Unsupervised Federated Graph Learning
Authors: Lele Fu
Tianchi Liao
Sheng Huang
Bowen Deng
zhangchuanfu
Shirui Pan
Chuan Chen
Venue: NeurIPS 2025 poster ()
Keywords: Federated Learning
Graph Learning
Unsupervised Learning
Primary area: general_machine_learning

Abstract: Federated graph learning (FGL) is a privacy-preserving paradigm for modeling distributed graph data, designed to train a powerful global graph neural network.
  Existing FGL methods predominantly rely on label information during training, effective FGL in an unsupervised setting remains largely unexplored territory.
  In this paper, we address two key challenges in unsupervised FGL: 1) Local models tend to converge in divergent directions due to the lack of shared semantic information across clients. Then, how to align  representation spaces among multiple clients is the first challenge.
  2) Conventional federated weighted aggregation easily results in degrading the performance of the global model, then which raises another challenge, namely how to adaptively learn the global model parameters.
  In response to the two questions, we propose a tailored framework named FedPAM, which is composed of two modules:
  Representation Space Alignment (RSA) and Adaptive Global Parameter Learning (AGPL).
  RSA leverages a set of learnable anchors to define the global representation space, then local subgraphs are aligned with them through the fused Gromov-Wasserstein optimal transport, achieving the representation space alignment across clients.
  AGPL stacks local model parameters into third-order tensors, and adaptively integrates the global model parameters in a low-rank tensor space, which facilitates to fuse the high-order knowledge among clients.
  Extensive experiments on eight graph datasets are conducted, the results demonstrate that the proposed FedPAM is superior over classical and SOTA compared methods.

Unsupervised Federated Graph Learning
Lele Fu', Tianchi Liao', Sheng Huang', Bowen Deng', Chuanfu Zhang',
Shirui Pan?, Chuan Chen
Sun Yat-sen University. Guangzhou, China
Griffith University, Brisbane, Australia
(fulle, liaotch, huangsh253, dengbw3 ) @mai 12 sysu. edu cn s. p.an@griffith edu. (zhangchf9, chenchuan @mail sysu. edu. cn
Abstract
Federated graph learning (FGL) is privacy-preserving paradigm for modeling distributed graph data, designed to train powerful global graph neural network. Existing FGL methods predominantly rely on label information during training, effective FGL in an unsupervised setting remains largely unexplored territory. In this paper, we address two key challenges in unsupervised FGL.: 1) Local models tend to converge in divergent directions due to the lack of shared semantic infor- mation across clients. Then, how align representation spaces among multiple clients is the first challenge. 2) Conventional federated weighted aggregation easily results in degrading the performance of the global model, then which raises an- other challenge, namely how to adaptively learn the global model parameters. In response to two questions, we propose a tailored framework named FedPAM, which is composed of two modules: Representation Space Alignment (RSA) and Adaptive Global Parameter Learning (AGPL). RSA leverages a set learnable anchors to define the global representation space, then local subgraphs with them through the fused Wasseesteenn optimal transport, achieving the representation space alignment across clients. AGPL stacks local model parameters into third-order tensors, and adaptively integrates the global model parameters in low-rank tensor space, which facilitates fuse the high-order knowledge among clients. Extensive experiments on eight graph datasets are conducted, the results demonstrate that the proposed FedPAM is superior over classical and SOTA compared methods.
aligned
Introduction
In response to growing concerns over privacy protection [1, 2, 3], the storage of graph data is becoming increasingly decentralized, where individual clients hold their own private subgraphs. Nevertheless, distributed subgraphs result in the emergence of data silos, triggering data unavailability and impeding the generalization federated graph learning (FGL) [9, 10. 11, 12, 13, 14] has emerged collaborative training of a powerful GNN across multiple clients, while maintaining the usability of private subgraphs without compromising their confidentiality.
graph neural networks (GNNs) [4, 5, 6, 7, 8). this context, paradigm that enables the
FGL has made substantial progress in recent years. the context of graph-level tasks, studies [15, 16, 17] primarily concentrate on handling the challenges arising from distributional heterogeneity. For nove-level tasks, considerable attentions [18. 9, 19] are devoted to defining the topological heterogeneity in distributed subgraphs and overcoming its adverse effects on the effectiveness of federated training. The existing approaches provide innovative insights and solid solutions for FGL.
"Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Oke 2 :los
Clan Omn
cliet Orr
Triel
Glacs
Gime?
liyer
AGDT Coss sare co
Client D
Clarr. Weigl
(a) FodAvg
(b) FodRAM
Statistics CiteSeer (b) Comparison of client weights Figure Comparison of client weights on CiteSeer dataset for FedAvg and FedPAM. FedPAM learns an adaptive weight for cach client according to the data heterogencity while FedAvg assigns a fixed weight for each client based on the number of local samples.
Figure Scatter comparison on Physics dataset for FedAvg and FedPAM. It can be seen that FedPAM achieves a better inter-class separation, aligning the representation spaces of different clients, while FodAvg fails to do so.
However, they inherently depend on label information during training, indicating that they adhere to supervised or semi- supervise learning paradigms. In practice, data labels are frequently inaccessible, and the lack of consistent semantics across clients further exacerbates the difficulties of federated learning (FL) [20, 21, 22, 23, 24, 25]. As result, enabling effective FGL in an unsupervised setting poses a significant challenge. For unsupervised FGL, one of the most straightforward solutions is to leverage mature FL frameworks in conjunction with self-supervised learning (SSL) strategies [26, 27, 28, 29]. For example, clients perform local training using the SimCLR [30], while the server aggregates model parameters by FedAvg [31]. However, naively transplanting existing approaches fails to effectively cope with the common issue of heterogeneous data in FL. Moreover, some unsupervised FL algorithms originally designed for non-graph data are also worth mentioning. Studies [32, 33] used the clustering strategies to enhance the consensus of global space. Zhuang et al. [34, 35] proposed to dynamically update the local models based on the parameter differences between the global and local models. Liao et al. [36] addressed the problem of representation collapse in unsupervised FL.
These methods are insightful for unsupervised FL. but they exhibit two primary limitations: 1 Inability to align the representation spaces of various clients. Due to the absence of labels, there are no consistent signals among clients, which can casily lead to shifts in the representation spaces and weaken the generalization for global model. Although studies [32, 36] adopt clustering or optimal transport (OT) to enhance the separability of representations, they merely alleviate representation collapse and do not achieve unified representation space across clients. Especially, when faced with graph data, their learned representation spaces are prone to fall into suboptimality due to complex topological networks. @ Unable adaptively learn the optimal parameters of global model. Most existing federated aggregation methods use weighted averaging based on number of local samples, which might degrade the performance of global model. Studies [34, 35, 36] introduce divergence- aware and multi- multi-objecttiv aggregation strategies, but their matrix-level designs fall short in effectively capturing the complementary information among local models. From Fig. (()), we can see that the scatter output by FedAvg is dispersed throughout the space, the nodes of the same categories from various clients cannot be clustered together, indicating that representation spaces of different clients are unaligned. The subgraph statistics for different clients are reported in Fig. 2(a). The total numbers of nodes in the different clients are almost the same, but the numbers of nodes in each category vary considerably, showing different heterogeneous situations. If the traditional FedAvg is used, each client assigned almost a same weight as shown in Fig. 2(b), which is clearly unreasonable.
In light of the above concerns, we propose a tailored framework FedPAM for unsupervised FGL, which mainly consists of two key modules: Representation Space Alignment (RSA) and Adaptive Global Parameter Learning (AGPI.). Specifically, RSA aims to learn a set of cross-client anchors, which are used to define the global representation space. To align the representation spaces of various -assoosstssrrtnnnn optimal transport (FGW-OT) [37] is employed to establish
clients, the fused mapping between local subgraphs and the anchor graph with minimal cost. Additionally, the anchors also serve as global space projector, projecting local embeddings into the global space to facilitate the contrastive learning, thereby mitigating the biased training caused by data heterogeneity. AGPL departs entirely from the conventional weighted aggregation paradigm. It stacks parameters of local models into third-order tensors, and leverages the low-rank tensor decomposition to capture high-order correlations among clients. Further, the optimal parameters of global models are adaptively learned in the low-rank tensor space, enabling effective integration of local knowledge. From Figs. 1(b) and 2(b), it can be observed that the proposed FedPAM can effectively align the representation spaces

