OASIS: One-Shot Federated Graph Learning via Wasserstein Assisted Knowledge Integration
Authors: Guancheng Wan
Jiaru Qian
Wenke Huang
Qilin Xu
Xianda Guo
Boheng Li
Guibin Zhang
Bo Du
Mang Ye
Venue: NeurIPS 2025 poster ()
Keywords: Federated Learning
Graph Learning
Primary area: social_and_economic_aspects_of_machine_learning

Abstract: Federated Graph Learning (FGL) offers a promising framework for collaboratively training Graph Neural Networks (GNNs) while preserving data privacy. In resource-constrained environments, One-shot Federated Learning (OFL) emerges as an effective solution by limiting communication to a single round. Current OFL approaches employing generative models have attracted considerable attention; however, they face unresolved challenges: these methods are primarily designed for traditional image data and fail to capture the fine-grained structural information of local graph data. Consequently, they struggle to integrate the intricate correlations necessary and transfer subtle structural insights from each client to the global model. To address these issues, we introduce **OASIS**, an innovative one-shot FGL framework. In OASIS, we propose a Synergy Graph Synthesizer designed to generate informative synthetic graphs and introduce a Topological Codebook to construct a structural latent space. Moreover, we propose the Wasserstein-Enhanced Semantic Affinity Distillation (WESAD) to incorporate rich inter-class relationships and the Wasserstein-Driven Structural Relation Distillation (WDSRD) to facilitate the effective transfer of structural knowledge from the Topological Codebook. Extensive experiments on real-world tasks demonstrate the superior performance and generalization capability of OASIS. The code is available for anonymous access at https://anonymous.4open.science/r/OASIS-NeurIPS25.

OASIS: One-Shot Federated Graph Learning via Wasserstein Assisted Knowledge Integration
Guancheng Wan It Jiaru Qian't, Wenke Huang Qilin Xu Xianda Guo',
Boheng Li", Guibin Zhang', Bo Du', Mang Ye
Sccooo of Computer Science Wuhan University 2NTU NUS
{guanchengwan, yemang)@uhu.edde edu.cn
Abstract
Federated Graph Learning (FGL) offers a promising framework for collaboratively training Graph Neural Networks (GNNs) while preserving data privacy. In resource- constrained environments, One-shot Federated Learning (OFL) emerges as an effective solution by limiting communication to single round. Current OFL approaches employing generative models have attracted considerable attention; however, they face unresolved challenges: these methods are primarily designed for traditional image data and fail to capture the fine-grained structural information local graph data. Consequently, they struggle to integrate the intricate correlations necessary and transfer subtle structural insights from each client to the global model. To address these issues, introduce DASIS, innovative one-shot FGL framework. In DASIS, we propose a Synergy Graph Synthesizer designed generate informative synthetic graphs and introduce Topological Codebook to construct a structural latent space. Moreover, we propose the Wasserstein- Enhanced Semantic Affinity Distillation (WESAD) incorporate rich inter-class relationships and the Wassersteinn-rivvae Structural Relation Distillation (WDSRD) facilitate the effective transfer of structural knowledge from the Topological Codebook. Extensive experiments on real-world tasks demonstrate the superior performance and generalization capability of OASIS, with an average improvement of 15.81 1 over the baseline. The code is available for anonymous access at https://gitthbb com/ wwt//iaaariaaa/AASSS.
Introduction
Federated Learning (FI.) 19 20 enables decentralized model training, allowing collaboration across clients while preserving privacy. Many real-world applications involve non- Euclideen data structures, such as graphs, where entities are interconnected through complex relationships. These graph-structured data are common in various domains, including biological networks [69], urban mobility systems [68), and online social platforms [47]. To learn from such structures from multiple participants, Graph Neural Networks (GNNs) (25 58 have been integrated with FL, leading to Federated Graph Learning (FGL) 9 (T. This approach combines both paradigms, ensuring privacy while enabling efficient learning distributed graph data.
Although most research in FGL has focused on personalized learning, where each client has tailored model [3] [73 30), there is growing demand for global model that can generalize across diverse graph data from multiple clients (52). This especially important in scenarios with limited data or where consistency between clients is necessary, such as in medical data analysis across
Equal Contribation. Corresponding Author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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hospitals or regional traffic network predictions [57 71. However, training a global model in traditional FGL requires multiple rounds of communication, which can be burdensome for edge devices with limited bandwidth and computational power 13 2 To address these challenges, One-shot Federated Learning (OFI.) has been proposed, restricting communication to single round 12 34), thus reducing communication costs and potentially enhancing privacy [62]. However, existing OFL methods are primarily designed for traditional data types like images and do not tackle the unique challenges of graph-structured data. We define the task of Generalizable One-Shot Federated Graph Learning as developing a global model capable of generalizing well for graphs distributed across multiple clients using only one round of communication. A key research question is:
How design generalizable OFI. framerok specifically tailored for graph-structured data?
Several OFL approaches have proposed using public datasets to train a global model. However, this approach may be impractical because of the limited availability of suitable datasets or due to stringent data sensitivity constraints [46] (36). Consequently, generative methodologies have emerged as compelling alternatives. For in- stance, DENSE [66] employs strategy in which each client trains a generator to produce syn- thetic data that reflects its distribution. In con- trast, FedDEO [6] utilizes diffusion models as generators, leveraging client- traine descriptors to guide the server training a global model. Nevertheless, these methods are primarily de- signed for image data and do not capture the fine-grained structural information inherent in nodes, particularly substructure variations such as connectivity patterns and neighbor distribu- tions, as illustrated in Figure 2 The direct ap- plication of these generators fails to encapsulate the spatial intricacies of graphs, thereby limit- ing their effectiveness. This limitation raises a critical question: 1) How can client genera- tors better capture the fine-grained structural knowledge of local graphs?
(1) Client: Fail to capture fine grained structural knowledge
Connectivity
Pattern
Substructure Diffeerencs
Intra-
Graph Comstrunt
$
Class
Ege- Graph
Data
Generator
DNna
Neighbor
Degradted
Distribation
Representativeees
(I)) Server. Strugple to transfer subtle structural knowledge
Crowy Entryyy
Structure
Varistions
Ciient
Transmit Knowfedge
Global
A
Model
Traditiona!
Degraded Gemeraligtion
Clentt
KL Divergence
Figure 1: Problem Illustration. For current one-shot FGL. scenarios: I) at the client stage, existing generative methods fail to capture the fine-grained structural knowl- edge of local graphs: II) at the server stage, conventional KD techniques are unable to effectively transfer the sub- tle structural characteristics from diverse clients, thereby hindering the global model generalization.
After training the local generator, several methods have leveraged knowledge distillation (KD) on the server to train the global model [6. 4T For example, DENSE first ensembles local models and then distills the resulting ensemble, while FedCVAE- KD [15] employs KD from local decoders to train the server decoder. However, these distillation methods depend excessively on class-related semantic signals when aggregating client knowledge, and the class space may not be sufficiently expressive to represent the diverse local graph structures of nodes. Consequently, they fail to capture the intricate correlations and struggle to transfer the clients' structural awareness to the final global model. This observation raises an important follow-up question: II) How can we effectively transfer subtle structural knowledge during server distillation?
To address the aforementioned challenges, this paper presents the first systematic study on the problem of generalizable one-shot FGL. We introduce the OASIS framework: One-Shot Federated GrAph Learning via Wasserstein AsSISted Knowledge Integration. To address problem D) we introduce the Synergy Graph Synthesizer, which is designed to generate informative synthetic graphs. This approach is further refined by incorporating alignment constraints based on the Fused Gromov- Wasserstein distance, which effectively harmonizes the reconstruction of both features and structures. More importantly, inspired by 555 64 we construct a novel structural latent space tailored for graph data. This enables the learning of a structure- tokenizer that encodes each node along with its substructure as discrete code, thereby capturing its spatial characteristics. By utilizing the established Topological Codebook, we can precisely characterize subtle structural variations, thereby enabling the meticulous integration of fine-grained structural knowledge from client graphs. On the server side, address issue II) we first propose the Messerstein-Enhancee Semantic Affinity Distillation, which models the rich inter-class relationships by distilling logits from the teacher local model to the
2

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student global model, while learning more general structural differences. At the intra-class level, we introduce the Wasserstein-Drie Structural Relation Distillation, which dynamically measures the distance between samples and the discrete codes derived from the Topologicul Codebook, thereby facilitating the sophisticated transfer of subtle structural knowledge. Our principal contributions are summarized as follows:
Problem Identification. We identify the key challenges in generalizable one-shot FGL:: how to effectively capture fine- fine-graines grained structural knowledge of local graphs, and then how to transfer this knowledge during distillation to enable global model generalization.
Practical Solution. We propose novel approach that integrates the Synergy Graph Synthesizer with Topological Codebook capturing subtle structural variations, then we introduce hierarchical Wasserstein- based distillation skillfully transfer structural knowledge server. Experimental Validation. We examine OASIS through extensive experiments on various graph datasets, demonstrating superior generalization ability for one-shot FGL.
2 Preliminaries
2.1 Notations
Graph Neural Networks. Consider a graph 9 = (V,E), where denotes the of N nodes, and represents the edges. Each node o is associated with an F-dimensional feature vector 2;, and these vectors collectively form the feature matrix X T2. .....,,, .NN] The structure of & is encoded in the adjacency matrix A E IR N where the entries are defined such that Aii,j) 1 if nodes i and j are connected, and A (i. j) = Ó otherwise. Graph Neural Networks (GNNs) iteratively build node representations by aggregating features from neighboring nodes and then applying an update function. Specifically, the representation h/+1 of node Us at layer 1 + 1 is computed as:
1+1 Update (h4, Aggregate ((h) by e ((u)))))
(1)
where hy is the current representation at layer I, NV;,)) denotes the set of neighbors of node U., Aggregate(-) .) combines the neighbor features, and Update(..-) updates the node's representation.
Initially, hQ Vig.
Problem Formulation. In a One-shot Federated Graph Learning framework, a central server coordinates K clients (collectively denoted as C, where each client indexed by k). Each client k holds its own graph g = (V* e with the corresponding adjacency matrix A For every node 2/ E VA there exists an associated feature vector at and a label y if it is available. Within this setup, each client trains model Fps parameterized by H and then transmits them the server. The objective is to minimize:
N
min
())
(2)
This is computed as the weighted sum of the K local nodes N*, where N denotes the total number of samples across all clients. The local objective (-) is typically defined as the expected error over all nodes from the local graph g*, with the global model parameterized by Φ. Unlike conventional FGL, one-shot FGL. restricts the exchange to a single communication round.
2.2 Motivation
This paper systematically explores the challenge of maximizing the generalization ability of the global model in one-shot FGL. Traditional OFL approaches rely on generative models such as Variational Autoencoders or Generative Adversarial Networks--rriinnllly designed for image-based data fail to deliver competitive performance on graphs, as demonstrated in Sec. 4 The primary issue is their inability to effectively capture the spatial and relational dynamics between nodes. These models emphasize coarse, class-base based semantic signals while overlooking the underlying substructural variations among nodes. To better illustrate this, we present the t-SNE visualization of the learned graph representation space after training the teacher GNN and the Topological Codebook. (Detailed methods are illustrated in Sec. 3 We select three categories with their representative nodes and corresponding two-hop substructures. Results are shown in Figure 2 where nodes within
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the same class can exhibit significantly different connectivity patterns and neighbor distributions. Consequently, the key objective in the local training stage of one-shot FGL identify knowledge
repository that optimally encapsulates both local semantics and fine-greeed fine-grained structural insights. We formally define this pursuit follows:
Leiter Class
Taken 924
Taken 1550
Tokes 773
Disci eppiiic
R. arg min Bg ~R (G'IAA) 9))";9, (3)
(X.A)~G
Token 14665
19223
Here, R denotes the optimal knowledge repository obtained by minimizing the expected distance Q he- tween the generated graph 9" and the original graph G. In this formulation, Rup is parameterized func- tion that produces G given the node features X and the adjacency matrix A. After obtaining the power- ful knowledge repository R. our goal is to integrate these granular insights into the global model F& dur- ing the server distillation phase without erosion of knowledge:
Token 612
Toloon 15399
Token 1045
A
F = arg min Σ
E RA T T(Fo()"); For (9*))
(4) For each client k. R represents a specialized knowl- edge repository. The global model Fo trained to align its outputs with those of the correspond- ing client models, while T quantifies the knowl- edge difference between them. Minimizing this ex- pected transfer loss ensures that the ultimate global model Fx effectively assimilates the granular infor- mation encapsulated within the knowledge reposi- tories. Based on these discussions, we present the principle for designing the ideal Generalizable one- shot FGL pipeline:
l+
Intra- Clars Diserspancy
Figure 2: Motivation. In the t-SNE visualiza- tion, nodes from different categories (colored red, yellow and green ) are scattered, representing inter-class discrepancy. However, nodes within the same category are also mapped to different topological codebook tokens (We visualized three intra-class discrep-
each color), indicating ancy in terms of connectivity patterns and neighbor distributions. Please refer to Sec. 2.2 for details.
Generalizable One-shot FGL Design Principle: Communication Efficiency: Achieve robust generalization via a single, streamlined communication round, thereby reducing overhead without compromising performance. Knowledge Extraction: Precisely capture both local semantic signals and fine- ine-grained structural variations from graphs. Knowledge Integration: Seamlessly integrate these granular insights into the global model, ensuring minimal knowledge loss during the server distillation.
In following sections, will elaborate on how OASIS adheres to these principles, encapsulates fine-grained knowledge and then effectively distills it to the global model.
3 Methodology
3.1 Framework Overview
In this section, we present an overview of OASIS. On the client side, we perform Local Graph Knowledge Extraction, where a synergy graph synthesizer and topological codebook are con- structed to capture fine-grained structural knowledge. After the one-shot communication, we employ Wasserstein-Ennhare Semantic Affinity Distillation and Dasserstein-Drivard Structural Relation Distillation to transfer intricate topological knowledge during server distillation. The framework illustration is provided in Figure

