical grounding solidifies the foundation.\n\nS2. Logical flow from problem analysis to solution design, supported by illustrative figures and comprehensive ablation studies.\n\nS3. Evaluated on 5 datasets, 9 GNN architectures, and 4 task types (node/link classification, clustering, regression). Includes sensitivity analysis, interpretability studies.\n\n\n**Concerns**\n\nC1. Graph diffusion and optimal transport are established techniques. While their integration is novel for GC, the authors are expected to emphasize unique research challenges addressed, e.g., spectral coverage completeness in diffusion for architecture-agnostic GC, transport plan matching for task-agnostic semantic alignment.\n\nC2. This submission needs to focus on strengthening the task-constrained claim verification via more experiments\n\nC2-1: It would be better if authors can expand task diversity to demonstrate broader generalization beyond node/edge-level tasks, including subgraph/graph-level tasks.\n\nC2-2: Based on the pre-trained model with fine-tuning, it is a very classic pipeline to migrate or adapt to different tasks. The basic argument for this pipeline is that the performance of the model trained only on a certain task is not as good as this migration mode. For this reason, the author needs to focus on verifying this point. However, if understand correctly, some results in table 2 are quite conflicting:\n(1)\tGCDM’s Y→T (T→Y) vs. T→T (Y→Y) suggests GCDM handles some cross-task transfers better, contradicting PreGC’s claims. As shown in Figure 1, the authors argue that existing GC works cannot generalize to other downstreaming tasks.\n(2)\tPreGC’s Y→T (T→Y) vs. T→T (Y→Y) indicates fine-tuning instability. It is proved that the performance of the preGC model obtained by migration is not as good as that obtained by training only for this task. So, why should we follow the pipeline of “pretrained-finetuned”.\n\nC2-3: Claims like \"PreGC uses no task labels\" (Line 298) are insufficient. Add experiments: Compare PreGC vs. baselines under limited labeled data (e.g., 10%, 30%, 50% labels) to prove robustness to annotation scarcity.\n\nC3. Please fix some typos, such as “[22, 21, 45] distill...” and [53, 48] replicated”. In addition, the author did not standardize the tense.", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your recognition of our work. In particular, the theoretical analysis and summary of the existing limitations represent one of the most significant contributions in our study. Below, we will address your concerns and questions point by point to clarify any potential confusion.  \n\n**W1:** We are deeply grateful for your perceptive and constructive feedback. We fully agree with your perspective. In particular, the spectral coverage completeness guarantees that the condensed graph faithfully preserves the spectral properties of the original graph, while the task-agnostic semantic alignment enables unsupervised graph condensation. To achieve these two pivotal objectives, we devise two elegant mechanisms, i.e., graph-diffusion augmentation and transport-plan matching. This also ensures that PreGC strictly conforms to our formulated generalized GC paradigm (Definition 3.1).  \nWe appreciate your concise summary of our core contributions. We will incorporate the above content in the next version to further highlight our key innovations.  \n\n**W2-1:** We sincerely appreciate your insightful comment. Subgraph-level tasks are indeed pivotal for assessing the generalization capability of node-level graph condensation. This constitutes a novel task and a previously unexplored scenario in GC field. Nevertheless, owing to time constraints, it is non-trivial to modify each GC method to adapt to this new task.   \nMoreover, the intrinsic discrepancy between node-level and graph-level graph classification makes directly applying PreGC to graph-level tasks not easily achievable. Whereas node-level GC condensed a single large graph into a small graph, graph-level GC aggregates multiple graphs into a compact set while preserving the size of individual graphs. The former primarily aims to capture inter-sample relationships, whereas the latter emphasizes intra-sample associations. Therefore, graph-level GC bears a closer resemblance to typical dataset condensation.   \nIn future work, we will endeavor to transplant this study into the subgraph/graph-level tasks to investigate the generalizability of our proposed method.\n\n**W2-2:** Thank you for your insightful comments and careful evaluation of our work. We appreciate the opportunity to clarify the reason for following the pipeline of pretrained-finetuned. As noted in [1], we think that an ideal pre-trained model should satisfy two fundamental criteria. (**C1**) achieving competitive performance across diverse downstream tasks without fine-tuning, and (**C2**) demonstrating further performance improvement after task-specific fine-tuning. These criteria are equally applicable to graph condensation. First, without any fine-tuning, graphs condensed by PreGC already achieve comparable or sota performance across various tasks (as shown in Tables 1, 2, and 3), confirming that PreGC satisfies (**C1**). Second, we can observe from Table 2 that after fine-tuning (PreGC_ft), the condensed graphs exhibit additional performance gains on specific tasks. For example, rising from 60.55 to 60.81 on “T→T” and from 52.34 to 52.47 on “Y→Y” in Table 2, respectively. This demonstrates that PreGC_ft fully aligns with (**C2**).  \nFurthermore, regarding the concern about the fine-tuning of PreGC (Q2), this does not indicate instability in PreGC's fine-tuning. On the contrary, the condensed graphs achieve comparable performance after fine-tuning on the same task. For instance, for the OGB-Arxiv dataset in Table 2, the initial performance of the condensed graph directly migrated from the year task to the topic task (“Y→T”) is only 59.62 (PreGC). After fine-tuning, the performance improved to 60.86 (PreGC_{ft}), reaching similar performance (60.81) as fine-tuning directly on the topic task (“T→T”). Similarly, under all cross-task settings (“T→Y”, “P→C”, and“C→P”), PreGC_{ft} ultimately attains performance nearly identical to that achieved by fine-tuning directly on the source task. This demonstrates the effectiveness of the fine-tuning strategy proposed in this work.  \nRegarding (Q1), we indeed observe that, in certain cases, GCDM achieves better cross-task performance than its performance on the source task. This phenomenon that contradicts most GC baselines and appears counterintuitive. However, as mentioned in Section 5.2, the lack of explicit correlations leads to insufficient interpretability in existing GC methods. Moreover, since GCDM performs worst in most scenarios, it fails to satisfy (**C1**). Therefore, our focus remains on emphasizing the aforementioned viewpoint, i.e., PreGC is sufficient to achieve optimal performance in most tasks even without fine-tuning. And fine-tuned PreGC_ft can further enhance the quality of the condensed graph, particularly in cross-task settings.  \n\n**W2-3:** Incorporating your suggestions and **Reviewer 3v1U**’s remarks (**W2**), we supplement the performance evaluation of various GC methods under different training ratios. The following table demonstrates the performance of condensed graphs on the OGB-Arxiv dataset at the 1.25% condensation ratio. It is noteworthy that, despite the absence of any supervisory signals during the condensation phase, PreGC consistently achieves state-of-the-art performance across varying training ratios. This demonstrates the effectiveness of using semantic alignment as a substitute for supervisory signals. More importantly, existing GC methods require recondensing the graph whenever the training ratio is changed. In contrast, PreGC only requires condensing the original graph once. When the label distribution changes, PreGC can effortlessly transfer the label signals from the original graph to the condensed graph via Eq. (14). This clever strategy demonstrates PreGC's superior flexibility and reusability.  \nThank you again for your constructive feedback. We will incorporate these experimental results into the next version of the manuscript, which will further highlight the unique advantages of PreGC.\n\n\n| Training Ratio | 0.15       | 0.30       | 0.45       | 0.60       | 0.75       | Condensation times |\n| -------------- | ---------- | ---------- | ---------- | ---------- | ---------- | ------------------ |\n| Whole          | 66.71±0.43 | 69.53±0.11 | 70.44±0.51 | 71.69±0.20 | 71.72±0.57 | $~~~-$             |\n| GCDM           | 35.98±1.31 | 36.32±1.08 | 36.09±0.89 | 36.94±0.60 | 38.35±0.67 | $~~\\times$ 5       |\n| GCond          | 56.17±0.30 | 56.95±0.31 | 56.83±0.49 | 57.47±0.36 | 58.00±0.34 | $~~\\times$ 5       |\n| SFGC           | 59.09±0.42 | 59.85±0.38 | 60.04±0.53 | 60.46±0.51 | 60.59±0.48 | $~~\\times$ 5       |\n| SGDD           | 58.01±0.77 | 58.79±0.40 | 59.03±0.44 | 59.54±0.53 | 60.24±0.46 | $~~\\times$ 5       |\n| GDEM           | 52.23±0.32 | 54.20±0.57 | 54.81±0.44 | 54.98±0.48 | 55.41±0.31 | $~~\\times$ 5       |\n| CGC            | 58.39±0.33 | 59.04±0.49 | 58.92±0.72 | 59.53±0.56 | 60.05±0.29 | $~~\\times$ 5       |\n| **PreGC**      | 60.53±0.32 | 61.96±0.56 | 61.72±0.63 | 63.37±0.49 | 63.58±0.82 | $~~\\times$ **1**   |\n\n**W3:** Thanks for your feedback. The noted issues will be corrected in the next version. We believe that it will further enhance the readability of this manuscript.\n\n[1] Wang, Zehong, et al. Graph Foundation Models: A Comprehensive Survey. ArXiv:2505.15116, 2025."}, {"review_id": "fzrSgmy7Yw", "reviewer": "Reviewer_BLZo", "summary": "This paper develops a pre-training graph condensation method to achieve both architecture and task agnostic condensation, and fine-tuning for test-time adaptation, demonstrating the effectiveness both theoretically and experimentally.", "questions": "- Typo: line 216 explicittly->explicitly.\n- The scalability of the proposed method should be verified on large-scale graphs, such as Reddit, Yelp, and Ogbn-papers100M.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- Theoretical insight. The task decomposition of SGC convolution obtains the reconstruction and supervised signal terms, sufficiently illustrating the motivation of the proposed method.\n- Comprehensive design. The design of the proposed PreGC involves the parameter-free node representations based on graph diffusion and the semantic alignment to replace the task bias of supervised signals.\n- Sufficient experiments. The experimental tasks contain both node and edge tasks, and the compared baselines involve recent state-of-the-art methods.\n\nWeaknesses:\n- Validity of assumption. The first proof assumes the existence of an analytic filter $g(\\cdot)$, however, it necessitates a discussion on the validity of the assumption.\n- Validity of semantic alignment. This paper proposes the semantic alignment to replace the biased supervised signals. Therefore, the gap between semantic alignment and supervised signals requires an in-depth analysis.\n- Validity of fine-tuning. The proposed fine-tuning stage at test-time may challenge the fairness of the claimed pre-training technique, compared to other methods.", "was_revised": true, "final_justification": "I am satisfied with the thoughtful response of the authors. I decide to keep my positive score.", "rebuttal": "Thank you for your positive comments and recognition of our work, particularly for your high appraisal of the design motivation, methodology, and experimental results of PreGC. In what follows, we will address each of your concerns point by point.  \n\n**W1:** We thank the reviewer for pointing out this question. We think that the assumption is both theoretically mild and practically common.  \nFirstly, any continuous spectral response can be approximated arbitrarily well by an analytic function on the compact set $\\sigma(\\mathbf{L})\\subset [0,2]$. Therefore, it is reasonable to assume that $g(\\cdot)$ is analytic, as it merely rules out pathological, non-smooth filters that are rarely considered in graph learning. Considering a GNN with an analytical filter $g(\\cdot)$ is a more common setting [14] [30].  \nIn addition,  widely used GNN filters satisfy this assumption, such as graph diffusion ($g(\\mathbf{L})=e^{-T\\mathbf{L}}$) [28], SGC ($g(\\mathbf{L})=\\mathbf{I}-\\mathbf{L}$) [39], and PPNP ($g(\\mathbf{L})=\\alpha (\\mathbf{I}-(1-\\alpha )\\mathbf{A})^{-1} $) [24], among others. It can more easily generalize the problem to more general scenarios (i.e., Eq. (4)).\n\n**W2:**  Thanks for your profound insights. In fact, we argue that there still exists a discrepancy between semantics and supervisory signals. This is because downstream tasks are manually designed, whereas node semantics are inherent properties of the data itself. To mitigate this discrepancy, we further introduce a fine-tuning strategy, where PreGC_{ft} can be regarded as fitting from semantic alignment to supervisory signal alignment. However, a more intriguing observation is that even without fine-tuning, the graphs condensed by PreGC already achieve sota performance (as shown in Table 2, PreGC). This phenomenon strongly suggests that alignment with the supervisory signal can be effectively approximated by semantic alignment alone. Additionally, in the experiment “Performance Gap between Condensed Graph and Original Graph”, we further validate the differences between various condensed graphs and the original graph via the labeled reconstruction error (LRE). The extremely low LRE of PreGC in Figure 3 indicates that semantic alignment is not only remarkably effective but also superior to existing GC methods that rely on pre-defined labels.  \n\n**W3:**  Thank you for your feedback. We ensure that the entire process of PreGC is completely fair. Firstly, PreGC operates without any label guidance during the condensation phase, which guarantees that the condensed graph is aligned with the original graph solely in terms of representation distribution and semantics. However, existing GC methods require assigning labels to each condensed node based on the training set of the original graph before condensation, and explicitly aligning the condensed graph based on its labels. Clearly, this naive approach limits the reusability of the condensed graph in other supervised tasks (especially regression tasks, as shown in Table 2). In contrast, when a graph condensed by PreGC needs to be evaluated on a specific downstream task, it only requires mapping the original training set to the condensed graph by Eq. (14). Even without further fine-tuning, graphs condensed by PreGC still achieve significant performance on most tasks, as demonstrated in Tables 1, 2, and 3 (note that in Tables 1 and 3, we did not employ the fine-tuning at all).  \nAs you mentioned in the previous question, even for the same graph data, the supervision signals for different tasks may deviate from the semantics. To address this issue, we introduced a fine-tuning strategy (PreGC_{ft}), which is similar to test-time adaptation [1]. Specifically, **(1)** during fine-tuning, PreGC_{ft} adjusts based solely on predicted logits, avoiding any label leakage issues. **(2)** In addition, PreGC_{ft} keeps the condensed graph fixed (i.e., maintaining its structure $\\tilde{\\mathbf{A}}$ and features $\\tilde{\\mathbf{X}}$ unchanged) during fine-tuning and only updates the semantic assignment matrix $\\mathbf{M}$, making the process highly efficient. **(3)** More importantly, it only requires fine-tuning once for per task.  \nAnother intriguing finding is that PreGC demonstrates remarkably strong performance even without fine-tuning, as shown in Tables 1 and 3. The improvement of PreGC_ft becomes more pronounced in cross-task transfer settings (such as “Y→T”, “T→Y”, “P→C”, and “C→P” in Table 2). Therefore, we think that in scenarios with a single supervised task requirement, the vanilla PreGC is sufficient, as the graph condensed by PreGC has already effectively captured the essential properties of the original graph with high fidelity.  \n\n**Q1:** Thanks for your feedback. The noted issues will be corrected in the next version.  \n\n**Q2:** We appreciate your valuable suggestions. Guided by the comments from both you and the other reviewers (**JS8B** and **3v1U**), we have newly incorporated experiments on the large-scale graph Reddit and the heterophilic graph Flickr [1] to systematically assess the generalizability of PreGC. The results are reported in the table below.  It can be clearly observed that on the Reddit dataset, the graph condensed by existing GC methods does not consistently demonstrate ideal performance across each GNN. This is particularly evident in the cases of k-GNN, GAT, and GraphSAGE. In contrast, our proposed method achieves competitive results across all nine representative GNNs and demonstrates compatibility with any GNN architecture, as further evidenced by the average performance metrics.  \nNotably, we observe that Yelp [2] is a multi-label graph dataset, which deviates from the typical GC setting (existing baselines must assign a single, unique label to each condensed node before condensation). Additionally, the scale of OGBn-Papers100M is so immense that no baseline has yet reported experiments on it. Owing to these two factors, we are unable to evaluate the performance of existing baselines on these datasets in a short period of time.  \nHowever, it cannot be denied that you have proposed a very innovative setting. Validating the performance of condensed graphs in the multi-label setting would further underscore the value of graph condensation, and we leave this exploration to future work.\n\n\n| Reddit (r=2.5%) | SGC  | GCN | APPNP | k-GNN| GAT  | SAGE    | SSGC  | Bern.   | GPR.   | Avg.  |\n| --------------- | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ |\n| Whole           | 86.1±0.1     | 86.5±0.2     | 85.9±0.2     | 83.7±0.1     | 83.1±0.5     | 83.0±0.2     | 86.8±0.1     | 85.7±0.2     | 84.5±0.3     | 85.0±1.4     |\n| GCDM            | 21.9±1.5     | 24.0±0.6     | 56.3±0.8     | 17.4±3.6     | 15.1±1.4     | 30.6±0.4     | 57.5±0.7     | 53.0±1.4     | 40.9±0.9     | 35.2±16.1    |\n| GCond           | 70.4±1.0     | 69.3±0.5     | 61.8±1.2     | 31.2±0.7     | 23.3±2.6     | 40.8±1.0     | 63.8±0.6     | 64.5±0.5     | 47.2±1.3     | 52.5±16.4    |\n| SFGC            | 65.2±1.0     | 67.7±0.7     | 68.9±3.0     | 33.9±1.3     | 19.6±2.9     | 36.7±2.7     | 59.9±0.3     | 68.9±2.2     | 51.6±2.1     | 52.5±17.2    |\n| SGDD            | 75.5±0.9     | 77.0±0.6     | 68.6±2.2     | 24.6±2.0     | 38.9±2.7     | 46.7±2.8     | 69.6±0.2     | 68.9±1.9     | 51.2±2.2     | 57.9±17.3    |\n| GDEM            | 78.3±0.8     | **79.2±1.0** | 79.1±1.0     | 19.2±2.7     | **77.4±2.4** | 42.4±0.7     | **82.2±0.4** | 69.6±2.2     | 57.2±1.1     | 65.0±20.3    |\n| CGC             | 76.9±0.4     | 78.4±0.3     | 72.5±0.9     | 18.5±6.0     | 50.3±4.3     | 41.3±0.3     | 75.6±0.4     | 75.0±1.1     | 58.8±1.2     | 60.8±19.5    |\n| PGC             | **79.2±0.2** | 74.1±0.4     | **79.9±0.3** | **73.7±0.3** | 72.9±4.1     | **70.3±0.4** | 78.5±0.3     | **75.9±1.3** | **70.3±1.2** | **75.0±3.4** |\n\n\n| Flickr (r=2.5%) | SGC          | GCN          | APPNP        | k-GNN        | GAT          | SAGE         | SSGC         | Bern.        | GPR.         | Avg.         |\n| --------------- | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ |\n| Whole           | 48.3±0.5     | 47.9±0.1     | 44.4±0.6     | 41.8±2.2     | 43.5±0.9     | 46.9±0.2     | 48.0±0.3     | 46.5±0.3     | 46.2±0.3     | 45.9±2.1     |\n| GCDM            | 46.4±0.8     | 45.5±0.5     | 46.5±0.1     | 42.1±0.1     | 44.9±1.1     | 44.4±0.2     | 45.9±0.3     | 45.4±0.5     | 44.8±0.2     | 45.1±1.3     |\n| GCond           | 47.6±0.2     | 46.5±0.1     | 47.3±0.3     | 43.0±1.7     | 41.9±0.7     | 44.9±1.0     | **47.6±0.1** | **46.3±0.7** | 46.4±0.2     | 45.7±1.9     |\n| SFGC            | 46.0±0.9     | 44.1±0.6     | 45.3±1.6     | 46.8±0.9     | 44.6±1.2     | 44.4±0.2     | 45.8±0.2     | 36.1±6.6     | 45.5±0.2     | 44.3±3.0     |\n| SGDD            | 48.0±0.2     | 43.7±0.9     | 44.7±1.1     | 42.1±0.0     | 30.6±5.9     | 42.1±0.0     | 45.6±0.8     | 42.5±0.7     | 43.6±0.5     | 42.6±4.6     |\n| GDEM            | 42.2±0.0     | 42.1±0.1     | 42.3±0.1     | 42.0±0.1     | 42.4±0.4     | 42.6±0.4     | 42.3±0.2     | 41.5±0.6     | 42.3±0.1     | 42.2±0.3     |\n| CGC             | 47.7±0.5     | **46.9±0.4** | 47.0±0.4     | 47.0±0.4     | 46.8±0.5     | 45.8±0.2     | 46.8±0.3     | 46.0±0.7     | 45.9±0.4     | 46.7±0.6     |\n| PGC             | **48.0±0.4** | 46.5±0.1     | **47.7±0.2** | **47.7±0.7** | **47.4±0.3** | **45.9±0.2** | 47.1±0.2     | 46.0±1.0     | **46.8±0.4** | **47.0±0.7** |\n\n[1] Boudiaf, Malik, et al. Parameter-free online test-time adaptation. CVPR, 8344-8353, 2022.  \n[2] Zeng, Hanqing, et al. GraphSAINT: Graph Sampling Based Inductive Learning Method. ICLR, 2020."}, {"review_id": "aAtkIUuFm8", "reviewer": "Reviewer_8BSw", "summary": "This paper introduces an innovative pre-training framework PreGC for graph condensation, which overcomes the limitations of existing approaches that are tied to specific tasks and architectures. Utilizing optimal transport, PreGC develops a generalized condensation objective that synchronizes both structural and semantic information between the original graphs and their condensed counterparts. Key contributions encompass a hybrid-interval graph diffusion augmentation technique to enhance architectural generalization, as well as a transport plan matching mechanism to guarantee task-independent semantic consistency. Through comprehensive experiments spanning diverse datasets, tasks, and GNNs, the effectiveness, generalizability, and reusability of PreGC are verified.", "questions": "See weaknesses.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**  \n\n- The paper re-examines the goal of graph condensation and formulates a generalized optimization framework that integrates existing graph condensation methods under a unified paradigm.  \n\n- PreGC is intentionally engineered to be decoupled from both downstream tasks and GNN architectures, rendering it highly reusable and scalable across a wide range of scenarios.  \n\n- By incorporating optimal transport, the framework enables semantically meaningful alignment between original and condensed graphs, which has the potential to boost interpretability and traceability.  \n\n- PreGC demonstrates consistent superiority over state-of-the-art approaches across various datasets, tasks, and GNN backbones.  \n\n\n**Weaknesses**  \n\n- While the paper highlights enhanced performance, it lacks a detailed breakdown of key practical metrics such as training time, memory consumption, and inference latency.  \n\n- Despite including ablation study results, these analyses are limited to a single dataset (H&M), which restricts the generalizability of the findings.  \n\n- The paper compares PreGC with several recent GC methods but largely overlooks discussions of relevant work in neighboring fields—for instance, data condensation techniques tailored to graph-level tasks [1].\n\n[1] Efficient Graph Continual Learning via Lightweight Graph Neural Tangent Kernels-based Dataset Distillation", "was_revised": true, "final_justification": "Thank you, my concerns have been addressed.", "rebuttal": "Thank you for your recognition of this work. Below are our point-by-point responses to your concerns. We believe that the additional experiments and survey will further consolidate and refine our study.  \n\n**W1:**  We sincerely appreciate your valuable feedback. In response to your suggestions and those of **Reviewer 3v1U**, we conduct additional experiments to compare the computational efficiency of PreGC with existing baselines on the Cora dataset. It covers peak memory usage and runtime across the pre-processing, condensation, and fine-tuning stages (CGC is excluded as it is a training-free method). The results are shown in the table below (* denotes the result of a single execution. Note that in practical implementations, multiple preprocessing often be necessary depending on specific settings.).   \nIt can be observed that PreGC achieves significantly faster condensation time compared to most GC methods. Even when fine-tuning is required, it only increases a few additional time costs. Although GDEM spends the least time during the condensation process, it requires extra preprocessing time for eigenvalue decomposition. Overall, both PreGC and GDEM exhibit substantially lower training times than other GC methods. The underlying reason is that existing GC methods employ a nested bi-level optimization: an outer loop updates the GNN parameters, while an inner loop optimizes the condensed graph. In contrast, PreGC leverages graph diffusion, a parameter-free message passing mechanism, eliminating the time-consuming outer loop. This not only reduces condensation time but also prevents the condensed graph from overfitting to architecture-specific parameters.    \nFurthermore, as elaborated in our responses to reviewer **NPxj**’s C2-3 and reviewer **3v1U**’s W2, when downstream tasks or label distributions change, existing baselines must re-condense the graph to capture this updated knowledge. In contrast, PreGC is task- and label-agnostic, thus requiring only a single condensation and can be reused multiple times.   \nA dedicated section analyzing condensation efficiency will be incorporated in the subsequent version to offer a more thorough demonstration of our research.\n\n| r=1.3% | Memory (MB) | Pre-processing (S) | Condensation (S) | Fine-tuning (S) | Total (S)     |\n| ------ | ----------- | ------------------ | ---------------- | --------------- | ------------- |\n| GCDM   | 1,276       | -                  | 1542.38          | -               | 1542.38       |\n| GCond  | 1,280       | -                  | 2,304.41         | -               | 2,304.41      |\n| SFGC   | 2,514       | 14.01*             | 1,923.64         | -               | 1,937.65      |\n| SGDD   | 1,644       | -                  | 2,364.18         | -               | 2,364.18      |\n| GDEM   | 2,658       | 4.56*              | 7.48             | -               | 12.04         |\n| PreGC  | 1,714       | -                  | 22.36            | 0.81 (optional) | 22.36 / 23.17 |\n\n**W2:** Thanks for your suggestion. We add ablation experiments of PreGC on the OGB-Arxiv dataset, with the results presented in the table below. Consistent with the observations on the H&M dataset, both key modules play a significant role in graph condensation. Notably, the removal of graph diffusion augmentation leads to increased performance fluctuations, which demonstrates that graph diffusion augmentation can enhance the generalization capability of the condensed graph. We will incorporate these results into the manuscript to ensure the generalizability of the findings.