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across clients and adaptively learn the global model parameters. The framework of the proposed FedPAM is presented in Fig. We summarize the contributions this paper from four-fold:
We are the first to identify the challenges inherent in unsupervised FGL and design unified framework to address them. This framework proposes the representation space alignment and the adaptive global parameter learning for unsupervised FGL..
We learn set of anchors to span the global representation space, and adopt the FGW-OT to match local subgraphs with the anchor graph, thus achieving the alignment of representation spaces across multiple clients.
We stack the local model parameters into third-order tensors and extract its low-rank components to capture the high-order correlations among clients. In the high-dimensional space, the optimal global model parameters are obtained by adaptive fusion.
To verify the effectiveness of the proposed FedPAM, we conduct the comparative exper- iments eight graph datasets. Compared traditional and SOTA methods, FedPAM achieves more superior performance.
Fiys
Tensor W
Low-rank Tensor
A
Global Anchors
P
Server
Fuse
P
Pwe
Upload/Distribtte Models
Upload/Distribute Anchors
Zx
Zi
H5
Client kn
Fwa
G
FGW-OT
Contrastive Learning
Contrastive Learning
Gx
FGW-OT
Fw
G
HR
Figure The framework of proposed FedPAM. client performs the RSA module and the server performs the AGPL module, models and anchors are transferred between the server and clients.
2 Preliminaries
Federated Graph Learning. Given a FGL system with a central server and K clients, each client possesses private subgraph Gk (Vr, Es, X&J. where Vs. Ex. and X, indicate the vector set, the edge set, and the feature matrix, respectively. Specifically. X E IR Nux has N₁. nodes and dimensions. The adjacency matrix A N is induced by the edge set Eg. If there an edge between the i-th node and the j-th node, then Ag(i,)
1. Otherwise, A (i. = 0. In the unsupervised FGL, the node labels Y, [0] je....., NN are all unavailable during the GNN's training. Notably, we assume that all subgraphs originate from a complete graph, namely G. E G. At the beginning of communication round in FL, the central server distributes the global GNN Fw parameterized by W to each client and starts the local training, the computation GNN can be summarized
++1
(Fw, (2), AGG(Z)) A.j))) [Vn E (V:)),
where 5(-) denotes the activation function, Fw, denotes the 1-th encoder layer, x the i-th node's representation at the 1)th ((()) the neighborhood node set for node v; AGG(-) denotes
3

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the embedding fusion manner. When the local training finishes, local GNNs are uploaded to the server for aggregating the global GNN, which formulated
No N
W
Wk.
where Ek= N₁ denotes the sum of local nodes.
Tensor Nuclear Norm. For third-order tensor X' E R Xmxxxs its the j-th frontal slice is denoted x06) E IR X22 The fast Fourier transformation (FFT) is used to transform tensor into the frequency domain, which is formulated as X fft X D. 3), its inverse operation is written as X iff 3). The product Z E paixxaxns of X E [ Xmaxng and y RguaXnaxng is defined as
2 X fold(beire (x) bvec()))),
where (V) [2()), .... (ms)) is the block vectorizing operation, and lold(bvec()))) is inverse operation. beire(X) is defined as
(1 x2)
(ms) x²
x1)
x3)
beirc(x()
(4)
X (ns)
(ma-1) 1)
x1)
Further, X * x x I, X is called orthogonal tensor. E IR Xn x is an identity
tensor, where the first frontal slice is identity matrix and the remaining slices all zeros. The tensor singular value decomposition (T-SVD) is a key operation for calculating the tensor nuclear norm, their definitions are introduced follows.
Definition 1. For a tensor X
IR X the T-SVD is formulated as
x u D V
(5)
where U E RR XnLRRs and V E Xngxn are orthogonal, D E &RRIXnnXns is f-diagonal.
its T-SVD based tensor nuclear norm [38] is defined as Definition 2. For tensor X E RR th Xngxnn
n3
min(ni.9) ng
1
1
11.
Σ
Σ
(i, (i,
(6)
j=1
where D (i,i) is obtained by x (1)
(i,
= =4t
DD)) 1,)
3
Methodology
In this section, we elaborate on the proposed FedPAM. Overall, FedPAM is comprised of two modules: RSA and AGPL. Specifically, RSA is used to align the representation spaces across clients through the FGW-OT between local subgraphs and anchor graph. AGPI, is employed to adaptively learn the global model parameters in the low-rank tensor space.
3.1 Fused Gromov- Wasserstein Optimal Transport for Aligning Representation Spaces
Local Self-Supervised Learning. Self-supervised learning (SSL) an effective technique for preventing representation collapse and enhancing the discriminative quality of learned representations. In the absence of node labels, local clients adopt a self-supervised strategy for training. For the k-th local subgraph GR. it is augmented to G and G via two different kinds of augment methods. Then, the local GNN W, encodes them into the embedding space:
Z Fw, (G Z PW, (G)).
A SSL loss between Z and Z is used for back propagation and generally written
Lsst EVI,JE(NN) (z, z Xeneg (z, (2)
(8)