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Client: Symerg Graph Synthesber
Server: Wasserstein Assisted
Knowledge Integration
Zahodday Engl
Cooo
Wasserseiin Eahanced Semantiic Aftinity Dicillation (WISSD))
Pover Goneet
Orenle
D
Laca
Chan-Co
Client: Topelogical Codchool:
Wissersteia Deiven Stractural Relation Distillasoon (WDSRD)
Gonstion Stite
Ben Greph
Presurry
Grapon
[azasey
Lahe
eNoy
GANN
GIIN
Figure 3: Architecture illustration of DASIS. (a) The left part shows the training process of Synergy Graph Synthesizer and Topological Codebook the client side. (b) The right part presents the knowledge integration part on the server side. The communication round limited to one, with codebooks and parameters of the local GNN and Synthesizer uploaded. The far right section displays the legend. Zoom for details.
3.2 Local Graph Knowledge Extraction
Synergy Graph Synthesizer. To construct high-fidelity graphs, we introduce novel Synergy Graph Synthesizer Qip' as part of the knowledge repository R2p (omitting i for brevity), which reconstructs both node features and graph topology in structured manner. Given a graph g (V,E), for each node e E V, a synthetic feature vector is generated by mapping Gaussian noise (sampled from a standard Gaussian distribution) and the corresponding label y/: through the generator: i = 2up (E) where 2pp' is parameterized by / This process constructs the synthetic feature matrix X-
RNF. Next, an activated similarity matrix is computed as H (XT) where denotes the sigmoid function. By applying a K-Nearest Neighbors strategy to each row of H, we obtain the synthetic adjacency matrix A:
[21,
1, if E Topk (H()),
(i.j) = 0. otherwise.
(5)
Thus, the synthesized graph is denoted as 9 = () while preserving the original labels y = 1/2,,/2..... WN] We adopt a transductive setting, where some nodes in the local data are unlabeled. Therefore, we utilize only the labeled nodes in the local training set to train the synthesizer. Meanwhile, to ensure that the synthesized graph faithfully reflects both the distribution of node features and intrinsic topological patterns of the original graph, design two-fold consistency mechanism to compute the synthesizing objective Lyyn. Details can be found in Appendix B
Topological Codebook. While the Synergy Graph Synthesizer effectively aligns global feature distri- butions and preserves large-scale structural patterns, it inevitably smooths out nuanced distinctions among fine- fine-grained -r-igrriinddddd substructures. To address this limitation, we introduce discrete structural latent space-terman termed the Topological Codebook to explicitly model subtle neighborhood variations. First, each node E V is mapped to contimuous embedding h; E IR via encoder. We then construct a learnable codebook Egg" = [ei,..., em] E RD where each code vector e, represents a distinct local substructure pattern. For each node U; the discrete token corresponding to its local substructure is determined by quantizing its continuous embedding to the nearest code:
2 arg min
||;; e;
(6)
(E(L.....M)))
This quantization process maps continuous embeddings into discrete tokens {......, 2x }. where each node is assigned token corresponding its most similar local substructure. Using these tokens, we derive the quantized representations Q = (c.................... .zv } for all nodes. These discrete representations enable the model to capture subtle variations in node neighborhoods more effectively.
Concurrent Optimization. To learn the Topological Codebook, we adapt the core principles of Vector Quantized- Guantized-Variationall Variational Autoencoders (VQ-VAE) for graph data. The model optimizes the
5

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following components: the reconstruction loss Lyyn and two additional losses: the consistency loss and the discretization loss. The consistency loss ensures that the selected code ez2 preserves original node attributes and topology information. Specifically, we design two decoders py: Py- to separately map the quantized representations the original dimension: e'z, = pr (ez,); ), Q' pay (Q). Then, we introduce our consistency loss:
N
le - A σQQ Q )el Lcon Σ
(7)
where denotes the sigmoid function. Meanwhile, the discretization loss consists of three key components: the codebook term, the commitment term, and the orthogonal term. The codebook term ensures that the selected code 6N aligned with the encoder output h; maintaining the coherence between continuous and discrete representations. The commitment term encourages the encoder output to stay close the chosen code, preventing excessive fluctuations between code vectors that could destabilize the learning process. By stabilizing the encoding process, the commitment term ensures that each node's ' representation remains consistent with its assigned code. The orthogonal term promotes diversity among codebook vectors by encouraging them to be independent and avoiding convergence in the same direction:
Cdise
sg[b.] ez:
12
Σ N
sg. sg( - 1111 Ao
G |e.le,
1 M
(8)
N
M
where sg[:] denotes the stop-gratde grpddiddient operator, and 17, A are hyperparameters controlling the strength of each term. The overall optimization objective combines these losses with negative log-likelhhood term LNLL (Vt. De..) to maintain node-label consistency with GNN Fg predictions fe.."
Loverall = Layn + CNLL Ne.;) Lcon + AcCaise
where d is balancing hyperparameter. By jointly optimizing these terms, we obtain Topological Codebook Egg that captures a rich of discrete tokens, each representing unique local substructure. This latent structural space empowers the model to integrate localized topological knowledge with global graph structures. The concurrent optimization of the Synthesizer 2pr and the Topological Codebook leads to the creation of a refined knowledge repository R. {2r, Egg") ). which effectively synthesizes both micro- and macro-level graph information. the subsequent section, we explore how this extracted knowledge can be transferred to construct generalizable global model.
3.3 Server Graph Knowledge Integration
Wasserstein-Enhancee Semantic Affinity Distillation. After local training, each client uploads its respective knowledge base R = O E } and local GNN F to the central server. At the server, for each local model k; the local GNN model Fø serves as the teacher, while the global GNN model Fg acts as the student. Specifically, a proxy graph G is generated via O (e; Duni), where Juni denotes a class-balanced distribution. This proxy graph retains general information from the client data distribution, thereby providing an effective signal for subsequent knowledge transfer.
A central challenge in this knowledge transfer process is minimizing the loss incurred between the teacher and student models. Traditional distillation techniques based on Kullback-Leible divergence comparisons, failing to capture nuanced cross-category semantic
perform only eategory-to-categnn affinities. To address this limitation, we propose the Wasserstein- Eensersteii-Enhhneeed Semantic Affinity Distilla- tion (WESAD) method, which employs the discrete Wasserstein distance to achieve a comprehensive alignment between the probability distributions of the teacher and student models. For the k-th teacher GNN model, let P denote the class probability distribution produced by F and ps the distribution produced by Fa. These distributions are computed via the softmax function a with temperature т. To measure their discrepancy, we define the discrete Wasserstein distance loss D DESAD as follows:
D ps) = min Σ
c: gab + 7abbb log Jab-
(10)
where qab represents the mass transferred from the teacher's category C2 to the student's 's category Cb, subject to the constraints:
gah 2 0, Σ qub pr.ar
Σ Yar Pssb-
(11)

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Here, is a hyperparameter controlling the entropy regularization term. A key component of this formulation is the cost matrix bb. which encapsulates the semantic dissimilarity between categories. We convert the semantic affinity, denoted as SA (Cas Cb), into a distance metric by defining
SAa C&) = [Ca]d [[] @ [C&]]]
(12)
where & denotes element-wise multiplication, and W is the weight matrix of the teacher's projection head after l2 normalization. The term SA (Ca, C6) reflects the intrinsic semantic affinity between categories in the local GNN teacher Fer. We then compute the cost matrix ex as
Cab = - KSA* (Ca. Cb).
(13)
where y controls the cost. When two categories are semantically similar, a higher SA (Ca, Cb) results a lower e thereby reducing the transportation cost between them; conversely, semantically dissimilar categories incur a higher cost. This design ensures that probability mass is effectively reallocated between semantically proximate categories during knowledge transfer, facilitating fine- grained semantic alignment. By minimizing WESAD (PT, Ps), the global student model F is not only aligned with the output distribution of each local teacher model F but also benefits from the incorporation of inter-class semantic correlations.
Wasserstein-Drrin Structural Relation Distillation. Building upon the semantic knowledge transfer described before, we now seek to integrate the fine-grained latent structural knowledge. Unlike semantic alignment, which primarily focuses on matching class-level distributions, structural distillation emphasizes the nuanced topological patterns underlying node interactions. We introduce a Meesserrtein-rriver Structural Relation Distillation ( (WDSRD) scheme that harmonizes the structural articulations by leveraging the topological codebook E = [e .ef...., e where each ex E IK is a learnable structural token that represents prototypical substructure. This codebook unifies latent structural information for both local and global models. Concretely, for each generated proxy graph G+, we compute two sets of node-level representations:
H = Fá (G*) E R
= Fo E 8* XD E
(14)
where Ñ denotes the number of nodes in g and D is the feature dimension. For each node v; let hy E IRP be its representation from the local model and h; E RDD from the global model. We then compare each h (or h4) against all M codes in E to derive soft assignment distribution:
B = a (Dist (h,, /T)) B = = (f.,, ET))/TT,
(15)
R.M distance- comparison function (e.g., Euclidean distance) that assigns each node v; to every code em with a probability reflecting their similarity. We then obtain structural code assignments By from global GNN and B from local GNN over the M structural codes from k-th topological codebook E.... We first define the mean the simple average of the assignment vectors across all nodes. Specifically, the local and global means are given by:
where Dist : RDD MxD
N
Plocl
Σ
B
Ngloball =
Σ B
(16)
Ñ
N
To characterize the dispersion of the assignments, we compute the covariance matrices as:
5
13 - /
(15t
Joeal
No
(17)
gi
S
Zgloball
B
Mggatal)
(B:
Mglaball
Nrk
1-1
Thus, the local and global assignments for the agent graph are approximated by the Gaussian distribu- tions: . (pican Etaaal .N (Hiinhal Eglobal) Inspired by the Wasserstein distance, we employ the closed-form Wasserstein distance between two Gaussian distributions. For two Gaussians, N N(H,, E,) and (u2,, Σ2), the Wasserstein distance is decomposed into a term measuring the difference in means and term capturing the difference in covariances:
DWD (N(p,, E,), Nu E2)) = || wa laa
+ tr Eg + 5 - x 2/51//35331111 1/))
(18)
7

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Table I: Comparison with the state-of -he---r-t---rt methods on eight real-world datasets. We report node classification accuracies (%) for downstream task performance. Green arrows denote advancements in accuracy metrics than FedAvg while red arrows I indicate regressions. OOM means out-of-memory error. The best and second results are highlighted with bold and underline, respectively.
Calgany
Metiads
Cora
Checser
Pulteed
Amu-PPasto
Countonns
Actor
Ronna-Einjire
OlganArsic
Fed/vg (ASTXTL7]
.1
32.88
57.91
23.12
22.50
24.40
18.49
14.58
FedProxx [MI 8yc30)
30.08ro.a av
35
(3.93
50.58, 55
24.16,1.04 .04
21.44, 1.06
23.75,0.0
15.46.0.0
13.00 on
FL.
FodNova [NeurIP520]
14:21 (00
18.08 17:50
20.48
6.15
10.07
18.83 (9.07
04
6.49 11 00
1.17, 1.17,18.41 (18.41
RedRCI [CVPR244
17.60 (10.0
12.78,20.15
28.
p9.70
4.02
118.00
14.76,1/75
10.72
13 68
11.47,9.02
2.56,122.0
RedPub (ICML223
30.52, 0.00
74.91
41.22
16.00
21.91,1.91
21.91
26.75
26.78
(2.17
13:71
10.02 4.00
FGL
POSSL (UCA123)
30.23 as
21.95 10.90
20.68
18.98
13.06
10.06
22.44,
[2.44...00.
19 (8.9)
14.62 (1:7
9.24 5.84
PedCTA [VLDB24]
14.02 100.5
17.75
81.45,
20.46
4.10
.2..02
10.80 11.79
25.255
4.65
1.15 15.43
FedTAD [JCAM14)
30.43,mm 18.1
30.88
00.88
30.32, 19.39
22.0
14.09.0.4
23,58,0
14.40,4.09
00M
DENSE [Neu
12:92 L19.9
7:87 01
20.84,22.07
p2:00
3,96 18.04
10:72 13
3.00 14.50
0.33 14.23
OFF.
FedCVAB [CLR22)
09.25
24.76
52,0115.00
31.62
va.50
14.00,7.0
19.38
28.99-11..8
13.71.0.47
FedS520 [Nem
17.78
0.00
26.
3.66
18.00
18.83 (.
3.92,
0.76
PENS (NculPP524)
an
20.97 11:00
1-
25.30
22.54.0..0
13.00, 10
0A513
118.0 os
#1
99. os
60.44:37
00
25.
02
30.07 on
15
Accordingly, the DWDSRD between the local and global assignments for the entire agent graph is
DÖDSRD-
|
Hglobal 1
(19)
tr Diiccal
2 2((22 1o (2lccal Ligokoal
>
2hocal
where the first term quantifies the discrepancy between the local and global means, while the second term reflects the differences in the distribution shapes captured by the covariances. Minimizing LWDSRD encourages the local and global GNNs to progressively align their output distributions of structural assignments, thereby achieving an effective distillation of fine-grained topological knowledge.
In our final server optimization, we integrate both the distillation losses into a single objective with addition of NLL loss on the proxy graph:
No (CALL + AADEAD + AuT DWNSRD))
Lyerver
(20)
where N denotes the overall size of all synthesized graphs and X2, A balance each term. This unified approach leverages the strengths of both methods: WESAD captures and aligns inter-class semantic affinities to transfer rich semantic signals, while WDSRD preserves the fine-grad aen--graindd topological structures inherent in local graphs. Minimizing this combined loss ensures that the global model assimilates both semantic and structural knowledge from the local teachers, resulting in a more generalizable global model.
4 Experiment
In this section, we comprehensively evaluate DASIS through four axes: Q1 (Superiority), Q2 (Effectiveness)). Q3 (Sensitivity). Q4 (Pricacy). The answers of Q1-Q2 are illustrated in Sec. 4.2 Sec. 4.3 and the analyses of Q3-Q4 can be found in Appendix K and Appendix
4.1 Experimental Setup
Datasets. To effectively evaluate the performance of our approach, we employed eight benchmark
graph datasets of various scales and distributions, including Cora (37. CiteSeer [10], PubMed (5),
Amazon-Phont Photo, Coauthor-CS [45) Actor [42), Roman-empire 4% and Ogbn-Arxiv [7]. Detailed
descriptions and splits for these datasets can be found in Appendix D
Counterparts. We compare OASIS against several traditional FL methods: (1) FedAvg [ASTAT17]
(38, (2) FedProx [MLSys20] (27]. (3) FedNova [NeurIPS20] (54) (4) FedRCL [CVPR24] 44];
four popular FGL approaches: (5) FedPub [ICML23] 4. (6) FGSSL [ICCAI23] 18; (7) FedTAD
[JCA124] 73. (8) FedGTA [VLDB24] 30 four One-shot FL methods: (9) DENSE [Ne2)
66.. (10) FedC [ICLR23] 15) (11) FedSD2C [NeurIPS24] 67], (12) FENS
[Neur]PS24]
Detailed descriptions can be found in Appendix E
8