\n\n| OGB-Arxiv           | r=1.25%    | r=1.25%    |      | r=2.5%     | r=2.5%     |\n| ------------------- | ---------- | ---------- | ---- | ---------- | ---------- |\n|                     | Topic      | Year       |      | Topic      | Year       |\n| PreGC               | 60.55±0.43 | 52.34±0.30 |      | 62.66±0.47 | 52.96±0.35 |\n| PreGC$_{w/o~Aug}$   | 59.21±0.75 | 51.10±0.64 |      | 59.93±0.68 | 51.06±0.79 |\n| PreGC$_{w/o~Lcost}$ | 58.84±0.51 | 49.42±0.58 |      | 59.99±0.42 | 50.55±0.49 |  \n\n**W3:**  We sincerely appreciate your insightful comments. We have noticed that the work [1] you pointed out focuses on the method of graph condensation in specific application scenarios. It innovatively integrated the lightweight graph neural tangent kernel condensation paradigm into graph continual learning, and markedly reduced computational overhead. Motivated by your suggestion, we have systematically surveyed the graph-level GC [1] [2] [3] [4] [5] and the extended applications of GC to additional research directions [1] [6] [10], such as graph continual learning [1] [6] [7] and federated graph learning [8] [9]. **We will devote a dedicated subsection in related work to these developments in the next revision**. We believe that such cross-scenario investigations will enrich the theoretical and methodological landscape of graph condensation and foster sustained advancement in this field.\n\n[1] Qiu, Rihong, et al. Efficient Graph Continual Learning via Lightweight Graph Neural Tangent Kernels-based Dataset Distillation. ICML, 2025.  \n[2] Wang, Yuxiang, et al. Self-supervised learning for graph dataset condensation. KDD, 3289-3298, 2024.  \n[3] Gupta, Mridul, et al. Mirage: Model-agnostic graph distillation for graph classification. ICLR, 2024.  \n[4] Xu, Zhe, et al. Kernel ridge regression-based graph dataset distillation. KDD, 2850-2861, 2023.  \n[5] Jin, Wei, et al. Condensing graphs via one-step gradient matching. KDD, 720-730, 2022.  \n[6] Liu, Yilun, Ruihong Qiu, and Zi Huang. Cat: Balanced continual graph learning with graph condensation. ICDM, 1157-1162, 2023.  \n[7] Liu, Yilun, et al. Puma: Efficient continual graph learning with graph condensation. TKDE, 2024.  \n[8] Yan, Bo, et al. Federated graph condensation with information bottleneck principles. AAAI, 39-12, 2025.  \n[9] Zhang, Hao, et al. Rethinking Federated Graph Learning: A Data Condensation Perspective. ArXiv:2505.02573, 2025.  \n[10] Chen, Dong, et al. Dynamic Graph Condensation. ArXiv:2506.13099, 2025."}, {"review_id": "HyxDo1wdTg", "reviewer": "Reviewer_JS8B", "summary": "The authors aim to expand the application of graph condensation algorithms to multi-task and multi-architecture scenarios. They first provided a theoretical summary of existing graph condensation methods, demonstrating their shortcomings and proposing a new theoretical framework. Based on the proposed framework, the authors proposed PreGC. They introduced a random diffusion based augmentation to enhance message-passing diversity, and a graph optimal transport based proxy task, which achieved unsupervised pre-training of graph condensation models for adaptability to any model architecture and any downstream task.", "questions": "Apart from the weaknesses mentioned above, the proposed method appears to work only on homophily graphs. Does it also work for heterophily graphs?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths.\n\n- The authors' theoretical analysis and contributions are concise and excellent. Proposition 3.1 clearly illustrates the limitations of existing work concerning GNN architectures and downstream tasks. Definition 3.1 circumvents these limitations by imposing constraints on representations and graph semantics.\n- The idea of introducing random diffusion based augmentation is innovative.\n- The experimental setup is interesting, particularly the multi-task configuration for node-level datasets.\n- The experimental results on Arxiv and H&M are strong.\n\nWeaknesses.\n\n- The results for cross-task and cross-GNN scenarios on the Cora, CiteSeer, and PubMed datasets are not satisfactory. Other baselines do not consider these scenarios, while PreGC is designed specifically for them. Therefore, I believe the performance improvement of PreGC on these datasets is marginal. The effectiveness of the proposed method may require further clarification.\n- The introduction of the methodology is somewhat unclear. Although Appendix B provides some explanation, I feel it is insufficient. The authors should provide more detailed information on how Equations 8 and 9 are solved.\n- I suggest the authors provide statistical information table about the datasets.", "was_revised": true, "final_justification": "The theoretical analysis is clear and interesting, and some components of the method are novel. Although the performance in Table 1 is somewhat limited, the proposed method performs well in other settings. Therefore, I maintain my rating of 4.", "rebuttal": "We sincerely appreciate the reviewers' positive evaluation of our work, particularly their recognition of the concise and excellent theoretical analysis and contributions, as well as the interesting and novel experimental design. In what follows, we provide detailed responses to each of the questions raised.  \n\n**W1:** Notably, PreGC is not tailored to a particular application scenario. Its primary objective is to distill a task-agnostic and architecture-flexible condensed graph.  \nLet us revisit the original motivation behind graph condensation. The original intention of GC lies in accelerating GNN training, which facilitates researchers in developing novel GNN architectures and rapidly testing their performance on certain tasks. However, existing baselines only consider NC task and are limited to specific GNNs (e.g., SGC and GCN). These methods are scenario-constrained and deviate from the original intention of GC.   \nIn contrast, our method does not focus on specific scenarios but instead follows the general GC paradigm (i.e., Definition 3.1) derived from the perspective of GNN optimization consistency. **(1)** First and foremost, PreGC only requires condensation once on a given dataset and does not need to be re-condensed when tasks or label ratios change. This flexible reusability significantly enhances the practical value of condensed graphs. **(2)** Despite the absence of label signals during condensation, PreGC still outperforms other baselines in most cases on NC task, as shown in Table 1. **(3)** More importantly, the graphs condensed by PreGC consistently outperform existing methods on LP task, further demonstrating that the general GC paradigm enables the condensed graphs to capture and inherit the properties of original graphs. **(4)** Additionally, experiments across different GNNs validate the generalization capability of the condensed graphs, with PreGC achieving the best average performance across nine GNNs.  \nOverall, in scenarios specifically tailored to existing GC methods, PreGC still achieves comparable or even sota performance without relying on supervised signals to guide condensation. Moreover, in more generalized scenarios (such as unsupervised tasks, node regression, and cross-task settings), PreGC demonstrates significant superiority over existing baselines. Therefore, the cross-task and cross-architecture evaluations demonstrate the generalizability and reusability of graphs condensed by PreGC, fully aligning with the original intention of graph condensation.\n\n**W2:** We thank the reviewer for this comment. Since Eq. (9) can be regarded as a special form of Eq. (8) (i.e., $\\gamma=1$), our discussion primarily focuses on solving for $\\pi_\\mathcal{D}$ in Eq. (8).  (**PS: Due to the fact that certain mathematical symbols cannot be properly compiled, we have appropriately employed line breaks for clearer representation.**)  \n**(1)** Eq. (8) decomposes into two terms: the former is a Wasserstein distance that operates solely on node features. This term can be efficiently approximated using the Sinkhorn-Knopp algorithm [1], which iteratively converges to the optimal transport plan $\\pi^* $. Specifically, the Sinkhorn-Knopp algorithm introduces an entropy regularizer and alternates between Sinkhorn projections, $\\pi^{(0)}=\\exp(-\\boldsymbol{\\mathcal{K}} (\\mathbf{X},\\tilde{\\mathbf{X}}) / \\lambda)$ and $\\pi^{(l+1)}=\\mathcal{F}_1(\\mathcal{F}_2(\\pi^{(l)}))$, where $l$ represents the number of iterations and $\\lambda$ denotes the weigth of regularization.   \nThe projections $\\mathcal{F}_1(\\pi)=\\pi\\oslash (\\mathbf{1}\\mathbf{1}^\\top \\pi)$ and $\\mathcal{F}_2(\\pi)=\\pi \\oslash(\\pi\\mathbf{1}\\mathbf{1}^\\top)$    \n\ndenote the column normalization and row normalization, respectively, where $\\oslash$ is element-wise division. They ensure that the rows and columns satisfy the given marginal distribution constraints. In the limit, this alternating projection will converge to a minimizer $\\lim_{t\\to\\infty}{\\pi^{(t)}}=\\pi^* $.  \n**(2)** The latter term corresponds to a Gromov-Wasserstein (GW) distance, which can be formulated as a nonconvex quadratic programming problem. We use semi-relaxed Gromov-Wasserstein divergence [2] and optimize $\\pi$ with the conditional gradient solver. The gradient of $\\pi$ with respect to the GW term is $\\nabla_{\\pi}=2\\boldsymbol{\\mathcal{J}} (\\mathbf{A},\\tilde{\\mathbf{A}}) \\otimes \\pi $. The conditional gradient algorithm consists in solving a linearization $\\langle \\mathbf{P}, \\nabla_{\\pi}\\rangle $ at each iteration, where $\\mathbf{P} \\in \\mathbb{R}^{N\\times M}$ the extreme-point solution that arises from the linearized sub-problem solved. It can be solved by gradient descent with a direction $\\mathbf{P}^{(l)} − \\pi^{(l)}$, followed by a line search for the optimal step.  \nIn the optimization process, we utilize the third-party library functions GeomLoss and Python Optimal Transport (POT) to solve Eq. (8) and Eq. (9), respectively. For further details, please refer to [1], [2], and [36]. We will add the above optimization specifics in the next version.\n\n**W3:** We will add statistical tables about the dataset in the next version, including the number of nodes, edges, feature dimensions, edge homophily, and so on.\n\n| Dataset   | Nodes   | Edges      | Features | Classes    | Edge Hom.       |\n| --------- | ------- | ---------- | -------- | ---------- | --------------- |\n| Cora      | 2,708   | 5,429      | 1,433    | 7          | 0.81            |\n| CiteSeer  | 3,327   | 4,732      | 3,703    | 6          | 0.74            |\n| PubMed    | 19,717  | 44,338     | 500      | 3          | 0.80            |\n| OGB-Arxiv | 14,167  | 33,520     | 128      | 40(T) / 5(Y) | 0.70(T) / 0.37(Y) |\n| H&M       | 15,508  | 920,284    | 191      | 21(C)      | 0.34(C)         |\n| Reddit    | 232,965 | 23,213,838 | 602      | 41         | 0.78            |\n| Flickr    | 89,250  | 899,756    | 500      | 7          | 0.32            |\n\n**Q1:**  As a GC method following the generalized graph condensation paradigm, PreGC remains applicable to heterophily graphs. In fact, for the year prediction task on the OGB-Arxiv dataset and the category prediction task on the H&M dataset, they exhibit strong heterophily (edge homophily is 0.37 and 0.34, respectively), and the condensed graph by PreGC still achieves state-of-the-art performance. This is because PreGC learns the spectral properties of the original graph through graph diffusion augmentation, rather than the task-relevant homophily or heterophily characteristics.\n\nAdditionally, inspired by feedback from other reviewers (Reviewers **BLZo** and **3v1U**), we have included experiments on the heterophily graph Flickr (edge homophily is 0.32), with the corresponding results available in the following.\n\n| Flickr (r=2.5%) | SGC          | GCN          | APPNP        | k-GNN        | GAT          | SAGE         | SSGC         | Bern.        | GPR.         | Avg.         |\n| --------------- | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ |\n| Whole           | 48.3±0.5     | 47.9±0.1     | 44.4±0.6     | 41.8±2.2     | 43.5±0.9     | 46.9±0.2     | 48.0±0.3     | 46.5±0.3     | 46.2±0.3     | 45.9±2.1     |\n| GCDM            | 46.4±0.8     | 45.5±0.5     | 46.5±0.1     | 42.1±0.1     | 44.9±1.1     | 44.4±0.2     | 45.9±0.3     | 45.4±0.5     | 44.8±0.2     | 45.1±1.3     |\n| GCond           | 47.6±0.2     | 46.5±0.1     | 47.3±0.3     | 43.0±1.7     | 41.9±0.7     | 44.9±1.0     | **47.6±0.1** | **46.3±0.7** | 46.4±0.2     | 45.7±1.9     |\n| SFGC            | 46.0±0.9     | 44.1±0.6     | 45.3±1.6     | 46.8±0.9     | 44.6±1.2     | 44.4±0.2     | 45.8±0.2     | 36.1±6.6     | 45.5±0.2     | 44.3±3.0     |\n| SGDD            | 48.0±0.2     | 43.7±0.9     | 44.7±1.1     | 42.1±0.0     | 30.6±5.9     | 42.1±0.0     | 45.6±0.8     | 42.5±0.7     | 43.6±0.5     | 42.6±4.6     |\n| GDEM            | 42.2±0.0     | 42.1±0.1     | 42.3±0.1     | 42.0±0.1     | 42.4±0.4     | 42.6±0.4     | 42.3±0.2     | 41.5±0.6     | 42.3±0.1     | 42.2±0.3     |\n| CGC             | 47.7±0.5     | **46.9±0.4** | 47.0±0.4     | 47.0±0.4     | 46.8±0.5     | 45.8±0.2     | 46.8±0.3     | 46.0±0.7     | 45.9±0.4     | 46.7±0.6     |\n| PGC             | **48.0±0.4** | 46.5±0.1     | **47.7±0.2** | **47.7±0.7** | **47.4±0.3** | **45.9±0.2** | 47.1±0.2     | 46.0±1.0     | **46.8±0.4** | **47.0±0.7** |\n\n[1] Distances, Cuturi M. Sinkhorn. Lightspeed computation of optimal transport. NeurIPS, 26. 