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where poss and Eeeg denote the losses for positive and negative pairs, respectively. Overall, the SSL loss Lsst. can be categorized into two classes: contrastive and non-contrastive. Contrastive methods require the involvement of negative samples during the computation of SSL, while non-contrastive methods do not rely on negative samples. We investigate the impacts of various SSL on the proposed framework in the experimental section.
FGW-OT between Local Graphs and Anchor Graph. Under our assumption, local subgraphs originate from a underlying global graph, implying that node representations across clients following unified distribution. However, due to the heterogeneous nature of local data, the embeddings learned by local clients tend to exhibit distributional discrepancies. Moreover, since data from other clients is unaccessible, local models lack awareness of the global distribution. To address this issue, we introduce Np learnable anchors P {Pi.-.. ..... P J in the server to span the global space, the relationships between anchors are recorded by the anchor graph p. These anchors also serve as the global projector, enabling the projection of local embedding into the global space. At the heginning of each communication round, the global model and the anchors are distributed to each client for local training. To align local subgraphs with anchor graph, the FGW-OT is introduced, whose definition is written as
Definition 3. Given two metric measure space 91 (V1, p1) and G1 (V2, A2, V1 and V22 denote the node sets, where the data sizes are respectively N1 and N2, A and A2 denote the relationship matrices, μu and U2 denote the data marginal distributions. Then, the FGW-OT distance between 91 and G2 is defined as
N2
inf
(() + +al[A]la - [Ag]mm IP) JT
REN
(9)
mnn
S.I. II = (TERNIN II,, = =,,, x IN, = P2)
where X1, E V1, X2,j E V2, D(.) measures the distance between two nodes, a is a trade-off parameter between the feature-level distance and the relationship-level distance, p is constant used to adjust the strength of the distance, IN, and 1N, denote the two vectors of all-one elements with N1 and N₂ dimensions. E RRNINN XN2 22 is the expected optimal transport plan.
In context of graph learning, FGW-OT can measure the sum of feature and topology distances between two graphs. Since anchors are derived from the global representation space, thus the anchor graph can be optimally transported to each local subgraph G. with minimal cost. The optimization objective is formulated as
N& NP
min
ΣΣ6 m7n
a-(-[P 7 +a |[A [Aplm.l)) [Tuls.m [xl cH(TT))
ITy ETT
(10)
IT {TT E TR..N NxN ( I M1, I p2).
where H(-) denotes the entropy of certain variable, denotes trade-off constant. The entropy regularization is introduced to ensure that the optimization problem has a unique solution. The product P Z. measures the similarity between the anchors and nodes, then -P; Zt is the distance between the two terms. Problem (10) can be optimized by the proposed method in [37], then an optimal transport matrix Ny N is obtained. 7 reflects the unnormalized transfer probability from nodes to anchors, and its normalized form is formulated as
[Taja Σ, [T
(11)
For the anchor-node similarity matrix P.Z.. its has two-fold meanings. First, it depicts the relationships between anchors and nodes, and its normalized form can also be viewed as their transfer probability. Thus, we have
exp( [Z P (i/T) [Clls ΣS expl ZR P /T))
(12)
Te closed-form solution obtained through algebraic computation and does not have gradients, whereas C2 is an inexact solution with gradients. In addition, C₁ is also used to conduct the anchor
5

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graph Ap Cg.
Cj. C₁ is expected to drive close to Tk, then the loss is written as:
Nu Np
LCB= - - (T log' log(Cela.)..
(13)
Notably, mentioned above, there two augmented latent embeddings: Z and Zg. We perform the OT mapping on the two views, then Le is reformulated as
Lc Lē Ci
(14)
Second, the anchor matrix P represents the entire representation space and is regarded as global space projector. From another perspective, PZ" is understood global-aware representation. In Eq. (8), performs the contrastive learning with local representations and is denoted as L'SSL- In addition, expect the global- representations to be more discriminative well. By projecting Z and Z into global space via anchors Pk.. the global-aware representations Hi ZiP. and H Z.P.P are obtained, then the contrastive learning between them is executed through Eq. (8), the loss is denoted as L'ss Thus, the local SSL loss is reformulated as
LSSL = L'ss + LSsL.
(15)
In summary, the training loss in local clients is concluded as
L LSSL + XCCE,
(16)
where \ denotes the trade-off parameter. Through the back propagation from loss C, the local GNNs and anchors are optimized. When the local training completes, the local GNNs and anchors are uploaded sever for aggregation.
3.2 Low-Rank Tensor Optimization for Adaptively Learning Global GNN Parameters
Traditional federated aggregation performs weighted parameter fusion based on number of local samples. However, this approach has two drawbacks. First, the matrix-level fusion cannot capture the high-order correlations between clients. Second, the fixed weights cannot adaptively determine the importance of each client. In light of the shortcomings, we propose to find the optimal global GNN's parameters in a low-rank tensor space. Concretely, when the server receives local GNNs' parameters, the 1-th layer's parameters (WR.. W. are stacked as third-order tensor W, We argue that the low-rank component L; in W; contains the globally shared information, and the noise component E contains the client-specific information. Then, the former needs to be retained, while the latter should be discarded. Further, we aim to adaptively fuse local GNNs' parameters in the low-rank tensor space for obtaining the optimal global GNN parameters A₁. Considering the above concerns, the following optimization objective is proposed (the subscript is omitted for concise expression):
K
min A,E,A,o
||[| + +1111|
+ Σ (ax)" (ILL: -Aliz)
(17)
K
a% = s.t. W = C C3 Σ A=1
1, az 2 0, 4k [K].
where /3 is a trade-off parameter, ag is the weight for the k-th client, r a constant to smooth the distribution of weights. The optimization problem (17) can be solved by the alternating direction method of multipliers (ADMM). The augmented Lagrangian function (17) conducted follows:
F L: A; (ax)
K |||| + BB|11111 + Σ (on))" L A + W.W- L --E)+ - L E1
(18)
k=1
+ (.. C) + カリ 2
where is an auxiliary variable, y and H denote the two Lagrange multipliers, l a trade-off parameter. The update rule for each variable is presented as follows.