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50
0
13
0.20
3.25
0.20
0.35
0.40
22
0.3
0.00
0.04
0.5
(a) Cora
(b) CiteSeer
(c) Analysis on
(d) Analysis on X
Figure 4: Sensitivity Study hyperparameters. (a) and (h), we vary r and A on Cora and CiteSeer respectively. In (c) and (d), we conduct sensitive study of and X on four datasets. Please refer to Appendix K for further analysis.
Implementation Details. We adopt two-layer GCN the backbone, with hidden layer size of 128. We set K = 10 clients and draw Pk: ~ Dir(a) from a Dirichlet distribution [40] and assign a fraction pK of class The codebook size is set in the range {26, 2?, 2²). More implementation details and experiments various client numbers can be found in Appendix and Appendix G
client k. Specifically, is set as 0.05 to simulate highly non-IlD senario.
Superiority
To address Q1, we analyze the superior performance of OASIS. We demonstrate the node classification performance with various real-world graph datasets and summarize the generalized test accuracy in Tab. From the table, several key observations can be made (Obs.): Obs. OASIS consistently outperforms other counterparts, with an average improvement of 15.81% % over the baseline. By capturing fine-grained structural information through our synthesizer and codebook, the global model able to acquire more intricate knowledge during distillation. Obs. Traditional FL and FGL methods heavily depend gradual local model updates multiple rounds, and some tailored for personalized optimization, such as FedPub and FedGTA. Therefore, clients fail to efficiently share generalized information within one communication round, leading to the poor model performance.
Obs. 2 One-shot FL approaches such as FedCVAE and FedSD2C perform well in some small graphs. However, they excessively rely class-related semantic signals and overlook the structural information, thus struggling with large-scale graphs.
4.3 Effectiveness
To address Q2, conduct ablation study the key components the Server Graph Knowl- edge Integration part: Semantic Affinity Dis- tillation (WESAD) and the Topological Code- book (WESRD). The experimental results Cora and Amazon- Phozoo are shown in Figure 5 From the bar chart, components improve the model performance significantly. WESAD employs the Wasserstein distance to align the probability distribution be- tween the teacher (local GNN) and the student (global GNN). highlighting the incorporation of inter-class semantic correlations. Meanwhile, WDSRD integrates the fine-grained structural knowledge from the topological codebook. When both WESAD and WDSRD are combined, the performance reaches its peak, with both semantic and structural knowledge effectively distilled to the well-generalin -elereenerlllzabee student global model.
40
observe that both
46
AC00 R0000
Em
(a) Cora
(b) Amazon Photo
Figure 5: Ablation Study of the Semantic Affin- ity Distillation (WESAD) and Topological Codebook (WESRD) on Cora and Amazon-Photo datasets. For an in-depth analysis, please refer to Sec. 4.3
5 Conclusion
In this paper, we introduce OASIS to address two key challenges in existing One-shot Federated Learning approaches with generative models: weak awareness of fine-grained structural knowledge and poor distillation capability from the topological aspect. We first establish Synergy Graph Synthesizer capture complex structural knowledge and then construct structural latent space by

---

introducing the Topological Codebook. On the server side, we propose Wasserstein EaasersteinEEnhanee Se- mantic Affinity Distillation model inter-class relationships and build Wassersteenn-Drvve Structural Relation Distillation to precisely transfer intricate topological knowledge from the codebook to the global model. Extensive experiments on diverse datasets demonstrate the effectiveness of OASIS.
Acknowledgement
This work is supported by National Natural Science Foundation of China under Grant (62361 166629, 622251 13, 623B2080), the Major Project of Science and Technology Innovation of Hubei Province (2024BCA003, 2025BEA002), and the Innovative Research Group Project of Hubei Province under Grants 2024AFA017. The supercomputing system at the Supercomputing Center of Wuhan University supported the numerical calculations this paper.
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A
Notations
We present comprchensive review of the commonly used notations and their definitions in Tab,
Table 2: Notation and Definitions
Notation
Definition
Graph data.
V
The node set of G.
The edge set of G.
X
The feature matrix of G.
A
The adjacency matrix of G.
F
The dimension of the node feature.
D
The dimension hidden embeddings and codebook tokens.
K
The number of clients.
The generated graph for client k.
X
The synthetic feature matrix.
H
The activated similarity matrix.
Á
The synthetic adjacency matrix. Node i in V.
W
The representation of v; at the I-th layer of GNN.
(u:))
The set of neighbours of node Uj
Fok
The local model of client k.
g
The parameters of the local model Fot of client k. The global model. The ultimate global model.
F
F
The parameters of the global model.
R
The optimal knowledge repository.
R
The specialized knowledge repository of client The Gaussian noise.
The topological codebook of client k. The number of tokens in the codebook. The discrete token index in the codebook.
2,
Q
The quantized representations.
The Synergy Graph Synthesizer client k.
Jumi
The balanned distribution.
p
The class probability distribution produced by Fot. The class probability distribution produced by Fzz The temperature.
ps
Ca
The teacher's category.
C
The student's category.
&
The k-th cost matrix. The mass transfered from C2 to Cb.
Gab
W
The weight matrix of the projection head of client k's teacher GNN after l2 normalization. The structural code assignments from the teacher GNN. The structural code assignments from the global GNN. The size of the synthesized graph of client k.
Bk
lkk
Nik
N
The overall size of all synthesized graphs.
Plocal
The local means of the assignment 1.
Pelobal
The global means of the assignment 13,5
The covariance matrix of Bk
The covariance matrix of B
global
14

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B Alignment of the Synergy Graph Synthesizer
To ensure that the synthesized graph faithfully reflects both the statistical distribution of node features and the intrinsic topological patterns of the original graph, we design two-fold consistency mechanism. First, to align the distributions of synthetic and original features, we introduce distribution convergence term:
I F ΣΣ [V] 1=1
s(()) &.(f)'
dist
(f)
(21)
where softmax(2;) and 8; softmax((;) represent the normalized representations of the original and synthetic features, respectively. Second, to enhance local feature consistency and mitigate mode collapse, we incorporate feature dispersion constraint:
I
Espread =
(22)
M
1
Moreover, to simultaneously reconcile structural and feature discrepancies, we employ a fused Kromov-Wasserster loss that quantifies the differences between the original and synthetic graphs within an optimal transport framework:
Lrew
min TUGA)
Σ (a(A(t,5)- A(u,)))
(23)
+ (1 +(-aa)(x(1) - (()) T.
where a E [0, 1] regulates the relative importance of structural and feature fidelity, and the transport plan I is determined under the assumption of uniform marginal distributions and ú. These components are integrated into unified synthesizing objective:
Lsyn = Lspead + de + Ay
(24)
where hyperparameters Aa and 'y control the relative contributions of each term. By leveraging the Synergy Graph Synthesizer, each client generates a refined synthetic graph. However, this approach relies continuous latent space and employs techniques such KNN-based adjacency construction, which primarily capture global statistical trends and ensure overall structural consistency. Consequently, these methods tend to aggregate partial connectivity patterns, inadvertently smoothing fine-grained details and diminishing subtle local nuances.
C Related Work
Federated Graph Learning, Federated Graph Learning (FGL) extends Federated Learning (FL. to graph-structured data, enabling decentralized training while preventing the exposure of raw graph data, thus enhancing privacy protection [14 ? J. Existing FGL methods can be categorized into
intra-graph FGL and inter-graph FGL (65). Inter-graph FGL approaches such as GCFL+ (59) and FedGNN [56] consider settings where clients possess disjoint graphs, such as molecular graphs or independent social networks, and focus on training separate graph models for each client. In contrast, intra- graph FGL assumes that each client holds a subgraph of a globally connected graph, with methods such as FedSSP (50) and FGGP [52] aiming aggregate local updates while maintaining connectivity. However, these approaches struggle to capture fine-grained structural variations and cross-client dependencies. Moreover, their dependence on iterative communication leads to high costs. To address these issues, we propose one-shot FGL approach that removes iterative communication while preserving fine-grain mine-grained structural knowledge through a novel hicrarchical knowledge distillation framework, thereby improving model generalization.
One-shot Federated Learning. One-shot Federated Learning (OFL) paradigm that reduces com- munication overhead by limiting the number of communication rounds to one, making it particularly advantageous for resource- constriennnt and privacy- sensitive environments 12 34.. Unlike traditional federated learning methods that require multiple iterative updates [38 28), OFL eliminates prolonged client-server interactions, significantly reducing latency and computational overhead. Several recent methods seek to enhance OFI. through generative modeling techniques, such as FedDEO [6] and Dense (66], or mnsemble-based strategies, such as FuseFI. (5T) and FENS M. Additionally, knowl- edge distillation techniques have been introduced to facilitate more effective cross-dee cross-client knowledge
15

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transfer in OFL, as seen in FedDF [3] and FedGEMS [6] However, existing OFL methods fail to effectively capture local semanties and fine-grained structural variations federated graph learning. Consequently, we introduce a Synergy Graph Synthesizer to align global features while preserving large-scale structures and a Topological Codebook to model neighborhood variations, improving graph representation in one-shot FGI..
Knowledge Distillation. Knowledge Distillation (KD) is widely utilized for model compression and knowledge transfer [б]. allowing smaller models to perform comparably to larger models while
reducing computational demands. federated learning, methods such FD-FAug [2 60 extend KD techniques to address von-independent and identically distributed (non-IlD) data. Similarly, KD has been leveraged in OFL, preserving knowledge transfer, where clients distill local knowledge into a compact representation for global aggregation. Existing OFL distillation methods can be classified into data-based and model-based distillation [34] Data-based approaches, including DOSFL 77],, FedD3 48 and FedSDD2C [67]. utilize synthetic data to transfer knowledge across clients. Conversely, model- based distillation methods transfer knowledge through latent feature compressions, such as Dense [66] and FedCVAE [13. However, existing methods fail to preserve intricate graph structures, while traditional KD techniques weaken inter-class semantic affinities, limiting their effectiveness in heterogeneous FL. Thus, this study introduces a Hierarchical Wasserstein-basee Distillation framework that aligns semantic affinities and preserves structural dependencies, facilitating efficient and privacy- preeeeriig knowledge transfer.
D Dataset Details.
To assess the effectiveness of, we conduct experiments on eight real-world graph datasets: Cora, CiteSeer, PubMed, Amazon-Phota CoAuthor-CS, Actor, Roman-Empire, and Ogbn-Arxis Each dataset is split into training, validation, and test sets in fixed 20%/40%/40% ratio. The key statistics of these datasets are summarized in Tab. 3 detailed description is provided below:
Cora, CiteSeer, and PubMed. These three citation network datasets are standard benchmarks in graph-based machine learning, especially for tasks like node classification and link prediction. In these datasets, nodes correspond to academic papers, while edges represent citation links. Each node is assigned class label, and its feature vector is constructed from textual information such as words in the title or abstract. These datasets exhibit sparsity and high dimensionality, making them well-suited for evaluating the effectiveness and scalability of graph neural networks (GNNs).
amazon-Phote This dataset is built from the Amazon product catalog, where nodes represent product images and edges indicate co-purchase relationships. Each photo is categorized into specific class, and node features are derived from image metadata. Amazon- serves as a benchmark for testing graph-based learning models in visual domains.
CoAuthor-CS. This dataset represents a co-authorship network in the field of computer science, where nodes correspond research papers, and edges denote co-au ao-auuhorship relations. Each paper is associated with a topic category, and features are extracted from the paper's is title and abstract. This dataset is commonly used evaluate node classification and community detection algorithms.
Actor. The Actor dataset is a heterophilic graph where nodes represent actors, and edges indicate their co-occurrence on the same Wikipedia page. Node features are derived from textual descrip- tions, and classification is performed based on predefined actor categories. This dataset presents unique challenges due to structural differences from traditional citation networks.
Roman-empire. The Noman-empin dataset captures historical relationships in an ancient setting, where nodes correspond different entities, and edges represent interactions between them. The dataset is particularly useful for studying graph-baser algorithms in non non-traditional network structures, offering a distinct perspective on real-world graph learning.
Ogbn-Arxiv. This large-scale citation network is constructed from arXiv papers, where nodes represent papers and edges capture citation links. Each paper belongs to a specific subject category, including physics, computer science, and mathematics. Node features are extracted from paper abstracts. Ogbu-Arxt Arxiv is widely used for benchmarking GNNs due to its diversity and scale.
E
Counterpart Details.
This section provides comprehensive overview of the bascline approaches employed in study.
16