2292-2300, 2013.  \n[2] Vincent-Cuaz, Cédric, et al. Semi-relaxed Gromov Wasserstein divergence with applications on graphs. ICLR, 1-28, 2022."}, {"review_id": "18yN47fFz3", "reviewer": "Reviewer_3v1U", "summary": "This paper proposes a new framework, PreGC, for pre-trained graph condensation that enables task-agnostic and architecture-agnostic usage. While prior methods rely heavily on fixed GNN architectures and supervised labels, PreGC addresses these limitations by leveraging optimal transport theory, graph diffusion augmentation, and transport plan matching. Extensive experiments across tasks, datasets, and GNN architectures demonstrate the effectiveness and flexibility of the proposed approach.", "questions": "I like the idea of using graph condensation for graph data valuation. It would be valuable to explore whether other graph condensation methods can also be applied in this context, and how the proposed approach compares to existing graph data valuation techniques.", "limitations": "N/A", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n1. The paper identifies and addresses critical limitations in existing graph condensation methods, particularly architecture dependency and task supervision.\n2. The use of optimal transport and graph diffusion augmentation in the context of graph condensation is novel and theoretically grounded.\n3. The experiments involving cross-task generalization are new to this domain and are likely to inspire further research.\n4. I also appreciate the exploration of the application of graph condensation to data valuation via node significance, which is an interesting direction that enhances interpretability and practical utility. \n\n\nWeakness:\n1. The paper lacks details on the computational cost of both the pretraining and fine-tuning phases. It is unclear how expensive it is in terms of runtime or resources compared to standard training or existing GC methods. It would be better to report training time, GPU usage, memory footprint, or number of epochs.\n2. The description of PreGC’s flexibility in Section 5.2 is not very clear. Specifically, the claim that existing methods are \"limited to certain proportions of labels\" needs more explanation or supporting examples.\n3. The datasets used are relatively small. It would strengthen the work to evaluate on larger or more diverse benchmark datasets, such as those in [1,2].\n\n[1] GC4NC: A Benchmark Framework for Graph Condensation on Node Classification with New Insights. 2024\n[2] GC-Bench: An Open and Unified Benchmark for Graph Condensation. 2024", "was_revised": true, "final_justification": "The authors have addressed my concerns and I am happy to maintain my scores.", "rebuttal": "We sincerely appreciate your high recognition of our work, particularly regarding the limitations of existing methods and the theoretical foundation of our approach. Furthermore, we are deeply grateful for your endorsement of our proposed node importance-based data evaluation strategy. We will address all your concerns point by point.  \n\n**W1:** In response to your recommendations and those of Reviewer **8BSw**, we conduct supplementary experiments to systematically evaluate the computational efficiency of PreGC with existing baselines on Cora dataset. The results are shown in the table below (* denotes the result of a single execution).   \nA dedicated section analyzing condensation efficiency will be incorporated in the subsequent version to offer a more thorough demonstration of our research. \n\n\n| r=1.3% | Memory (MB) | Pre-processing (S) | Condensation (S) | Fine-tuning (S) | Total (S)     |\n| ------ | ----------- | ------------------ | ---------------- | --------------- | ------------- |\n| GCDM   | 1,276   | -    | 1542.38    | -    | 1542.38    |\n| GCond  | 1,280    | -    | 2,304.41   | -     | 2,304.41   |\n| SFGC   | 2,514    | 14.01*    |1,923.64   | -     | 1,937.65 |\n| SGDD   | 1,644    | -      | 2,364.18    | -    | 2,364.18   |\n| GDEM   | 2,658    | 4.56*    | 7.48      | -    | 12.04    |\n| PreGC  | 1,714    | -    | 22.36   | 0.81 (optional) | 22.36 / 23.17 |\n\n**W2:**  Incorporating your suggestions and **Reviewer NPxj**’s remarks (C2-3), we conduct additional experiments with other baselines under varying training ratios, as shown in the following Table. Notably, the flexibility of PreGC lies in the fact that it only requires one-time condensation for a given dataset. When the task or label ratio changes, the updated information can be efficiently transferred to the condensed graph by Eq. (14). In contrast, existing methods must recondense the graph to capture new label knowledge, which significantly hinders the reusability of condensed graphs. We will further refine and expand this section in the next version.\n\n| Training Ratio | 0.15    | 0.30 | 0.45  | 0.60    | 0.75  | Condensation times |\n| -------------- | ---------- | ---------- | ---------- | ---------- | ---------- | ------------------ |\n| Whole   | 66.71±0.43 | 69.53±0.11 | 70.44±0.51 | 71.69±0.20 | 71.72±0.57 | - |\n| GCDM  | 35.98±1.31 | 36.32±1.08 | 36.09±0.89 | 36.94±0.60 | 38.35±0.67 | × 5|\n| GCond   | 56.17±0.30 | 56.95±0.31 | 56.83±0.49 | 57.47±0.36 | 58.00±0.34 | × 5|\n| SFGC  | 59.09±0.42 | 59.85±0.38 | 60.04±0.53 | 60.46±0.51 | 60.59±0.48 | × 5 |\n| SGDD | 58.01±0.77 | 58.79±0.40 | 59.03±0.44 | 59.54±0.53 | 60.24±0.46 | × 5  |\n| GDEM  | 52.23±0.32 | 54.20±0.57 | 54.81±0.44 | 54.98±0.48 | 55.41±0.31 | × 5  |\n| CGC  | 58.39±0.33 | 59.04±0.49 | 58.92±0.72 | 59.53±0.56 | 60.05±0.29 | × 5   |\n| **PreGC**  | 60.53±0.32 | 61.96±0.56 | 61.72±0.63 | 63.37±0.49 | 63.58±0.82 | × **1**   |\n\n**W3:**  We appreciate your valuable suggestions. Guided by the comments from both you and the other reviewers (**JS8B** and **BLZo**), we carefully reviewed [1][2] and existing research. We incorporate experiments on two large-scale graphs Reddit and Flickr to systematically assess the generalizability of PreGC. The results are reported in the table below.\n\n| Reddit (r=2.5%) | SGC          | GCN          | APPNP        | k-GNN        | GAT          | SAGE         | SSGC         | Bern.        | GPR.         | Avg.         |\n| --------------- | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ |\n| Whole           | 86.1±0.1     | 86.5±0.2     | 85.9±0.2     | 83.7±0.1     | 83.1±0.5     | 83.0±0.2     | 86.8±0.1     | 85.7±0.2     | 84.5±0.3     | 85.0±1.4     |\n| GCDM            | 21.9±1.5     | 24.0±0.6     | 56.3±0.8     | 17.4±3.6     | 15.1±1.4     | 30.6±0.4     | 57.5±0.7     | 53.0±1.4     | 40.9±0.9     | 35.2±16.1    |\n| GCond           | 70.4±1.0     | 69.3±0.5     | 61.8±1.2     | 31.2±0.7     | 23.3±2.6     | 40.8±1.0     | 63.8±0.6     | 64.5±0.5     | 47.2±1.3     | 52.5±16.4    |\n| SFGC            | 65.2±1.0     | 67.7±0.7     | 68.9±3.0     | 33.9±1.3     | 19.6±2.9     | 36.7±2.7     | 59.9±0.3     | 68.9±2.2     | 51.6±2.1     | 52.5±17.2    |\n| SGDD            | 75.5±0.9     | 77.0±0.6     | 68.6±2.2     | 24.6±2.0     | 38.9±2.7     | 46.7±2.8     | 69.6±0.2     | 68.9±1.9     | 51.2±2.2     | 57.9±17.3    |\n| GDEM            | 78.3±0.8     | **79.2±1.0** | 79.1±1.0     | 19.2±2.7     | **77.4±2.4** | 42.4±0.7     | **82.2±0.4** | 69.6±2.2     | 57.2±1.1     | 65.0±20.3    |\n| CGC             | 76.9±0.4     | 78.4±0.3     | 72.5±0.9     | 18.5±6.0     | 50.3±4.3     | 41.3±0.3     | 75.6±0.4     | 75.0±1.1     | 58.8±1.2     | 60.8±19.5    |\n| PGC             | **79.2±0.2** | 74.1±0.4     | **79.9±0.3** | **73.7±0.3** | 72.9±4.1     | **70.3±0.4** | 78.5±0.3     | **75.9±1.3** | **70.3±1.2** | **75.0±3.4** |\n\n| Flickr (r=2.5%) | SGC   | GCN  | APPNP | k-GNN  | GAT  | SAGE | SSGC  | Bern.  | GPR.    | Avg.   |\n| --------------- | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ | ------------ |\n| Whole           | 48.3±0.5     | 47.9±0.1     | 44.4±0.6     | 41.8±2.2     | 43.5±0.9     | 46.9±0.2     | 48.0±0.3     | 46.5±0.3     | 46.2±0.3     | 45.9±2.1     |\n| GCDM            | 46.4±0.8     | 45.5±0.5     | 46.5±0.1     | 42.1±0.1     | 44.9±1.1     | 44.4±0.2     | 45.9±0.3     | 45.4±0.5     | 44.8±0.2     | 45.1±1.3     |\n| GCond           | 47.6±0.2     | 46.5±0.1     | 47.3±0.3     | 43.0±1.7     | 41.9±0.7     | 44.9±1.0     | **47.6±0.1** | **46.3±0.7** | 46.4±0.2     | 45.7±1.9     |\n| SFGC            | 46.0±0.9     | 44.1±0.6     | 45.3±1.6     | 46.8±0.9     | 44.6±1.2     | 44.4±0.2     | 45.8±0.2     | 36.1±6.6     | 45.5±0.2     | 44.3±3.0     |\n| SGDD            | 48.0±0.2     | 43.7±0.9     | 44.7±1.1     | 42.1±0.0     | 30.6±5.9     | 42.1±0.0     | 45.6±0.8     | 42.5±0.7     | 43.6±0.5     | 42.6±4.6     |\n| GDEM            | 42.2±0.0     | 42.1±0.1     | 42.3±0.1     | 42.0±0.1     | 42.4±0.4     | 42.6±0.4     | 42.3±0.2     | 41.5±0.6     | 42.3±0.1     | 42.2±0.3     |\n| CGC             | 47.7±0.5     | **46.9±0.4** | 47.0±0.4     | 47.0±0.4     | 46.8±0.5     | 45.8±0.2     | 46.8±0.3     | 46.0±0.7     | 45.9±0.4     | 46.7±0.6     |\n| PGC             | **48.0±0.4** | 46.5±0.1     | **47.7±0.2** | **47.7±0.7** | **47.4±0.3** | **45.9±0.2** | 47.1±0.2     | 46.0±1.0     | **46.8±0.4** | **47.0±0.7** |\n\n\n**Q1:** Thanks for the positive feedback. Intuitively, leveraging the condensed graph to retrospectively evaluate the node value of the original graph should be a natural idea. This stems from the fact that graph condensation aims to distill the most informative (or valuable) node features and topological structures from the original graph while discarding redundancy or noise. Unfortunately, existing GC methods only implicitly learn certain characteristics of the original graph (e.g., gradients optimized by GNNs, distributions in the representation space, etc.), thereby lacking explicit associations with the original graph. This undoubtedly hinders the transparency or interpretability of the condensation process. PreGC addresses this limitation by introducing a transport plan whose transport probabilities explicitly bridge the condensed and original graphs, providing clearer explanations (i.e., higher transport probabilities indicate greater node contributions). Consequently, existing GC baselines still leave the evaluation of original graph data value an open question.  \nAs far as we know, current approaches for graph data valuation can be categorized into two types. **(1)** Effective resistance (ER) estimation [3]. This method focuses on quantifying node proximity within graphs and has been widely applied in tasks such as the detection of low-conductance sets in graph clustering [4] and graph spectral sparsification [5]. However, due to its emphasis on pairwise node relationships, ER estimation essentially functions as an edge valuation method rather than a node valuation method. **(2)** Migration of existing data valuation methods (e.g., Shapley, Banzhaf, Datamodels) to GNNs [6]. To adapt to structured data, researchers explicitly incorporate graph structures into the valuation process. For example, when evaluating node $i$, one must first construct an induced subgraph by removing node $i$. Subsequently, a GNN is trained on this induced subgraph to assess the impact of node $i$ on model performance. In short, for a graph with $N$ nodes, this procedure requires generating $N$ distinct $(N−1)\\times(N−1)$ subgraphs and performing $N$ complete GNN training evaluations. This renders the approach prohibitively expensive in practice.  \nWe sincerely appreciate your recognition of our idea and the valuable research direction you have proposed. In future work, we will explore the potential of other GC methods in graph data valuation to further advance the broader applicability of graph condensation.\n\n[1] Gong, Shengbo, et al. GC4NC: A benchmark framework for graph condensation on node classification with new insights. *ArXiv:2406.16715*, 2024.  \n[2] Sun, Qingyun, et al. GC-Bench: An open and unified benchmark for graph condensation. NeurIPS, 37, 37900-37927, 2024.  \n[3] Lai, Yurui, et al. Efficient topology-aware data augmentation for high-degree graph neural networks. KDD, 1463-1473, 2024.  \n[4] Alev, Vedat Levi, et al. Graph Clustering using Effective Resistance. ITCS, 94, 2018.  \n[5] Spielman, Daniel A., and Nikhil Srivastava. Graph sparsification by effective resistances. STOC, 563-568, 2008.  \n[6] Antonelli, Simone, and Aleksandar Bojchevski. Data Valuation for Graphs. 2025."}]

Author rebuttal:

Meta-review: The reviews agree on the technical merits and applicability of PreGC. Therefore, I recommand acceptance.
Key strengths highlighted by reviewers includes "The task decomposition of SGC convolution obtains the reconstruction and supervised signal terms, sufficiently illustrating the motivation of the proposed method." (Reviewer BLZo); "The use of optimal transport and graph diffusion augmentation in the context of graph condensation is novel and theoretically grounded." (Reviewer 3v1U); "PreGC demonstrates consistent superiority over state-of-the-art approaches across various datasets, tasks, and GNN backbones." (Reviewer 8BSw); "The experimental setup is interesting, particularly the multi-task configuration for node-level datasets." (Reviewer JS8B); "Clearly identifies limitations of existing graph condensation methods" (Reviewer NPxj).
Decision: Accept (poster)