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Update G: Fixing L, E, A, fax 1= the subproblem with respect to g is written as
min 9
| @ +
(H,
g L) + | マタト و|g C
(19)
g (c (.-H1 H |
min g @ + #
-
2
For the solution of low-rank tensor 9, study [39] provided a tensor tubal-shrinkage method R, the updated g is obtained by
== R1/ -x). 1
(20)
Update L:
When optimizing C, the update rule for its each slice Lg is same. Fixing g. E, A, {ox) the subproblem for Lg is formulated as
min (axs) )" L All + &Y We L1 E Lu
1 I LL Exll
(21)
H. G - L + 2 G L
Taking the partial derivative with respect to La and setting to zero, the update rule for Lj is derived:
L 2(0x)* A + n. - Er + Y - n + H 2(ax)* + 2p
(22)
Update A: Masking variables unrelated A, we obtain the following subproblem:
K
min
Le -
(23)
The update rule for can be obtained by computing the partial derivative with respect A and equating it to zero, then we have
A km (ax)
(24)
Iam("a))
Update E: When & is updated, the other variables viewed constants, the subproblem is formulated as
min 8|| + (y.W-
W - C E
-
1 Hlw E11
E
(25)
min 3| 8| +
-W - ///1//
E
Let B = W - L 1/H, the update method with respect to E is
E" max A/μ, 0) min A/ 0).
(26)
Update ag: Neglecting variables unrelated to ag,, the subproblem about ag written as
min (ax) (I A...). s.t. Σ = 2 0, Ak [K].
(27)
Taking the partial derivative with respect to ag and setting it to zero, we have
[ A103))/001)))
of K [1 A 1/1-))/(1-)
(28)
Update y, H. μr:
H' = H - w( -- min(w μ. (mma);
(29)
where and jlnar are two predefined parameters. The optimal global GNN parameters are ob- tained through the iterative optimization based on ADMM described above. Thus, the global GNN parameters and anchors are distributed to clients in the next communication.

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Table 1: Performance comparison on Cora, CiteSeer, PuhMed, and Ogbn-Arxiv datasets with three different SSL losses, where the optimal results bolded and the suboptimal results underlined.
Cora
CiteSeer
PubMed
Oghn-Arxiv
SSL
Method
ACC
Fscore
ACC
Fscore
ACC
Fscore
ACC
Fscore
FedAvg
54.38
39.27
36.30
20.34
63.81
50.88
35.26
20.49
FedProx
55.35
40.83
36.80
21.34
63.81
50.88
34.61
20.03
Simsiam
MOON
54.03
39.81
35.94
20.41
63.80
50.87
35.46
21.20
FedU2
56.04
42.65
39.00
25.77
61.27
48.09
35.39
22.00
FedPAM
61.40
52.85
49.53
38.73
64.23
51.92
35.97
23.07
FedAvg
54.65
39.95
35.48
19.62
64.08
51.46
45.58
33.08
FedProx
55.26
40.68
35.56
19.78
63.95
51.17
44.40
31.86
MOON
54.56
39.81
36.30
20.34
64.42
52.17
46.23
33.96
SimCLR
FedX
56.73
43.11
37.10
22.37
61.77
49.09
46.69
34.77
FedU²
57.09
43.45
40.89
27.54
66.79
56.25
48.41
36.46
FedPAM
58.62
45.21
43.90
31.61
69.22
59.42
48.19
37.25
FedAvg
63.01
50.51
42.65
29.72
61.85
49.62
35.29
20.73
FedU
64.06
51.92
46.81
34.98
62.08
49.82
38.07
24.15
FedEMA
63.53
51.23
43.59
30.97
64.47
53.39
38.12
24.07
BYOL
Orchestra
54.47
39.63
37.54
24.39
61.27
48.09
35.26
20.49
FedU²
65.11
54.16
47.32
35.79
66.51
55.86
39.95
26.03
FedPAM
66.01
55.23
48.81
38.69
66.67
56,25
40.55
26.79
4 Experiments
4.1 Experimental Setups
Eight graph datasets are selected as benchmark datasets, including Cora, CiteSeer, PubMed, Ogbn-Arxiv, Computers, Photo. Physics, Amazon-ratings. The eight datasets vary in terms of scales, and cover different types, such as citation network, co-purchase network, co-author network, rating network. Each graph is divided into ten subgraphs via the Louvain method [40].
Graph Datasets.
Compared Methods. We compare the proposed FedPAMM with typical and SOTA FL algorithms, including FedAvg [31], FedProx [41], MOON [42], FedX [43], FedU [34], FedEMA [35]. Orches- tra [32], U2 [36]. local training, three SSL strategies are adopted, including Simsiam [44],
SimCLR [30], BYOL [45]. Specifically, Simsiam and BYOL are non-contrastive methods while SimCLR contrastive method. We adopt the standard linear probing to evaluate the performance of algorithms, ACC and Fscore are used as the evaluation metrics.
4.2 Performance Comparison
We evaluate the performance of FL algorithms using three SSL strategies eight graph datasets, the results are reported in Tables 1 and 2. Some interesting phenomenon can be observed. First, FL algorithms exhibit varying performance when equipped with different SSL losses. For instance, Cora dataset, employing B BOO leads to improvements by 8.63% and 8.36% in ACC compared to the other SSL losses for FedAvg, respectively. Encouragingly, the proposed FedPAM consistently achieves superior performance with different SSL methods, validating the contributions of RSA and AGPL. Second, compared baseline methods, the proposed FedPAM demonstrates more significant improvements on homogeneous graphs than on heterogeneous graphs. Node attributes and edge types homogeneous graphs are less diverse, then the learnable anchors are more likely to span the global representation space. Furthermore, homogeneous graphs induce local models to be more similar, facilitating low-rank tensor optimization find better global model parameters.
4.3 Ablation Study
RSA and AGPL paly critical roles for the proposed FedPAM. The results of their ablation studies reported in Table 3. When both modules are removed, the model degenerates into FedAvg and