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Table 3: Statisties of datasets used experiments.
Dataset
#Nodes
#Edges
#Classes
#Features
Cora
2,708
5,278
7
1,433
Citeseer
3,327
4,552
6
3,703
Pubmed
19,717
44,324
3
500
Amz-Photo
7,650
287,326
8
745
Coauthor-CS
18,333
327,576
15
6,805
Actor
7600
30,019
5
932
Roman-empire
22,622
65,854
18
300
Obgn-Arxiv 169,343
1,166,243
40
128
FedAvg [ASTAT17]. foundational algorithm in Federated Learning, FedAvg operates by allowing clients to independently train models on their local datasets and subsequently transmit their model updates a central server. The server performs a weighted aggregation of these updates to refine the global model, which is then redistributed to the clients for further local training. By transmitting only model parameters instead raw data, FedAvg reduces communication costs and enhances privacy. However, it struggles with performance degradation in scenarios where client data distributions are highly non- 29 39].
FedProx [MLSys20]. As an enhancement of FedAvg, FedProx is specifically designed to address the challenges posed by statistical heterogeneity in federated learning. introduces an additional regularization term that constrains local updates, preventing excessive divergence from the global model. This proximal term mitigates the impact of local data distribution shifts, leading to more stable convergence. By ensuring consistency in updates across clients, FedProx demonstrates improved robustness in non-lID settings.
FedNova [NeurIPS20]. FedNova refines the FedAvg framework by introducing normalization local updates before aggregation. Unlike standard averaging methods, FedNova ensures that each client's s contribution the global model is proportional to the amount of data it possesses. This approach addresses the issue of unequal client influence, leading to more balanced and efficient convergence. FedNova is particularly beneficial federated environments where data distributions are skewed across clients.
FedRCL [CVPR24]. FedRCL incorporates contrastive learning to improve federated learning performance under data heterogeneity [63] 53.. It examines inconsistencies in gradient updates across clients and attributes them variations in feature distributions. To counteract this, FedRCL employs contrastive regularization strategy that penalizes overly similar samples within a class, ensuring diverse and transferable feature representations. This approach enhances collaborative learning among clients and leads to notable performance improvements.
FedPub [CCML23] Unlike traditional FL methods that focus on training a single global model, Fed- Pub adopts personalized approach by facilitating the interaction of local Graph Neural Networks (GNNs). It employs functional embeddings to quantify similarity between client models, enabling adaptive weighted aggregation the server. Furthermore, sparse mask mechanism allows parameters, improving both privacy preservation and learning efficiency in heterogeneous graph scenarios.
clients to selectively update
FGSSL [ICCI]23] FGSSL addresses local client distortion caused by both node-level semantics and graph-level structures. It improves discrimination by contrasting nodes from different classes, aligning local nodes with their global counterparts of the same class while pushing them away from different classes. To handle structural information, transforms adjacency relationships into similarity distributions and distills relational knowledge from the global model into local models. This approach preserves both structural integrity and discriminability, achieving superior performance on multiple graph datasets.
FedTAD [ICAI24]] Designed address subgraph heterogeneity in federated learning, FedTAD de- composes variations in local graphs into differences in label distributions and structural homophily. By analyzing these discrepancies, prevents misleading model aggregation, which can occur when local models contribute inconsistently. The framework enhances knowledge transfer through topology-aware knowledge distillation, improving both reliability and aggregation efficiency in FL settings.
FedGTA [VLDB24).. FedGTA tailored for large-scale graph federated learning, tackling issues of slow convergence and suboptimal scalability, Unlike prior methods that focus either optimization
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strategies or complex local models, FedGTA integrates topology-ave aware local smoothing with mixed neighbor feature aggregation improve learning efficiency 72.. By leveraging graph structures in aggregation, it enhances scalability and performance in federated graph learning.
DENSE [NeurPPS22]. framework designed to overcome limitations of conventional one-shot FI., DENSE eliminates the need for additional auxiliary datasets or model information by employing a two-stage learning process. It first synthesizes data representations and then applies model distillation to refine the global model. This approach ensures that a federated model can be effectively trained in a single round of communication while accommodating heterogeneous client architectures.
FedCVAE [ICLR23]. A Hata-free shot FL method, FedCVAE utilizes a Conditional Varia- tional Autoencoder (CVAE) [8] 24 23 to improve generalization under statistical heterogeneity. The approach reframes local learning objectives, allowing effective global aggregation despite distribution disparities. An extended variant, FedCVAF-KD. incorporates knowledge distillation
11 to consolidate local decoders into a unified global model. FedCVAE outperforms traditional baselines, particularly in highly heterogeneous settings.
FedSD2C [NcurIPS24]. FedSD2C is a novel one-shot FL framework that mitigates performance degradation caused by data heterogeneity. It leverages distillation-based strategy to synthesize informative data representations directly from local distributions, bypassing the inconsistency issues present in conventional model aggregation. By sharing distilled representations instead of raw model updates, FedSD2C enhances knowledge transfer and ensures greater consistency across federated clients.
FENS [NeurIPS24). FENS is novel approach to OFI. that aims to bridge the accuracy gap between standard federated learning (FI.) and OFI. while maintaining high communication efficiency. FENS employs two-phase learning process: first, clients train local models and send them to the server, as in OFL; second, clients collaboratively train lightweight prediction aggregator using FL. Extensive experiments demonstrate that FENS achieves performance close to FL while preserving the efficiency of OFL.
F Implementation Details.
The experiments are conducted using NVIDIA GeForce RTX 3090 GPUs as the hardware platform, coupled with Intel(R) Xeon(R) Gold 6240 CPU @ 2.60GHz. The deep learning framework employed was Pytorch, version 1. alongside CUDA version 12.1. We adopt a two-layer GCN as the backbone, with a hidden layer size of 128. Moreover, we utilize 3 hidden linear layers and projection head as the synergic graph synthesizer which concats random noise and one-hot label as the input and generates pseudo features as the output. We set K 10 clients and draw pk ~ Dir(a) from a Dirichlet distribution [40] and assign a fraction Pf of class to client k. As for optimization of the graph synthesizer, Adaptive Moment Estimation (Adam) was chosen, featuring a learning rate of 5e 3 and weight decay of 4e 4. The codebook size is set in the range (25; 27, 2²}. with the same optimizer and learning parameter. At the local training phase, we set the training epoch Ts of the synthesizer to 100 and epoch Tc of the teacher GNN and the codebook to 50. X and Xy are determined through a grid search [32] within {0.01, 0.05, 0.1, 0.5) and (0.1, 0.2, 1} respectively. 1J, Xo are set as 0.25, 0.01 and r is set to 1. To make sure that LFOW is on the same scale as other loss functions for Ano-Photo and Ogbn- Arxiv datasets, we set their ' scales to le - and le - 7, respectively. We set in Lrow 0.5 to balance the feature part and the structure part. The communication round is limited to one. At the server side, we determine the global distilling epoch in the range {10, 20, 30) and adopt Adam the optimizer for the global model with learning rate of le- - 2 and a weight decay of 4e -
Ogbn-Arxiv
4. The synthesized graphs generated by the synthesizer of each client have the same scale N as the corresponding local subgraph. For the large graph Ogbn-Arxiv, we set To to 1 and * to one-tenth of the local graph. Moreover, Ag, A is set to 0.01, 1 and X₂ is determined in range 51 10]. The distillation temperature 7 is set for all datasets.
G Ablation Study on Different Numbers of Clients.
In this section, we vary the number of clients (5. 10, 20) and conduct the node classification task on CiteSeer, Amazon-Phae Photo. Experimental Results are shown in Tab. + From the table, we can observe that our OASIS outperforms all counterparts with different numbers of clients, demonstrating the stability of OASIS across various data distributions and subgraph scales. As we simulate a highly
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Table 4: Comparison with the oftte-of-thh-art methods with different numbers of clients. We report node classification accuracies (%) for downstream task performance. Green arrows denote advancements in accuracy metrics than FedAvg while arrows indicate regressions. OOM means -emoor error. The best and second results are highlighted with bold and underline. respectively.
Datasets
CiteSeer
Amazon-Phoot
Category
Methods (4)
5 Clients
10 Clicuts
20 Clicuts
Clicats
10 Clients
Clients
FodAvg [ASTAT17]
35.43
32.58
38.13
19.93
23.12
21.79
FedProx Syx
40.32
35. 35.73
39.39, 11.28
46.54
24.16;1.04
23.58
FedNova [NeurIPS20]
18.43 (17000
18.58 114.50
16.17 L23.80
9.69,
6.15 (16.:16.0
8,57 (13.22
FedRCL [CVPR24]
15.88, (5.88.19.56
12.73. 17.3.20.10
7.05
1.0.,811.8
21.80
4.92, 1182,0
10.69. (11.10
FedPub [CML223]
20.07.116.1 36
34.9
14.00
32.27.5.80
32.27
16.88
42.00,7.98
21.91 .1/21
21.7610.08
FGSSL [ICA123]
20.82, (14.01
21.95 (1.5.10.00 93
34.79.a.a 1a.33
41.64
13.06 10.00
22.288oo.o
FGL
FodGTA [VLDB24)
16.93 (18.80
17.76.16.18
15.73, /22.40
5.39.44.54
(44.54
4.10,
6.17
FodTAD [UCA124]
19.25
33. 33.8660099
28.34,9.79
25.29
22.01 22.01.1.11
21.37,0.22
DENSE [NcurPS22]
7.64 1ar.
7.87 125
7.06 (81.07
4.96
1.93, (1..5.
4.90
FedCVAF, [ICIR23]
40.33, 4.90
34.76-1.s8
18.10.20.08
52.39 (2.64
31.62,8.50
24.87
OFI.
FedSD2C [NeruIPS24]
20.97 114.46
29.96 (2.02
23.66 .14.47
11.75
1.775,38.0
8.73, 114.89
23.3841.09
[NemiPS24]]
11..98
.13.91
17.95_20.15
22.26
127.67
25.30
25.53
OPGL
0ASIS
40.45 40.45:5.02
45.69
47 -7.92.9.77
71.18 -21.25
63.73
54.11,32.8
non-IID scenario using a Dirichlet distribution with small concentration parameter a 0.05, increasing the number of clients may lead to situations where some clients possess very limited data or even no training samples at all, which does not fully align with real-world settings. Nevertheless, shown in Tab. OASIS consistently maintains stable and effective performance trend, even under the presence of a larger number clients.
H Performance of OASIS under moderate or mild heterogeneity
To demonstrate the performance of OASIS under moderate or mild heterogeneity (or even i.i.d. data), we have conducted additional experiments with varying levels of data heterogencity, Specifically, we evaluate the performance of DASIS under different values of the Dirichlet distribution parameter E {1, 10, 100, 1000, 10000} to simulate range of heterogeneity scenarios. These experiments are performed on the Cora dataset, and we adopt FedAvg, FedPub, FedCVAE and FedGCN(I-hop 2-hop) for comparison. Notably, when o= 1=10000, the data distribution approximates i.i.d. setting.
Table 5: Performance of OASIS under milder heterogeneity (or even i.i.d.) scenarios.
100
1000
10000
FedAvg
49.50
56.23
57.93
57.84
45.11
FedPub
39.87
44.96
60.13
52.70
58.55
FedCVAE
60.45
56.50
59.58
55.27
56.63
FedGCN (1-hop)
52.06
41.39
50.60
51.97
56.91
FedGCN (2-hop)
58.66
52.11
55.82
57.56
61.76
OASIS
63.15
75.64
76.35
69.11
68.44
As shown Tab. 5 OASIS consistently outperforms the existing methods across various heterogeneity levels, including strong, moderate, mild, and even i.i.d. conditions. The advantage of OASIS remains robust even the heterogeneity decreases, demonstrating that its performance is not diminished by a shift towards more homogeneous data. We will ensure that this additional experimental result is included in the revision.
The reason we choose to focus on strong heterogeneity our experiments that, real-world scenarios, non-uniform data distribution is more prevalent and presents greater challenge. We aim to showcase that DASIS not only performs well in typical i.i.d. and mild heterogeneity settings but also excels in more challenging scenarios with high heterogeneity.
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I Performance of OASIS on other prominent GNN models
To demonstrate the performance of OASIS on other GNN models, conducted experiments on the Cora and CiteSeer datasets using GAT and GraphSAGE respectively. We compare OASIS against five baselines, Results are shown in Tab. 6
Table 6: The performance of OASIS on GAT and GraphSAGE.
Methods
Cora-GAT
Cora-GrraphAAGE: CiteSeer-GAT StteSeer-GraphSAree
FedAvg
30.98
29.88
30.49
37.23
FedProx
31.71
29.97
27.94
39.40
FGSSL
29.79
27.45
20.52
20.22
FedCVAE
29.61
25.94
22.05
23.52
FENS
29.70
29.88
21.05
21.50
OASIS
42.62
39.96
38.73
44.87
From the table, we can observe that our DASIS consistently outperforms other baselines in both GAT and GraphSAGE backbones, demonstrating the capability of OASIS to generalize to other prominent GNN models.
Mathematical Analysis of OASIS
Here we provide thorough analysis on the mathematical bound on information retention of the synthesizer and the impact of global distillation.
J.1 Mathematical Bound on Information Retention
Let Gi = (Vi, Et, X1) be the local graph, where Vi is the node set, & the edge set, and X₁ E RIM xd the node feature matrix. The synthesized graph is G. (V., E., X.). with X, E IR [ and adjacency matrix A.. The goal is to ensure that Gs preserves the critical information in Gi, including node feature distributions and topological structures.
To quantify the retention of critical information, we derive bound the divergence between the local graph G. Di and the synthesized graph G, ~ D., using the Synthesizer's 's loss as a proxy. We measure the divergence between their distributions using combined metric that accounts for both feature and topological differences.
Step 1: Feature Distribution Divergence. The KL divergence loss Laist = DRI. (Px. Px.)) directly measures the feature distribution mismatch. By Pinsker's inequality, the total variation distance bounded by:
8rv Siv(Px,,P)) S DKa (xx, P
V Laia-
V22 -
Assuming Ldist S ed, the feature distributions are close in total variation:
cd
STv Px.) S V22
Step 2: Feature and Topological Alignment via FGW. The Fused Gromov- Wasserstein loss Ligw bounds the combined feature and topological mismatch. The FGW distance can be decomposed
Lign = FGW G.) aW2(Px, (1-
where W is the squared 2- Wasserstein distance between feature distributions, and GW is the Gromov-Wassersten Wasserstein distance between structures. Assuming Ligw S cf. we have:
aPx, (LA Sec.
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Since both temms are non-negative, we get:
Vx,, 1 GW(A, & cf 1
The Wasserstein distance further bounds the feature distribution divergence, and the Gromov- Wasserstein distance bounds the structural discrepancy.
Step 3: Feature Diversity. The cosine similarity loss Lspread ensures that the synthesized features do not collapse. We assume Lspred 2 -1, where > is a constant reflecting sufficient feature spread. This prevents degenerate solutions where X X for all i.j.
Step 4: Combined Bound. To derive bound on the overall divergence between Dj and D,, we consider joint metric that combines feature and topological differences. Define graph divergence metric:
AG 9.) = W2(Px, Px.) GW(A,,
From the FGW loss:
A,) & C$ + 1-a (Vaaaaa
Cf =q
A9 S GW(A,
The KL divergence provides an additional constraint on features. Combining with Pinsker's 's inequality, the total variation distance features is:
Cd 2
OTv S
Since Wasserstein and total variation distances are related (e.g., via transport inequalities in bounded spaces), we focus on the FGW-based bound for simplicity, as it captures both features and topology.
Theorem Assuming the Synthesizer's loss is bounded as Layn with Ldist S Ed, Legw S Et, and Lepread 2 -7) the divergence between the local and synthesized graph distributions is bounded as:
D.)
(Va a-a) + TV2
where A(D,, D.) = EE,,D,, W2 Px.) + GW(AI, A.)].
Proof. The total synthesis loss is:
Layn Lapread + >DaLdist + Ay
Since Lipread 2 -77, have:
AaLdist + Ly S 1). Assume T-iist 6ds Tegw ey, with Aaea + Ayef S + The feature divergence is bounded via
Pinsker's inequality:
8TV Pxx(Px,PPx))
The FGW loss bounds the combined feature and topological divergence:
aPx Px.) (1- Seg.
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Thus:
_ GW(A, i
Ef 1-a
Px. V
The total divergence is:
D.), & Px.) + GW(,,, A.)] =
of cf
1-a
Incorporating the feature distribution bound from Luist we add the total variation term for complete- ness, yielding:
A(D, c
(Va+1-a) +
(Va 1 a
V2
Ed
This bound quantifies the retention of critical information: small ef and ensure that the synthe- sized graph's feature and topological distributions are close to those of the local graph.
The divergence between the local and synthesized graph distributions is bounded by:
T (Va+1-0)
A(DI, D.) D,) sc
where (D; D.) combines feature and topological differences. This bound rigorously quantifies the retention of critical information, ensuring that the synthesized graph effectively captures the local graph's semantic and structural properties when eœ and ε, are small.
J.2 Impact on Global Model Generalization
The retention critical information and the distillation mechanisms directly influence the generaliza- tion performance of the global model. Our OASIS employs two distillation mechanisms to transfer local knowledge to the global model:
WESAD: Transfers inter-class semantic relationships.
WDDRDD: Transfers fine-grained topological structures.
Generalization Bound. quantify the impact on generalization, we derive a bound on the global model's expected error using the Rademacher complexity framework, adapted for federated graph learning.
Let D, denote the local data distribution for client I, and D, the distribution of the synthesized data. The global model fy F hypothesis class of graph neural networks) is trained on synthesized data to minimize the empirical risk:
(.)) = Ep, (G.,
where l is the loss function (e.g., cross-entropy), and y the label. The risk is:
R(fg) ED, ((
The generalization gap - Assuming the synthesized graph retains critical information (i.e.. W1 (D,, D.) c), bound the generalization error using the Wasserstein distance and Rademacher complexity.
Theorem 2. For a hypothesis class F with Rademacher complexity (F) over samples, and assuming the loss function T.-Lipschitz, the expected generalization error of the global model bounded as:
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log(1/8) n
[(R(Fg)] S + Le + CV
with probability at least 1 - 6, where == (i(D,, P.) is the Wasserstein distance between local and synthesized distributions, and C is constant.
The term Le quantifies the impact of information retention: if is small (i.e., the synthesized graph closely matches the local graph), the generalization error tightly bounded.
Proof. By the Wasserstein distance property, the difference expected loss is bounded:
( Eb, (U,, & (W,(P,,D)) - Le.
Using standard generalization bounds for empirical risk minimization:
log (1/8)
B[R(G)] (.2))) + (F) + Cy
n
Combining these, we obtain:
B[R(G)) S RR,))) (F).((F) + Le + +CV log log(1/8)
K Sensitivity
To address Q3, conduct analyses on hyperparameters of OASIS. Specifically, compare the model performance under different values of temperature 7 and Ae- We vary and X in range [1, 4] and [0.25, 1] with and 0.25 as the step size respectively. Moreover, as for the Topology Codebook. we vary and A in range [0.2, 0.4] and [0.01, 0.05] with 0.05 and 0.01 as the step size. Results shown in Figure demonstrate that the performance of OASIS stays consolidated under different hyperparameter values, proving the robustness of DASIS.
L Privacy Security
L.1 How to prevent malicious attackers from stealing communication data?
Privacy security plays crucial role in FGL systems. In Sec. 3.2 propose novel Synergy Graph Synthesizer Q.p' to generate powerful synthesized graphs with labels and Gaussian noise e. For simplicity, a standard normal distribution is typically utilized, where the center Ω is we introduce shift to the center 52 at the client side and communicate ç to the server either offline or through eneryption methods [5] 70 We consider the worst-case scenario where an eavesdropping attacker intercepts all the parameters. However, without knowledge of the specific shift c, the attacker can only utilize the original center Ω which is distant from the true center (Ω + s). Alternatively, the attacker might attempt overlap the center using a wide uniform distribution.
However,
To simulate these situations, we first set the original noise distribution center Ω = with shift varies in {30, 60, 90} while the attacker still takes = O. We conduct all experiments here on the node classification task with the knowledge distillation part excluded explicitly demonstrate the influence of the generated data. Results are shown in Figure ba From the bar chart, we can observe that the model performance sharply declined by 60%, proving that the data generated with center Ω are totally different from local data. Moreover, to simulate the overlapping attempt, we set Ω + ş = and the overlap range in (0(-100, (40),4((-20) 200, 200), (00,0000300 and conduct the experiment the same setting. Results are shown in Figure (b]] with the same phenomenon observed. Therefore, prove the security of data privacy of our OASIS.
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18
(a) Shifted Center
(b) Overlap Attempt
Figure Privacy Study of DASIS. Original here means the server generates the graphs with the distribution center as clients. For an in-depth analysis, please refer to Appendix L..
L.2 How to prevent a curious server from inspecting client privacy?
Although we design an encrypted shift ş for the distribution center S2, a curious server can still utilize the correct distribution to synthesize data. However, the true local data information remains protected and is not exposed the server, due to the following reasons:
(1) Discrepancy in Label Distributions. In practical FGL scenarios, the label distribution across clients is typically highly non-identical. In experiments, simulate extremely non-IID setting by sampling labels from a Dirichlet distribution with concentration parameter a = 0.05. Importantly, clients do not transmit their actual label distribution the server. Instead, the data synthesis is conducted under the assumption of a uniform label distribution Juni (Sec. 3.3). Consequently. the overall label distribution of the synthesized graph the server deviates significantly from that of any individual client, thereby mitigating the risk of direct data leakage.
(2) Structural Dissimilarity of Nodes. Even for synthesized nodes that share the same label as some local data, their structural context (e.g.. neighborhood connectivity) is distinct. As described in Equation (5). each node in the synthesized graph is connected to its five most similar nodes based on feature similarity. This fixed K --nearstt-niihhbor (KNN) construction introduces structural differences compared to the true graph topology of local data. Although our alignment module (Appendix B encourages the preservation of structural semantics via Lraw, it does not enforce strict local topological isomorphism. Therefore, structural privacy is preserved
considerable degree.
(3) Feature Perturbation via Two-Fold Consistency Mechanism. The synthesizer is trained with a two-fold consistency objective Lsyn involving both feature and structure alignments (Equation (24)). This objective steers the optimization away from directly replicating raw node features. As a result, even nodes with the same labels as in local data will exhibit distinct feature representations in the synthesized graph.
In summary, privacy preservation in our framework is inherently balanced with the learning objectives of the Synergy Graph Synthesizer. Through the structural regularization from the KNN construction and the two-fold consistency mechanism, we ensure that the synthesized data avoids leaking sensitive client information, while still capturing high-level latent knowledge required for effective global model learning. This design satisfies the fundamental goals of federated graph learning.
M Discussion on Limitations.
While DASIS achieves notable success in efficiently capturing fine- fne-grained structural knowledge of local graphs and effectively transfer the knowledge during server distillation in the one-shot scenario, still has inherent limitations as sythesizer-basee method [49] In particular, the presence of noise in local data [22] can impair the ability of the synthesizer to effectively extract and learn local-specific patterns, which may in turn impede the distillation module. Improving the robustness of the Synergy
Graph Synthesizer against such noise interference 35 26 remains promising direction for future work.
N Discussion on Broader Impacts
Our proposed DASIS framework contributes to the broader field FGL by enabling more efficient and
collaboration across decentralized graph datasets, especially under stringent communication constraints. By introducing domain-specific generative strategies and novel structural distillation techniques, DASIS opens possibilities for applying OFL to graph-structured data such
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as social networks, biomedical graphs, and knowledge graphs, where data are inherently sensitive and dispersed. This advancement can benefit applications involving privacy-critical domains like healthcare, finance, and cybersecurity, empowering institutions to jointly learn high-quality models without exposing private graph data. As FI. technologies become more widely adopted, continued attention to the ethical implications of synthetic data generation and the interpretability of structural knowledge transfer will be essential to fostering responsible AI deployment.
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Reviews:
[{"review_id": "Rd9qbsPqEH", "reviewer": "Reviewer_BhNM", "summary": "This paper investigates a novel approach to training Graph Neural Networks (GNNs) within the One-shot Federated Learning (OFL) paradigm. While prior OFL methods have primarily focused on image-based tasks, their direct application to graph-structured data often proves ineffective due to fundamental differences in data characteristics. To address this gap, the authors propose a new framework that synthesizes informative graphs and introduces a topological codebook to better capture graph-specific structural patterns. Furthermore, the framework leverages WESAD and WDSRD mechanisms to enable efficient structural knowledge transfer at the server level. Experiments are conducted to demonstrate the effectiveness of the proposed method.", "questions": "Please address my concerns in the weaknesses above.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n1. This work tackles the novel and challenging task of One-shot Federated Graph Learning (FGL), introducing a dedicated framework tailored to this setting.\n2. The paper is well-written with detailed and informative figures (e.g. Figure 2 & 3) to clarify the motivation and the framework. \n3. The mathematical formulations are detailed and precise, offering a rigorous description of the model components and training objectives, which facilitates reproducibility and theoretical understanding.\n4. The proposed method is rigorously evaluated through comprehensive experiments, benchmarked against a variety of baselines, including conventional FL, existing FGL techniques, and other OFL approaches. Experimental results consistently demonstrate the superior performance of the proposed framework over competing methods.\n\nWeaknesses:\n1. The authors need to further clarify the relationship between WESAD and WDSRD on the server side. Why is it necessary to combine these two distillation schemes?\n2. Most existing and widely used knowledge distillation methods adopt KL divergence as the loss function. In contrast, this work utilizes a combination of WESAD and WDSRD for knowledge distillation. What are the fundamental differences between these approaches and KL divergence? In other words, for Eq. (10) and Eq. (19), one could theoretically replace them with KL divergence. I hope the authors can explain the distinctions and potential advantages of their proposed distillation strategy over the standard KL-based approach. If possible, including a small comparative experiment would strengthen the argument.", "was_revised": false, "final_justification": "", "rebuttal": "***Dear Reviewer BhNM:***\n\nWe sincerely appreciate your insightful review and encouraging evaluation of our work. We are thankful for your acknowledgment of the novelty and significance of OASIS in advancing One-Shot Federated Graph Learning. Below, we provide detailed responses to your comments and questions.\n\n\n\n**`Weakness 1`: The relationship between WESAD and WDSRD needs to be further clarified.**\n\nThank you for your insightful question. We clarify that WESAD and WDSRD serve two complementary but fundamentally distinct purposes in server-side knowledge integration, and their combination is essential for fully transferring fine-grained knowledge to the global model.\n\nSpecifically, WESAD distills semantic knowledge, focusing on the alignment of class-level output distributions between local teacher and global student models. By leveraging a discrete Wasserstein distance informed by semantic affinity, WESAD allows **soft** matching across semantically related categories, which is particularly useful in federated settings where label distributions can vary across clients.\n\nIn contrast, WDSRD aligns node-level structural patterns between the teacher and student models by comparing their soft assignments (computed by the distance between node-level representations and topological codebook tokens in eq.(15)). Through the closed-form Wasserstein distance between Gaussian distributions, WDSRD captures both mean-level and covariance-level discrepancies in the assignment distributions, which reflects richer structural variation than simply aligning logits or embeddings.