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Table 2: Performance comparison Computers, Photo, Physics, and Amazon-ratings datasets with three different SSL. losses, where the optimal results are holded and the suboptimal results underlined.
Computers
Photo
Physics
Amazon-ratings
SSL
Method
ACC
Fscore
ACC
Fscore
ACC
Fscore
ACC
Fscore
FedAvg
63.09
50.92
72.01
63.56
75.55
67.73
36.72
19.79
FedProx
64.05
51.70
73.26
65.08
75.33
66.22
36.64
19.83
Simsiam
MOON
63.92
52.16
72.46
64.20
77.82
70.79
36.69
19.86
FedU²
63.20
51.03
72.65
64.57
86.26
81.02
36.72
19.79
FedPAM
64.19
53.93
74.06
66.96
86.51
81.86
37.45
24.97
FedAvg
78.21
72.27
84.59
79.82
79.07
71.30
36.56
20.06
FedProx
78.41
72.56
84.66
79.82
77.05
67.74
36.46
19.92
MOON
78.90
72.53
84.72
79.96
80.23
73.10
36.72
20.05
SimCLR
FedX
80.60
75.09
85.49
80.72
80.31
73.60
36.70
19.79
FedU²
79.24
73.34
84.94
80.15
81.03
74.27
39.58
27.92
FedPAM
81.63
77.42
85.78
81.25
83.76
78.16
39.60
29.69
FedAvg
63.38
52.20
71.12
61.93
84,99
79.13
36.77
21.05
FedU
65.00
54.92
74.19
67.42
85.36
79.54
37.11
24.07
FedEMA
65.44
56.38
76.81
70.31
83.39
77.53
38.32
27.22
BYOL
Orchestra
62.84
52.67
73.74
67.22
72.16
62.10
36.81
20.12
FedU²
65.55
54.73
74.86
68.11
85.41
79.74
36.82
20.50
FedPAM
66.66
57.57
80.50
75.34
86.43
80.98
38.43
26.52
Table 3: Ablation study with respect key mod-
ules RSA and AGPL on Cora and CiteSeer datasets, where SimCLR used the local training.
Metrics
Cora
CiteSeer
RSA
AGPL
ACC
Fscore
ACC
Fscore
Number of Anchors
Number Anohors (b) CiteSeer
54.65
39.95
35.48
19.62
(a) Cora
X
/
56.65
44.18
41.35
27.80
X
58.03
45.27
37.89
22.65
Figure 4: Performance comparison with different num-
bers anchors on Cora and CiteSeer datasets, where
58.62
45.21
43.90
31.61
SimCLR loss is used in the local training.
naturally achieves the worst performance. When either RSA and AGPL module is present, the performance the model improved, indicating that both representation space alignment and adaptive global parameter learning are effective. Certainly, the performance is optimal with both modules available.
FRAI
ACC (56)
ACC
4.47
Cora
CiteSear
(a)
Cora
(b) CiteSeer
Figure 5: Sensitivity study with respect À on Cora and CiteSeer datasets, where SimCLR loss is used in the local training.
Figure 6: Scnsitivity study with respect to / on Cora and CiteSeer datasets, where SimCLR loss is used in the local training.
4.4 Parameter Sensitivity Investigation
We investigate the impacts of three key hyperparameters in the proposed FedPAM, including the number of anchors M, X in the local training, and 3 in the low-rank tensor optimization. From Fig. 4, it can be seen that either too few or too many anchors is detrimental to achieving optimal

---

performance. Excessive anchors tend to reduce discriminability, while too few anchors may lead to representation collapse. controls the strength of the optimal transport, large value of À facilitates better alignment of representation spaces as shown in Fig. 5. However, setting / too high is not advisable as shown in Fig. 6, because may hinder the capture of client-specific information.
Conclusion
In this paper, we address two key challenges in unsupervised FGL; the misalignment of representation spaces and non-adaptive federated aggregation schemas. For the first challenge, we propose the RSA module, which aims to learn a set of anchors across clients and use them to align representation spaces of different clients. For the second challenge, the AGPL, module is introduced, employing the low-rank tensor optimization adaptively learn global model. Finally, the experimental results on eight graph datasets verify the effectiveness of the proposed FedPAM. More details about related work, algorithm, dadasets, and experimental results can be referred in the Appendix.
Acknowledgments
The research is supported by the National Key R&D Program of China (2023YFB2703700) the National Natural Science Foundation of China (62176269).
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A
Summary of Appendix
We present the following sections in the appendix as supplement to this manuscript:
Related work about FGL, OT, and low-rank tensor optimization.
The algorithm flow of the proposed FedPAM.
Implementation details for the proposed FedPAM.
The analysis for the computational complexity.
The details of the eight graph datasets used.
The performance comparison using KNN as the evaluation metric.
The performance comparison with large-scale clients.
The performance comparison of plugging the AGPL module into existing works.
B
Related Work
B.1 Federated Graph Learning
The objective of federated graph learning is enable the collaborative training of GNN multiple decentralized graphs, thereby improving its generalization. In terms of training data, FGL is generally divided into two categories: graph-level and node-level. Graph-level FGL focuses addressing challenges related to label or feature heterogeneity. For example, [15, 16] proposed to cluster clients or share topology patterns for Non-IID graphs. To mitigate the domain gaps between private graphs, [46, 17] captured the cross-domain knowledge to enhance the local personalized inference. In contrast, node-level FGL primarily targets challenges arising from topological heterogeneity. [18, 9, 19] leveraged the global information to assist the local training. [47, 48) emphasized the impact of topological structures on federated learning and proposed the topology-awart aggregation method. While the aforementioned approaches provide valuable insights, they overlook the problem of FGL in unsupervised scenarios. We recognize the challenges of unsupervised FGL and propose novel approach to address them.
B.2 Optimal Transport
Optimal transport [49] is used to find the shortest distance for transforming one distribution into another, and has been widely applied in the fields of domain adaption and domain generalization. [50, 51] aligned the distributions between the source domain and the target domain via OT. As an extension of classical optimal transport, GW-OT is designed to align structural relationships between data distributions, making it particularly suitable for applications in graph match tasks. For instance, [52, 53] aimed to match the topological structures between various graphs via GW-OT. Furthermore, to simultancously align the distributions in terms of features and structures, the FGW-OT is proposed. [37, 5] adopted the FGW-OT to achieve optimal mapping from source graphs target graphs through features and topologies. For unsupervised FGL., how to explore the global representation space critical challenge, we leverage the FGW-OT learn a group of anchors across multiple private subgraphs for positioning the global coordinate.
B.3 Low-rank Tensor Decomposition
Low-rank tensor decomposition seeks to recover the underlying low-rank components from a given tensor, and has found extensive applications image restoration, data mining, and data compression tasks. [54, 55, 56] employed the tensor singular value decomposition to explore the principal information of multi-source data. Deep learning provides new solution for exploring the low- rankness, [57, 58] used neural networks to nonlinearly capture the low-rank structures. Although there have been some existing studies working on federated tensor computation [59, 60, 61], they mainly focused on distributed tensor decomposition with respect to data features. In contrast, we leverage low-rank tensor decomposition to exploit the complementarity of local models and solve for the global model parameters in the low-rank tensor space.
22