\n\nThe combination of WESAD and WDSRD enables a holistic transfer: WESAD ensures that the global model learns to make semantically consistent predictions across diverse clients, while WDSRD ensures that the global model preserves important topological patterns from local graph structures. Our ablation studies in Section 4.3 confirm that omitting either component significantly degrades performance, demonstrating their necessity and complementarity.\n\n**`Weakness 2`: The distinctions and potential advantages of WESAD & WDSRD strategy over the standard KL-based approach.**\n\nThank you for your concern. We appreciate the chance to clarify:\n\nKL divergence operates under a strict one-to-one alignment assumption, penalizing any discrepancy between predicted probabilities regardless of semantic proximity. This can be suboptimal when semantic overlap exists across categories, which is often the case in heterogeneous clients. In contrast, WESAD introduces a semantic-aware optimal transport (OT) formulation that allows probability mass to flow between semantically similar categories, guided by a learnable affinity matrix. This soft matching provides more flexibility and better generalization, especially under label distribution shift. \n\nFor structural knowledge, applying KL divergence directly to the node-to-code assignment distributions would only account for first-order differences and ignore second-order statistics, such as the shape and spread of the structural representations. In contrast, WDSRD models these assignments as Gaussian distributions and uses the closed-form Wasserstein distance, which simultaneously aligns both the means and covariances of local and global distributions. This richer alignment captures the inherent variability and uncertainty in node structure assignments, leading to better preservation of topological information.\n\nTo further support our theoretical analysis, we conduct a comparative experiment on Cora, CiteSeer, Coauthor-CS and Roman-Empire datasets. Specifically, we compare our proposed distillation strategy (WESAD & WDSRD) with the standard KL divergence-based distillation. The results are shown in Table 1.\n\n*Table 1: Comparison of Wasserstein-based distillation and KL divergence-based distillation.*\n|Distillation Methods|Cora|CiteSeer\t| Coauthor-CS | Roman-Empire |\n|------|------|-------| ---|----|\n|KL Divergence\t|48.12\t|38.20|59.18 |28.55 |\n|WESAD & WDSRD (ours)|**49.59**|**45.69**| **60.44**| **30.07**|\n\nAs shown, our distillation strategy enables more fine-grained and efficient transfer of both semantic and structural knowledge, leading to improved generalization of the global model.\n\n\nThank you again for your supportive feedback. We hope our explanations address your concerns. If you have any further questions or suggestions, please feel free to let us know.\n\nBest Regards,\n\nAuthors"}, {"review_id": "nKk3kRkfIO", "reviewer": "Reviewer_Eosq", "summary": "The paper introduces OASIS, a one-shot federated graph learning framework that enables participants to collaboratively train a global graph neural network in just a single round. The authors propose a series of innovations tailored to the two primary phases of OASIS: local data synthesis and global model distillation. Experimental results demonstrate that OASIS achieves better performance than existing works.", "questions": "1. The reported experimental results appear relatively low compared to existing methods. For instance, the baseline approach FedAvg achieves only about 30% accuracy on the Cora dataset, whereas under similar settings with ten clients, it typically achieves around 60% accuracy as reported in [3]. Similar performance gaps are also observed across other datasets and baseline methods. Could the authors provide an explanation for this observed disparity in performance?\n\n2. A significant challenge in federated graph learning is the issue of missing edges. Can the proposed approach address this problem?\n\n[3] FedGCN: Convergence-Communication Tradeoffs in Federated Training of Graph Convolutional Networks", "limitations": "1. The loss function used to train the Synergy Graph Synthesizer is crucial for comprehending the proposed approach. This important detail should be presented in the main text rather than relegated to the appendix.", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "- Strengths\n1. The paper addresses an important topic—one-shot federated graph learning—which extends federated learning methodologies to graph data.\n\n2. The paper is clearly written, making it easy to follow and understand.\n\n3. The experimental evaluation is thorough, covering 8 different datasets and comparing against 12 related state-of-the-art methods.\n\n- Weaknesses\n1. The overall computation complexity of the proposed approach remains unclear. The approach adopts Wasserstein loss repeatedly, which would incur substantial computation overhead, potentially restricting its application to larger graphs. it would be beneficial to explicitly clarify the overall complexity and provide a comparative analysis with existing methods.\n\n2. Lack of theoretical analysis. Specifically, it remains unclear how each component contributes individually to task performance, particularly in the Graph Knowledge Integration method.\n\n3. The experiments exclusively utilize a 2-layer GCN model. It is unclear how the proposed method would generalize to other prominent graph neural network models, such as GAT[1] and GraphSAGE[2].\n\n[1] Graph Attention Networks\n[2] Inductive Representation Learning on Large Graphs", "was_revised": false, "final_justification": "", "rebuttal": "***Dear Reviewer Eosq:***\n\nWe sincerely appreciate your engagement with our work. We address your concerns below and hope these clarifications will positively contribute to your assessment of our work.\n\n**`Weakness 1`: Lack of the clarification of overall computing complexity of OASIS and the comparative analysis with existing methods.**\n\nThank you for your thoughtful comments. We will provide a detailed explanation addressing the concerns raised regarding the computational complexity of our method, particularly with respect to the Wasserstein-based losses in OASIS.\n\n1. Fused-Gromov Wasserstein (FGW):\nIn our paper, we only introduced a preliminary version of the FGW distance. Actually, the existing method proposed by Qian et al. (2024) [1] already introduces an efficient approximation for the FGW distance, which drastically reduces its computational complexity. Specifically, their approximation method enables GPU acceleration, improving computational efficiency by reducing the complexity to \n$O(N+E)$, where $N$ is the number of nodes and $E$ is the number of edges . As shown in Table 1, this approach maintains high accuracy while being scalable for large graphs.  We will further discuss about [1] and add the reference in our revision.\n\n*Table 1: Performance of the approximated FGW distance in OASIS.*\n\n|Method|Cora|CiteSeer|Roman-Empire|\n|-|-|-|-|\n|FedAvg|30.61|32.88|18.49|\n|OASIS|49.59|45.69|30.07| \n|OASIS(approx FGW)|48.49|44.88|29.30|\n\n[1] Reimagining graph classification from a prototype view with optimal transport: Algorithm and theorem.\n\n2. Wasserstein-Enhanced Semantic Affinity Distillation (WESAD):\nFor WESAD, we note that its computational complexity is approximately \n$O(C^2)$ where $C$ is the number of classes. Given that \n$C$ is typically small in graph datasets (e.g. Cora: 7 classes; Ogbn-Arxiv: 40 classes), the overhead associated with this term is quite minimal. \n\n3. Wasserstein-Driven Structural Relation Distillation (WDSRD):\nThe complexity of WDSRD is primarily $O(M^2)$, where $M$ is the number of codebook tokens. In practice, the codebook size is kept small (e.g., 64 or 128 tokens) as shown in Section 4.1. \n\nIn conclusion, the overall computing complexity of OASIS is $O(eL(N+E)d + C^2 + M^2)$. Moreover, we also conduct a comparative analysis of the overall computing complexity with three baselines: FedProx, FedTAD, FedCVAE in Table 2. \n\n*Table 2: The comparative analysis of the overall computing complexity. ($e$: training epochs, $L$: number of layers, $N$: number of nodes, $E$: number of edges, $d$: hidden dimension, $B$: batch size, $C$: number of classes, $M$: number of codebook tokens)*\n|Methods|Overall Computing Complexity|\n|-|-|\n|FedRCL|$O(eL(N+E)d+B^2+Bd)$\n|FedTAD|$O((eLd+p)(N+E))$|\n|FedCVAE|$O(eL(N+E)d + Nd)$\n|OASIS|$O(eL(N+E)d + C^2 + M^2)$|\n\nAs shown in Table 2, OASIS has a computational complexity comparable to existing methods while achieving significant performance improvements. We hope the above explanation addresses your concerns.\n\n**`Weakness 2`: Lack of theoretical analysis of how each component contributes individually to task performance.**\n\nThank you for your valuable feedback. In **Section 4.3** and **Figure 5**, we have conducted a detailed ablation study to assess the individual impact of each component in the Graph Knowledge Integration module. From Figure 5, it is evident that both WESAD (Wasserstein-Enhanced Semantic Affinity Distillation) and WDSRD (Wasserstein-Driven Structural Relation Distillation) contribute significantly to the overall performance. \n\nMeanwhile, we appreciate the opportunity to further provide theoretical insights into how WESAD and WDSRD function individually and synergistically. \n\nWESAD  is designed to transfer semantic knowledge in a more fine-grained manner than traditional distillation approaches. WESAD leverages the discrete Wasserstein distance to reallocate probability mass across semantically related categories. The resulting optimization not only ensures that the global model mimics the local predictions but also encourages it to internalize the relational structure between categories, thereby enhancing generalization in heterogeneous scenarios.\n\nWDSRD, on the other hand, focuses on distilling structural knowledge by aligning latent codebook assignments between the local and global models. By encoding graph topology knowledge into a topological codebook and modeling their assignment distributions as Gaussians, WDSRD captures both the mean and covariance of structural patterns. The use of a closed-form Wasserstein distance between these Gaussian distributions enables an expressive yet computationally tractable alignment. \n\nTogether, these two components form a theoretically grounded and complementary mechanism: WESAD transfers soft semantic correlations across categories, while WDSRD preserves latent structural dependencies across nodes. Their combination ensures comprehensive graph knowledge integration and significantly boosts the robustness and generalization ability of the global model.\n\n**`Weakness 3`: The performance of OASIS on other prominent GNN models, such as GAT and GraphSAGE.**\n\nThank you for your concern. To demonstrate the performance of OASIS on other GNN models, we conducted experiments on the Cora and CiteSeer datasets using GAT and GraphSAGE respectively. We compare OASIS against five baselines. Results are shown in Table 3.\n\n*Table 3: The performance of OASIS on GAT and GraphSAGE.*\n\n|Methods|Cora-GAT|Cora-GraphSAGE|CiteSeer-GAT| CiteSeer-GraphSAGE |\n|-|-|-|-|-|\n|FedAvg|30.98|29.88|30.49|37.23|\n|FedProx|31.71|29.97|27.94|39.40|\n|FGSSL|29.79|27.45|20.52|20.22|\n|FedCVAE|29.61|25.94| 22.05|23.52|\n|FENS|29.70|29.88|21.05|21.50|\n|OASIS|**42.62**| **39.96**|**38.73**|**44.87**|\n\nFrom the table, we can observe that our OASIS consistently outperforms other baselines in both GAT and GraphSAGE backbones, demonstrating the capability of OASIS to generalize to other prominent GNN models.\n\n**`Question 1` : Explanation for the performance drop of compared baselines.**\n\n\nThank you for your insightful comment. In Section 4.2, we have already discussed this issue, and we would like to provide further clarification here.\n\nFirst, our experiments are conducted under **extreme data heterogeneity**, modeled by a Dirichlet distribution with a concentration parameter $\\alpha$ = **0.05**. This setting reflects highly non-IID data partitions across clients. In contrast, the FedGCN uses significantly **larger** $\\alpha$ values **(1, 100, and 10000)**, corresponding to much milder heterogeneity. Traditional FL and FGL methods tend to struggle under such extreme heterogeneity—especially in one-shot settings where there is no iterative training to mitigate the distribution shift. This explains the substantial performance drop compared to the results reported in FedGCN.\n\nSecond, although existing one-shot FL methods can effectively capture knowledge at the **feature level**, they typically fail to preserve **fine-grained structural information**. This often results in disorganized or noisy graph topologies when generating synthetic graph data. Such disordered structure can severely undermine the quality of downstream training.\n\nIn contrast, our proposed OASIS is explicitly designed to tackle this challenge, which is also acknowledged by Reviewer BhNM and a6gX.  During local training, it captures both semantic and structural knowledge in a fine-grained and efficient manner. At the server side, it incorporates a carefully designed distillation mechanism to integrate and transfer this knowledge to the global model. This dual-level knowledge handling enables OASIS to perform robustly even in highly challenging FGL scenarios. \n\n**`Question 2`: Can OASIS address  the issue of missing edges?**\n\nWe sincerely appreciate your thoughtful comment regarding the challenge of missing edges. We believe that our OASIS can  address this challenge in several ways.\n\n* Synergy Graph Synthesizer:\nThe Synergy Graph Synthesizer is designed to reconstruct both node features and graph topology at the client level. With the sophisticated alignment mechanism, OASIS ensures that the synthesized graphs retain structural integrity, closely mimicking the  topology of the original graph. Meanwhile, the synthesis of adjacency matrices can \nprovide a means of capturing the most relevant topological relationships in the absence of direct edge-sharing across clients. It potentially helps approximate missing edges. \n\n\n* Topological Codebook:\nThe Topological Codebook in OASIS introduces a latent space to represent graph structure at a granular level, capturing subtle neighborhood variations. By quantizing nodes based on their local substructures and encoding them as discrete tokens, we can ensure that even missing edges are represented through these structural tokens. This encoding helps in maintaining the  spatial characteristics of the graph without the need for explicitly shared edge information.\n\n\nIn conclusion, the combination of two modules provides a robust mechanism for mitigating the effects of missing edges.  We will update the manuscript to clarify these aspects, emphasizing how our framework can handle edge omission effectively.\n\n**`Limitation 1`: The loss function of the Synergy Graph Synthesizer should be presented in the main text.**\n\nThank you for your valuable comment. We appreciate your careful reading and for pointing this out. Due to the **strict page limit**, we had to place some technical details in the appendix to ensure the clarity of the main text. In the revision, we will move the description of the loss function into the main body of the paper to improve completeness.\n\nThank you for your constructive comments. We would greatly appreciate it if you could potentially reconsider the evaluation. Please let us know if further clarification is needed.\n\nBest Regards,\n\nAuthors"}, {"review_id": "9GO2yPAuGI", "reviewer": "Reviewer_a6gX", "summary": "In this work, the authors introduce a One-shot Federated Learning Framework tailored for graph-structured data. To effectively preserve the structural characteristics of local graphs, they develop a graph synthesizer in conjunction with a learnable topological codebook. For the purpose of transferring structural knowledge across clients, the framework incorporates Wasserstein-Enhanced Semantic Affinity Distillation (WESAD) to capture inter-client relationships, and Wasserstein-Driven Structural Relation Distillation (WESRD) to model both inter- and intra-class dependencies.", "questions": "How does the proposed framework perform under more severely non-IID data distributions?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. This study tackles the emerging challenge of One-shot Federated Graph Learning (FGL) by proposing a well-structured and principled framework tailored to this unique setting.\n2. A key innovation lies in the integration of the Synergy Graph Synthesizer with a Topological Codebook, enabling the capture of fine-grained structural nuances. Furthermore, the proposed WESAD and WDSRD mechanisms effectively facilitate the transfer of local graph knowledge to the student model. The visualizations—ranging from problem formulation to architectural overview—are informative and well-designed, and the mathematical formulations are both sound and clearly explained.\n3. The code is anonymously available, enhancing the transparency and reproducibility of the experiments.\n\nWeaknesses:\n1. I am curious about how the proposed framework performs under more severely non-IID data distributions.\n2. The meaning of the cost matrix and $q_{ab}$ in Equation (10) on Page 6 is not entirely clear to me. Specifically, I find it difficult to interpret the statement in Line 219 that \"$q_{ab}$ represents the mass transferred from the teacher’s category $C_a$ to the student’s category $C_b$.\" I would appreciate it if the authors could provide a more detailed explanation.\n3. The experimental section lacks a sensitivity analysis for the hyperparameters $\\lambda_d$ and $\\lambda_f$ in the loss term $\\mathcal{L}_{\\text{syn}}$.", "was_revised": false, "final_justification": "", "rebuttal": "***Dear Reviewer a6gX:***\n\nWe sincerely thank you for your positive evaluation and for recognizing the potential of our work. Your feedback is greatly appreciated, and we are encouraged by your overall support. We have carefully addressed the comments you provided, and we hope that our clarifications and improvements can help strengthen your confidence in the paper.\n\n**`Weakness 1` & `Question 1`: The Performance of OASIS under more severely non-IID data distributions.**\n\nThank you for your insightful question regarding the robustness of OASIS under more severe non-IID conditions.\n\nIn our main experimental setup, we already adopt a Dirichlet distribution with a concentration parameter $\\alpha$ =0.05, which is commonly recognized in the literature as a highly heterogeneous setting. To further demonstrate the effectiveness of OASIS in even more extreme non-IID scenarios, we conducted additional experiments with a smaller Dirichlet parameter $\\alpha$=0.01, which introduces an even higher degree of data heterogeneity. The results on Cora and CiteSeer datasets are shown below:\n\n*Table 1: The performance of OASIS under more extreme non-IID scenarios.*\n\n|Method|Cora|CiteSeer|\n|-------|------|-------|\n|FedAvg | 30.89 | 23.15|\n|FedProx | 31.53 | 35.32|\n|FedPub | 30.34 | 20.71 |\n| FGSSL| 30.80 | 20.54   |\n| FedCVAE | 26.97 | 23.53|\n|FENS | 30.71 | 20.01 |\n|OASIS | **38.96** | **39.06**|\n\nAs seen from the results, our OASIS continues to outperform the baselines by a substantial margin under the more severely non-IID distribution. This provides strong empirical evidence of the robustness and generalization capability of OASIS.\n\n\n**`Weakness 2`: The meaning of the cost matrix and $q_{ab}$ in Equation (10).**\n\n\nThank you for your valuable comment. In Section 3.3, we introduce the WESAD method and have explained the role of the cost matrix and $q_{ab}$.\nWe appreciate the chance for further clarification on the meaning of the cost matrix $c^k_{ab}$ and the transport plan $q_{ab}$ in Equation (10). \n\nThe cost matrix $c_{ab}^k$ can be intuitively understood as the penalty or cost of the student model misclassifying an instance as class $C_b$ while the $k$-th teacher assigns it to class $C_a$ . Rather than treating all classes as equally distinct, we aim to capture the nuanced semantic relationships between them. To construct it, we first compute a semantic affinity score $\\text{SA}^k(C_a, C_b)$, which reflects how similar two categories are in the local teacher model. This is done by computing the cosine similarity between the weight vectors corresponding to $C_a$ and $C_b$ from the projection head $W^k$, thereby capturing their semantic alignment in the embedding space. \nThe cost is then defined as $c^k_{ab} = 1 - \\kappa \\cdot \\text{SA}^k(C_a, C_b)$, where $\\kappa$ is a scaling factor. This definition ensures that categories with stronger semantic affinity incur a lower cost when aligning their probabilities, encouraging the model to treat similar classes as softer alternatives during knowledge transfer.\n\nAs for the transport plan $q_{ab}$, it represents the optimal solution to the entropy-regularized discrete optimal transport problem between the teacher and student class distributions. Specifically, $q_{ab}$ denotes the amount of probability mass that is optimally transferred from category $C_a$ in the teacher output $p^k_T$ to category $C_b$ in the student output $p_S$, under the constraint that the row and column marginals of $q$ match the respective class probabilities.  This mechanism enhances the generalization of the student model by enabling it to capture nuanced cross-category relations embedded in the teacher’s predictions.\n\n**`Weakness 3`: Lacks a sensitivity analysis of $\\lambda_d$ and $\\lambda_f$.**\n\nThank you for your valuable suggestion. To demonstrate the robustness of our OASIS under different $\\lambda_d$ and $\\lambda_f$, we select Cora and CiteSeer datasets to conduct sensitivity study. Specifically, we design the range of $\\lambda_d$ in [0.01,0.09] with a step size of 0.02 and the range of $\\lambda_f$ in [0.05,0.25] with a step size of 0.05. \n\n*Table 2: The sensitivity study of $\\lambda_d$ on Cora and CiteSeer datasets.*\n\n|Datasets| $\\lambda_d = $ 0.01| $\\lambda_d = $ 0.03| $\\lambda_d = $ 0.05|$\\lambda_d = $ 0.07|$\\lambda_d = $  0.09|\n|----|----|----|----|----|-----|\n|Cora| 46.38|\t48.4|\t49.59\t|48.3|\t47.85|\n|CiteSeer| 45.24|\t45.47|\t45.69|\t45.77\t|44.64|\n\n*Table 3: The sensitivity study of $\\lambda_f$ on Cora and CiteSeer datasets.*\n|Datasets| $\\lambda_f = $ 0.05| $\\lambda_f = $ 0.10| $\\lambda_f = $ 0.15|$\\lambda_f = $ 0.20|$\\lambda_f = $  0.25|\n|----|----|----|----|----|-----|\n|Cora| 47.57|\t49.59|\t46.65|\t46.08\t|48.21|\n|CiteSeer| 46.74 |\t45.69|\t46.37\t|47.49|\t45.92|\n\nFrom the table, we can observe that our OASIS performs consistently under different $\\lambda_d$ and $\\lambda_f$, demonstrating the robustness of our OASIS.\n\n\n**`Paper Formatting Concerns：` The use of inline equations in Lines 157 and 182 contrasts with displayed equations elsewhere.**\n\nThank you for pointing this out. We appreciate your careful reading. We will make sure to unify the equation formatting throughout the paper in our future revisions to ensure consistency and improve readability.\n\nWe would like to express our deepest gratitude for your constructive suggestions. Your support and recognition of our work mean a lot to us.\n\nBest Regards,\n\nAuthors"}, {"review_id": "QZaKOA8oel", "reviewer": "Reviewer_seeB", "summary": "This paper proposes a new framework for one-shot federated graph learning. It combines a Synergy Graph Synthesizer to create useful synthetic graphs and a Topological Codebook to capture structural patterns. The framework also includes two distillation methods for knowledge integration: one for inter-class relationships and one for structural alignment. Experiments show the framework performs better than existing methods.", "questions": "1. Why do the baseline methods in Table 1 exhibit such low accuracy, with most results (even some OFL methods) falling below 30%?\n\nOther questions see Weaknesses.", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\n1. The proposed framework for One-Shot FGL is innovative and interesting to me. It diverges from conventional multi-round protocols and opens up new avenues for low-cost yet effective collaborative graph representation learning.\n\n2. A key highlight of the paper is the introduction of the Topological Codebook. This component is not only conceptually sound but also well-motivated in the context of capturing structural heterogeneity and relational dependencies in decentralized graph data. The paper also provides solid intuition and theoretical grounding for its use.\n\n3. The figures in the paper are highly detailed, and the overall methodology is rigorously derived. The entire framework is presented in a clear and coherent manner.\n\nWeaknesses:\n\n1. The paper lacks sufficient details regarding the local training epochs used in other traditional FL/FGL or one-shot FL baselines. Additionally, it does not clearly explain how graph data is generated in the OFL methods compared in Table 1.\n\n2. I’m a bit confused by Eq. (12); the authors should provide a more detailed explanation of how the Semantic Affinity (SA) and the cost matrix are computed.\n\n3. In Section 4.1, the authors mention using a Dirichlet distribution with $\\alpha =0.05$ to partition the data. However, more experiments under extreme heterogeneity (e.g., with a smaller α\\alphaα value such as 0.01) are needed to demonstrate the model's robustness in highly non-IID scenarios.", "was_revised": true, "final_justification": "My concerns have been addressed. I recommend accepting this paper.", "rebuttal": "***Dear Reviewer seeB:***\n\nThank you very much for your thoughtful feedback and valuable suggestions. We sincerely appreciate your support and the time you have taken to review our work. Below, we provide point-by-point responses to the concerns you raised.\n\n**`Weakness 1`: Lack of sufficient training details of compared baselines and explanations of the graph generation process.**\n\nThank you for your observation regarding our experimental setup. We would like to clarify the training details of all the compared baselines and further explain the process of graph generation in One-Shot FL methods.\n\nFor all the compared baselines, each client performs local training for 100 epochs to ensure convergence before uploading the local model. The communication round is limited to one.\n\nAs for the One-Shot FL methods, we follow the same procedure as described in the original papers to generate the feature matrix $\\boldsymbol{X}$ of the graph data. The graph structure is then constructed according to Section 3.2 and Equation (5) of our paper. Specifically, we compute the activated similarity matrix $\\boldsymbol{H}$ based on the feature matrix, and apply the k-nearest-neighbor algorithm to build the graph topology, with $k = 5$ consistently used across all methods. This ensures  fairness across all evaluations.\n\n\n**`Weakness 2`: Lack of more detailed explanation of how the Semantic Affinity (SA) and the cost matrix are computed.**\n\nThank you for pointing this out — we appreciate your careful reading of our WESAD module, and we would like to explain the Semantic Affinity and the cost matrix in more detail.\n\n\nThe Semantic Affinity $\\text{SA}^k(C_a, C_b)$ is computed as the cosine similarity between the weight vectors corresponding to class $C_a$ and class $C_b$ in the projection head of the teacher model. These weight vectors can be viewed as semantic prototypes for each class, and their cosine similarity reflects how semantically close two classes are in the representation space learned by the local GNN. \n\nBased on this affinity, the cost matrix $c^k_{ab}$ is defined as $1 - \\kappa \\cdot \\text{SA}^k(C_a, C_b)$, where $\\kappa$ is a hyperparameter that scales the influence of semantic similarity. This cost matrix plays a central role in the discrete Wasserstein distance computation: it quantifies the cost of transferring the predicted probability mass from class $C_a$ (in the teacher model) to class $C_b$ (in the student model). Intuitively, assigning a sample from a semantically dissimilar class $C_a$ to $C_b$ incurs a higher cost, while transferring between semantically related classes is penalized less. This design allows the distillation process to tolerate semantically meaningful mismatches while discouraging alignment between unrelated categories.\n\n**`Weakness 3`: More experiments under extreme heterogeneity are needed.**\n\nThank you for the valuable suggestion regarding the evaluation under extreme heterogeneity. To demonstrate the performance of  our OASIS  under higher data heterogeneity, we conduct experiments with Dirichlet parameter $\\alpha$=0.01 with six baselines, as shown in Table 1. \n\n*Table 1: The performance of OASIS under extreme heterogeneity ($\\alpha = 0.01$).*\n\n|Method|Cora|CiteSeer|\n|-------|------|-------|\n|FedAvg | 30.89 | 23.15|\n|FedProx | 31.53 | 35.32|\n|FedPub | 30.34 | 20.71 |\n| FGSSL| 30.80 | 20.54   |\n| FedCVAE | 26.97 | 23.53|\n|FENS | 30.71 | 20.01 |\n|OASIS | **38.96** | **39.06**|\n\n As the results show, our method OASIS significantly outperforms all baselines by large margins. These results clearly highlight the effectiveness of OASIS in handling extreme heterogeneity.\n\n**`Question 1` : Why do some baseline methods exhibit low accuracy?**\n\nThank you for  your insightful question about the experimental results. In Section 4.2, we have already discussed the poor performance of some baselines, and we would like to provide further clarification here.\n\nFirst, it is important to highlight that our experiments were conducted under extreme data heterogeneity conditions. Specifically, we employed a Dirichlet distribution with a concentration parameter $\\alpha = 0.05$, which leads to highly skewed data distributions across clients. This level of heterogeneity makes it extremely difficult for FL and FGL methods to effectively capture local knowledge during training. As a result, these methods perform poorly, especially in a one-shot setting, where only a single communication round is allowed.\n\nSecond,  OFL methods often struggle to retain fine-grained structural insights. This limitation leads to the generation of  disorganized graph structures, with unclear relationships between nodes. As a result, the generated graph data often causes confusion, which negatively impacts the overall training process.\n\nMeanwhile, our OASIS addresses these issues by focusing on efficiently capturing both feature and fine-grained structural knowledge during local training. Additionally, we have designed an effective distillation strategy to facilitate knowledge transfer on a global scale. This allows OASIS to outperform all the baselines under challenging conditions.\n\n\nThank you again for your valuable feedback and supportive review of our work.