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C The Algorithm Flow of the Proposed FedPAM
Algorithm The main steps FedPAM
Input: Number of clients K, communication rounds T. local training epochs E, learning rate 7,
trade-off parameters À and B, local subgraph Ga- = (Vk, Ex, Xa), local GNN lw..
Output: Global model Fw and Global anchors P.
1: Client Side:
2: for k == = K in parallel do
3: for epoch e 1: E do
4:
Augment the original local subgraph Ga into G and 9 via two different methods; 5: Encode the two views of subgraphs: zg Fw, (G Zz F (9));
6: Calculate the SSL loss for local representations: L'ss Z Z by Eq. (8);
7:
for view i in (a, b) do Calculate the optimal transport matrix T between Zj: and Px by Algorithm 2; Calculate the normalized transport matrix: Ti < (xt) by Eq. (11);
8:
9:
Calculate the transfer probability matrix: C f- (Z Pk) by Eq. (12);
11:
Calculate the CE loss C + (T : C+.) by (13); end for
12:
Calculate the CE loss for two views: LCE t (Ce C. C):
14:
Obtain the global-aware representations: H. Z.P H. Z.P.
15:
Calculate the SSL loss for global- representations: S + (He. H by Eq. (8);
16:
Calculate the total SSL loss LESL (L'ssl: s by Eq. (15);
Update local GNN W W VL LCE); Update local anchors P +- pe-1 v/V(Ls + LCE):
19: end for
20: Upload the local GNN Fw and local anchors Pş to the server,
21: end for
22: Server Side:
23: for :ot=1::2 do
24:
fo do
25:
Stack the [-th layer's parameters {Wr.)) of local model into third-order tensor Wi; Obtain the optimal parameters A₁ from Wi by Algorithm 4;
26:
27:
end for
28: Obtain the global anchors P=
Σ N: Ps:
N 29: Distribute the global GNN Fw and the global anchors P clients;
30:
end for
Algorithm 2 The main steps of fused GW-OT
Input: Node marginal distribution RR- anchor marginal distribution μ E R-M adjacency matrix A E IR xN anchor adjacency matrix S E Ma the transport matrix E R X. in terms of feature level, trade-off parameters and e, maximum iteration round T..
Output: The transport matrix π E RN N
π + rp=:
2: G₁ + S V IT)T:
3: 14-1(A
4: fo
5: G 2a A + Gi (1-
(-2S=A
6: + OT(p, G, m, via Algorithm 3;
7: end for
D Implement Details
2-layer graph convolutional network [62] used as the backbone, and the latent embedding
dimension is set as 128. Following [18, 48], the Louvain method [40] is used to divide the initial
23

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Algorithm 3 The main steps of OT via Sinkhorn algorithm
Input: Node marginal distribution E TR., anchor marginal distribution l E TM cost matrix CN initialized transport matrix я,, trade-oft parameter e, maximum iteration round T', Output: The transport matrix и E NR
1: + e eCee
2: for = do
3: V ju
4: p. p p/Kv;
5: end for
π diag(v) diag(u);
Algorithm 4 The main steps for low-rank tensor optimization
Input: Observed tensor W, trade-off parameters and μr, step w, maximum Mmaz 10 threshold
6. Initial the auxiliary variable g O, the Lagrange multipliers H 0, 0.Y- 0.
Output: Consistent matrix A.
1: while not convergent do
2: Update g-+ <- (C', 71) by Eq. (20);
3: for = 1 K do
4: Update L f- (A Z E Y. G H og by Eq. (22);
5: end for
6: Update 1+1 f- (L ) by Eq. (24);
7:
Update *h + (B', 3/ 1/33²)))) by Eq. (26):
8:
Update O'K ++1 < (Li A by Eq. (28);
9:
Check the convergence conditions:
max([) [[x[| L g 1 C
2'
E Se
10: D- When the conditions are satisfied, the iteration ends, otherwise it continues;
11:
t= 1+
12:
end while
graph to multiple local subgraphs, the number of clients is set 10. We use Adam the optimizer with the learning rate set to 0.001. The numbers of communication round and local training rounds set 100 and 5, respectively. When the federated training completes, we use the standard linear probing to evaluate the performance of algorithms. The values of ACC and Fscore on test sets are reported. Some trade-off parameters are required to be predefined, a and in FGW-OT is fixed as 0.5 and 1. À is tuned in {0.1, 5, 10, 50, 70, 100}. 3 is varied in {0.1, 0.5, 1}. the number of anchors tuned (30, 60, 100, 500, 1000).
E The Analysis for the Computational Complexity
The computation complexity mainly originates from two aspects: the FGW-OT for aligning represen- tation spaces and the low-rank tensor optimization for adaptively learning global model parameters. For the FGW-OT, the computation of G in Algorithm 2 takes O(NeM + M2 where Ng de- notes the number of edges, M denotes the number of anchors, and N denotes the number nodes. Notably, the adjacency matrices often sparse, the computational complexity cantly decreased. Likewise, the OT based on Sinkhorn algorithm in Algorithm 3 takes O(M N). Then, the FGW-OT takes O(Ng M M MN. For the low-rank tensor optimization, the update of G with Dt,1 x D1.2 X K in Algorithm 4 occupies the dominant computational complex- ity. First, the FFT and the inverse FFT require O ODD2, Di,2 log(K)) (K)). where D1,1 and Di,2 are the dimensions for the 1-th layer's 's parameters, is the number clients, and Dt,1 is the higher di- mension. Second, the T-SVD costs O(Dr,1 D2 K). Then, the low-rank tensor optimization takes O(D?, Dr. (og(K)= + Dr, K). Overall, the computational complexity for the proposed FedPAM is ONNE M + MN + Di,, Dr. log (K + Dr, Di,KK.
be signifi-
24

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The Details of Eight Graph Datasets
We conduct the experiments on eight graph datasets, including Cora, CiteSeer, PubMed, Ogbn-Arxiv. Computers, Photo, Physics, Amazon-ratings Their detailed information about the numbers of nodes, features, edges, classes, the proportions of dataset divisions, and the categories of datasets are described in Table 4. The statistical information of local subgraphs for eight graph datasets is presented in Fig. 7.
Table Descriptions of eight graph datasets.
Dataset
Nodes
Features
Edges
Classes
Train / Val / Test
Category
Cora
2,708
1,433
5,429
7
20% 40% / 40%
Citation Network
CiteSeer
3,327
3,703
4,732
20% 40% / 40%
Citation Network
PubMed
19,717
500
44,338
3
20% 40% / 40%
Citation Network
Ogbn-Arxiv
169,343
128
231,559
40
60% 20% / 20%
Citation Network
Computers
13,381
767
245,778
10
20% 40% / 40%
Co-purchase Network
Photo 7,487
745
119,043
8
20%
40% / 40%
Co Co-purchase Network
Physics
34,493
8,415
247,962
5
20%
40% / 40%
Co-author Network
Amazon-rating
24,492
300
93,050
5
50% 25%
25%
Rating Network
Toda
2000
TOSaI
X
10
Closs ID
Class ID
Class ID 2
Ciien
"Cienti ID
ID PubMed
Clant (d) Ogbn-Arxiv
(a) Cora
(b) CiteSeer
3318
Total
Teal 001
Toal
Chaes ID
Class
Class
Clent
(e) Computers
(f) Photo
(g)
Physics
(h) A mazon-ratings
Figure 7: Statistical information of local subgraphs for eight graph datasets when the number of clients is fixed
as 10.
G The Performance Comparison Using KNN as the Evaluation Metric
Standard linear probing is sensitive to linearly differentiable features and may not accurately evaluate the model if the data distribution complex. In addition using the standard linear probing, we use KNN as the evaluation metric, the results are reported in Table 5. When using Simsiam and BOYL as the SSL losses on CiteSeer dataset, the values of evaluation metries are much better than that when using standard linear probing. Hence, employing different evaluation methods provide more comprehensive understanding of models. Fortunately, can be seen that the proposed FedPAM achieves optimal performance in most cases, proving that embedding output by FedPAM has better discriminative property.
H
The Performance Comparison with Large-Scale Clients
Performance validation on large-scale clients is an important measure for the scalability of FL methods, we test the performance of various methods with and 100 clients on PubMed and Ogbn-Arxiv (OA) datasets, the results are presented in Fig. 8, where SimCLR is used as the SSL loss.
25

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Table 5: Performance comparison on Cora and CiteSeer datasets with three different SSL losses. The KNN is used evaluation method, where the optimal results are bolded and the suboptimal results underlined.
Cora
CiteSeer
SSL
Method
ACC
Fscore
ACC
Fscore
FedAvg
54.55
42.90
36.66
25.71
FedProx
55.17
41.80
39.59
28.01
Simsiam
MOON
56.28
44.83
36.75
26.75
FedU²
54.90
41.92
40.99
30.44
FedPAM
58.93
47.27
50.40
38.01
FedAvg
52.49
38.05
37.77
24.60
FedProx
53.28
39.10
36.37
22.73
MOON
53.01
38.74
37.68
24.80
SimCLR
FedX
54.31
40.62
42.74
31.16
FedU²
54.30
40.68
38.78
25.78
FedPAM
57.71
44.53
42.07
30.43
FedAvg
64.74
53.63
44.88
34.17
FedU
63.35
52.59
46.25
35.33
FedEMA
64.20
52.53
43.30
30.62
BYOL
Orchestra
54.12
41.25
36.89
24.65
FedU2
59.87
48.18
44.58
33.33
FedPAM
64.99
53.08
49.25
39.68
can seen that the proposed FedPAM stays ahead the curve even with large number of clients, showing that representation space alignment and adaptive global parameter learning are effective. Notably, the superiority of the proposed FedPAM is more obvious terms of Fscore, indicating that the proposed FedPAM not only has high overall prediction accuracy, but also shows better performance on the minority classes.
2213
6558
7247
70A4 7343
50
coll
rud
(1d)
PubMed: 50 Clients
PubMed: 50 Clients
PubMed: 100 Clients
PubMed: 100 Clients
50.0
4273
17.5
(4.
7ull"
40
0-2
FuBM
OA: 50 Clients
OA: 50 Clients
OA: 100 Clients
OA: 100 Clients
Figure 8: Performance comparison with more clients on PubMed and Ogbn-Arxiv (OA) datasets, where the number of clients is set to 50 and 100, respectively.
I The Performance Comparison of Plugging the AGPL Module into Existing Works
Notably, the proposed AGPL module is plug-and-play module, which can be incorporated into existing works and promote their performance. Specifically, we plug AGPL into FedAvg, FedProx,
26