\n\nBest Regards,\n\nAuthors"}, {"review_id": "5WHUevxaJA", "reviewer": "Reviewer_4jz9", "summary": "The paper introduces OASIS, a novel one-shot Federated Graph Learning (FGL) framework designed to address the challenges of capturing fine-grained structural knowledge from local graphs and effectively transferring this knowledge to the global model during knowledge distillation. By integrating the Synergy Graph Synthesizer, Topological Codebook, and Wasserstein-driven distillation techniques, the framework aims to enhance both the performance and generalization ability of federated graph models in resource-constrained settings. The authors present a comprehensive set of experiments, showing that OASIS outperforms existing methods in terms of node classification accuracy across a variety of real-world datasets.", "questions": "1. Why the accuracy is lower than other works?\n\n2. Can OASIS be compared to fedsage+?\n\n3. Can you provide a more detailed analysis of the communication cost compared to methods like fedGCN? or fedsage+?\n\n4. Can you added formal theoretical justification or guarantees on the possible information losses?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1.The paper presents a well-structured framework, OASIS, which integrates novel techniques such as the Synergy Graph Synthesizer and Wasserstein-based knowledge distillation to address fine-grained structural knowledge transfer in federated learning. This is a significant contribution to the field of One-shot Federated Graph Learning (OFL).\n\n2.The paper conducts a comprehensive set of experiments on several real-world graph datasets, including Cora, CiteSeer, and PubMed, demonstrating the potential of OASIS in improving global model performance. However, discrepancies between the results reported here and in other works may need further investigation.\n\nWeaknesses:\n1. A key concern is the substantial drop in accuracy compared to the original works. For example, the accuracy reported for FedPub on the Cora dataset is only 30%, compared to 79% in the original paper. This discrepancy requires careful analysis and clarification.\n\n2. Can OASIS be considered a data enhancement method? It should be compared with FedSAGE+ [1], an elegant method that also trains the model on generated graphs. A thorough analysis of the differences between these approaches, or an explanation of why OASIS is superior is preferred.\n\n3. Although the paper presents OASIS as a one-shot federated learning framework, the results do not convincingly demonstrate how the communication overhead is reduced in practical settings. The need to upload both the model parameters and the Topological Codebook could lead to higher communication costs compared to other methods, such as FedGCN [2], which communicate with the central server in a single pre-training step. A more detailed analysis of communication efficiency is needed to assess the practical scalability of the method.\n\n4.Despite OASIS incorporating multiple components (e.g., Synergy Graph Synthesizer, Topological Codebook, and WDSRD), the paper does not provide sufficient theoretical justification for why these blocks improve performance. A theoretical analysis of the difference between the OASIS-generated graph and the original graph is needed to better bound the performance of OASIS. This could either provide insights into how much OASIS outperforms the original method or offer a theoretical understanding of the potential performance loss compared to directly transferring the locally trained model on the original clean graph.\n\n[1] https://arxiv.org/pdf/2106.13430\n\n[2] https://arxiv.org/pdf/2201.12433", "was_revised": true, "final_justification": "Most of my concerns are resolved. However, the theoretical analysis is still limited on strict assumptions and the assumptions lack realistic justification. The analysis in response is like mechanically stacking blocks together, lacking deeper insight.\nAlthough I dont unserstand why reviewer BhNM appricate this mathematic description and say that  it facilitates theoretical understanding even thougn this paper dose not have any theoretically analysis before, I agree that this paper has superior experimental preformance over existing methods. \nSince I am not expert in FGL, I am not sure whether a FGL paper without theoretical justification can be accepct or not. Therefore, I will slightly raise my score.", "rebuttal": "***Dear Reviewer 4jz9:***\n\nWe appreciate your time and thoughtful feedback. We have addressed your concerns below and hope that the clarifications will positively contribute to the evaluation of our work.\n\n**`Weakness 1` & `Question 1`: Why the accuracy is lower compared to the original works (e.g. FedPub) ?**\n\nThank you for  your insightful question about the experimental results. In Section 4.2, we have already discussed the poor performance of some baselines, and we would like to provide further clarification here.\n\nFirst, it is important to highlight that our experiments were conducted under extreme data heterogeneity conditions. Specifically, we employed a Dirichlet distribution with $\\alpha = 0.05$, which leads to highly skewed data distributions across clients. This level of heterogeneity makes it extremely difficult for FL and FGL methods to effectively capture local knowledge during training. Moreover, they heavily rely on gradual updates over multiple rounds. Therefore, the one-shot setting further limits opportunities for iterative knowledge exchange.\n\nSecond, existing OFL methods struggle to retain fine-grained structural insights. This limitation leads to the generation of disorganized graph structures,  with unclear relationships between nodes. As a result, the generated graph data often causes confusion, which negatively impacts the overall training process.\n\nMeanwhile, our OASIS addresses these issues by focusing on efficiently capturing both feature and fine-grained structural knowledge during local training. Additionally, we have designed an effective distillation strategy to facilitate knowledge transfer. This allows OASIS to outperform all the baselines under challenging conditions.\n\n**`Weakness 2` & `Question 2`: Can OASIS be considered a data enhancement method? Can OASIS be compared to FedSage+ in theoretical and experimental way?**\n\nThank you for your question. We would like to clarify whether OASIS can be considered a data enhancement method, and how it compares to FedSAGE+ below.\n\n* Nature of OASIS: \nWe would more accurately describe it as a **data-centric** approach at the client level. Rather than aiming to improve local graph representations, OASIS is designed to train a synthesizer that effectively **captures the feature and structural information of local data**, which can then be uploaded to  the server  to train a generalized global model.\nMoreover, at the server side, OASIS focuses primarily on **model-centric** improvements, emphasizing the efficient transfer of knowledge from clients to the global model. Our approach aims to achieve better generalization capability by leveraging a topological codebook and Wasserstein-based distillation, which differs from traditional data enhancement techniques that focus on directly augmenting the dataset with additional synthetic data.\n\n* Theoretical Comparison with FedSAGE+:\nFedSAGE+ addresses the issue of missing neighbors by introducing a local neighbor generation mechanism. Its goal is to reconstruct the neighborhood of each node to improve local learning. However, it lacks a centralized mechanism to explicitly guide structural knowledge sharing across subgraphs. In contrast, OASIS captures both semantic and structural correlations among subgraphs. This allows for a more holistic transfer of knowledge from all the clients to the global model.\n\n* Empirical Advantages:\nIn terms of empirical performance, we compare OASIS against FedSAGE+ on Cora, CiteSeer, Coauthor-CS and Roman-Empire datasets. To ensure fairness, we utilize GraphSAGE as the  backbone for OASIS.   Results from Table 1 demonstrate that our OASIS outperforms FedSAGE+ in the one-shot setting. Notably, FedSAGE+ is a strong contender in handling missing neighbors, but OASIS provides a more robust and efficient exploration of local knowledge under extreme data heterogeneity. \n\n*Table 1: Performance comparison between OASIS and FedSAGE+ on four datasets.*\n|Methods|Cora|CiteSeer|Coauthor-CS|Roman-Empire|\n|-|-|-|-|-|\n|FedSage+|28.87|36.18|22.94|25.86|\n|OASIS|**39.96**|**44.87**|**52.51**|**26.73**| \n\nIn conclusion, OASIS should be considered more as a data-centric approach that emphasizes effective knowledge transfer across local models to achieve global generalization. It offers several advantages over FedSAGE+, particularly in handling structural variations and ensuring a more comprehensive integration of knowledge. We will ensure that the relevant results and a more detailed comparison with FedSAGE+ will be included in the revised manuscript, with additional discussions on the structural transfer mechanisms and empirical findings.\n\n**`Weakness 3` & `Question 3`: A more detailed analysis of the communication cost compared to methods like FedGCN or FedSage+.**\n\nThank you for pointing out this important aspect. We appreciate the opportunity to clarify the communication efficiency of our proposed OASIS.\n\nCompared to FedSage+, one of the key advantages of OASIS lies in its one-shot design. While FedSage+ requires multiple rounds of model uploading and distribution, it also introduces additional communication overhead by transmitting parameters of the NeighGen module, which includes both an encoder and a generator—components that are comparable in size to the GNN itself. In contrast, OASIS performs only a single round of communication. As a result, OASIS achieves up to ~100× reduction in communication cost compared to FedSage+.\n\nRegarding FedGCN, we acknowledge that it also adopts a single-round pretraining setting. Moreover, our OASIS introduces additional elements to enhance server-side knowledge integration. Importantly, the communication cost of these components remains manageable and justifiable. Specifically, the codebook size is $M×d$, where \n$M$ (typically 64 or 128) is the number of tokens and \n$d$ (set as 128) is the hidden dimension. The synthesizer we employ is a simple linear layer, further keeping the overhead minimal. Despite this modest increase in communication, the benefits are substantial. These components enable high-quality knowledge distillation and improve global model performance, representing a qualitative leap in effectiveness that we believe is well worth the cost.\n\n**`Weakness 4` & `Question 4`:  (1) Theoretical justification for why each module improves performance.  (2)The difference between the OASIS-generated graph and the original graph. (3) Theoretical justification  on the possible information losses.**\n\nThank you for your concerns. We would like to provide detailed explanations to address your concerns below.\n\n(1) Theoretical justification for why each module improves performance.\n\nIn **Section 4.3** and **Figure 5**, we have conducted a detailed ablation study to assess the individual impact of each component in the Graph Knowledge Integration module, which is also acknowledged by Reviewer a6gX. Here, we would like to further explain why each module improves performance in theoretical way.\n\n* Synergy Graph Synthesizer: Our Synergy Graph Synthesizer effectively captures both feature and structural knowledge during local training. It adopts Fused-Gromov Wasserstein distance to optimize the transport plan for aligning the local and synthesized graph, reducing the discrepancy in both feature and structural domains.\n\n* Topological Codebook: The Topological Codebook quantizes node representations into discrete tokens and enables the model to learn fine-grained information of graph substructures. The discrete approach forces the model to focus on small, intricate structural differences that might otherwise be overlooked in continuous representations, ensuring that even subtle variations in graph neighborhoods are captured and differentiated effectively.\n\n* WDSRD: WDSRD aligns structural patterns between local and global models by comparing their soft assignments (computed in eq.(15)). Through the closed-form Wasserstein distance between Gaussian distributions, WDSRD captures both mean-level and covariance-level discrepancies in the assignment distributions, which reflects richer structural variation than simply aligning logits or embeddings.\n\n(2) The difference between the OASIS-generated graph and the original graph.\n\nAccording to the fundamental setting of Federated Learning, it is impractical to directly upload local data to the server. As detailed in Appendix I.2, the synthesized graph generated on the server does not violate local data privacy, since it adopts a uniform label distribution and constructs node topology using the k-nearest-neighbor algorithm. As a result, the synthesized graph is explicitly different from the original local graph. Nevertheless, it sufficiently retains the implicit structural and semantic information, which is crucial for effective global model learning.\n\n(3) Theoretical justification on the possible information losses of OASIS compared to directly transferring the locally trained model on the original clean graph.\n\nIn Section 4, our experimental results have already demonstrated that OASIS significantly outperforms traditional FL/FGL methods, which directly transfer the locally trained models on the original clean graphs. Within one communication round, these methods fail to efficiently extract  information from local data. Moreover, due to the **extreme data heterogeneity** across clients, local models are highly prone to overfitting their respective data distributions, leading to poor aggregation performance. Although our Synergy Graph Synthesizer may not be able to fully preserve all the information from the original local data, it effectively captures the high-level semantic and structural distribution knowledge. As evidenced by our experiments, this distilled knowledge is sufficient to train a well-generalized global model.\n\nThank you again for your thoughtful and insightful feedback. We are deeply grateful for the opportunity to have the strengths of our work reconsidered by you.\n\nBest Regards,\n\nAuthors"}]

Author rebuttal:

Meta-review: This work proposes OASIS, a one-shot federated learning framework for graph data. The framework introduces a Synergy Graph Synthesizer and a Topological Codebook to capture structural patterns, along with Wasserstein-based distillation methods (WESAD and WDSRD) to transfer knowledge to a global model.

Reviewers generally leaned positively with only one exception, and evaluated the work favorably.  Authors were able to engage in a fruitful discussion period during which several reviewers improved their scores and maintained positive assessments.

Main feedback:

- several reviewers raised concerns around low baseline accuracy compared to original works, but the authors addressed this by explaining experiments were conducted under high data heterogeneity (low alpha values) (Eosq, 4jz9, seeB)

- several reviewers criticized the lack of theoretical justification for the proposed components, but the authors addressed this by providing a new bound on the information retention of the synthesized graph on global model generalization (Eosq, 4jz9)

- reviewers pointed out scalability concerns re: the Wasserstein-based losses, though the authors clarified that there are efficient approximations to scale up the method and time is saved also by the one-shot design (Eosq, 4jz9).
Decision: Accept (poster)