---

Table 6: Performance comparison Cora, CiteSeer, and Ogbn-Arxiv datasets when the AGPI. module is plugged into existing works, where the optimal results bolded.
Cora
CiteSeer
Ogbn-Arxiv
SSL
Method
ACC
Fscore
ACC
Fscore
ACC
Fscore
FedAvg
54.38
39.27
36.30
20.34
35.26
20.49
FedAvg+AGPL
58.56
46.67
43.29
30.60
35.33
21.21
FedProx
55.35
40.83
36,80
21.34
34.61
20.03
Simsiam
FedProx+AGPL
60.15
47.07
43.75
31.58
35.26
20.49
MOON
54.03
39.81
35.94
20,41
35.46
21.20
MOON+AGPL
58.92
46.22
41.34
30.42
35.44
21.59
FedAvg
54.65
39.95
35.48
19.62
45.58
33.08
AGPP
56.65
44.18
41.35
27.80
47.10
37.13
FedProx
55.26
40.68
35.56
19.78
44.40
31.86
FedProx+ AGPL
56.79
46.29
41.54
28.61
47.36
35.01
SimCLR
MOON
54.56
39.81
36.30
20.34
46.23
33.96
MOON+AGPL
56.49
43.74
41.82
29.26
47.91
39.04
FedX
56.73
43.11
37.10
22.37
46.69
34.77
FedX+AGPL
57.97
44.56
42.92
30.75
48.76
39.56
MOON, and FedX with Simsiam and SimCLR losses, respectively. Table 6 reports the experimental results. It can be observed that the performance of existing works after equipped with AGPL is improved, demonstrating its effectiveness.
27


Reviews:
[{"review_id": "Ksuozwuwgh", "reviewer": "Reviewer_cZbY", "summary": "This paper addresses the unsupervised Federated Graph Learning (FGL) problem and proposes a novel framework named FedPAM, which consists of two key modules: (1) Representation Space Alignment (RSA), which aligns the representation spaces of local clients using learnable anchors and Fused Gromov-Wasserstein Optimal Transport (FGW-OT); and (2) Adaptive Global Parameter Learning (AGPL), which fuses local model parameters in a low-rank tensor space via tensor decomposition. The authors validate the method through extensive experiments on eight graph datasets using various self-supervised learning (SSL) strategies.", "questions": "See weaknesses.", "limitations": "The author should include the limitation section to discuss limitations and potential negative societal impact of this work.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. *Novel and well-motivated problem setting*: The paper focuses on unsupervised FGL, a relatively under-explored yet practically important setting where label scarcity and data heterogeneity present serious challenges.\n2. *Well-organized methodology*: The proposed method is intuitive and reasonable, including both RSA and AGPL from two perspectives.\n3. *Clear writing and presentation*: Despite the technical depth, the paper is generally well-written with clear motivation, formulation, and visual illustrations (e.g., Figure 1–3).\n\nWeaknesses:\n1. *Limited theoretical support*: While the paper proposes technically rich modules, it lacks formal analysis (e.g., convergence, generalization bounds, or even complexity analysis in the main text), which would have strengthened the methodological contribution.\n2. *Computational overhead and scalability unclear*: The main paper lacks discussion on training cost, communication overhead, or scalability in practical federated environments.\n3. *Out-dated baselines*: Only one baseline, FedU$^2$, was proposed after 2023. More advanced baselines should be included to demonstrate the advantage of proposed method. Moreover, some remarkable related works, such as [1-3], are missing, which need to be added and discussed.\n\n[1] Federated Contrastive Learning of Graph-Level Representations.\n\n[2] Federated Self-Explaining GNNs with Anti-shortcut Augmentations.\n\n[3] Subgraph Federated Unlearning.", "was_revised": true, "final_justification": "In the rebuttal period, the authors made good clarification and improved this paper, such as the theoretical analysis. After consideration, I give a boardline acc to this work.", "rebuttal": "**R1 to W1**: Thank you for this constructive advice.\nFor a comprehensive understanding of the proposed FedPAM, we provide the analysis for the **generalization bounds** and **complexity analysis**. Given a FL system, its generalization bound for the global model is defined as **Lemma 1**.\n\n>**Lemma 1**. Given a FL system with global distribution ${\\mathcal{D}}$ and local distribution ${\\mathcal{D}}_{k}$, the generalization error for any hypothesis with the probability at least $1-\\delta$ ($0<\\delta\\leq 1$) is\n\n$$\n\\mathcal{R}\\_{\\mathcal{D}}(h) \\leq \\frac{1}{K} \\sum\\_{k \\in[K]} \\hat{\\mathcal{R}}\\_{\\tilde{\\mathcal{D}}\\_k}\\left(h\\_k\\right)+\\frac{1}{K} \\sum_{k \\in[K]}\\left(d\\_{\\mathcal{H} \\Delta \\mathcal{H}}\\left(\\tilde{\\mathcal{D}}\\_k, \\tilde{\\mathcal{D}}\\right)+\\lambda\\_k\\right)\n+\\sqrt{\\frac{4}{m}\\left(d \\log \\frac{2 e m}{d}+\\log \\frac{4 K}{\\delta}\\right)}.\n$$\n\nFor the $\\mathcal{H}$-divergence $d_{\\mathcal{H}\\Delta\\mathcal{H}}(\\tilde{\\mathcal{D}}_{k},\\tilde{\\mathcal{D}})$, we further have\n\n$$\n\\begin{aligned}\n\td\\_{\\mathcal{H} \\Delta \\mathcal{H}}\\left(\\tilde{\\mathcal{D}}\\_k, \\tilde{\\mathcal{D}}\\right) & \\leq d\\_{\\mathcal{H} \\Delta \\mathcal{H}}\\left(\\tilde{\\mathcal{D}}\\_k, \\tilde{\\mathcal{D}}\\_l\\right)+d\\_{\\mathcal{H} \\Delta \\mathcal{H}}\\left(\\tilde{\\mathcal{D}}\\_l, \\tilde{\\mathcal{D}}\\right) (K-1) d\\_{\\mathcal{H} \\Delta \\mathcal{H}}\\left(\\tilde{\\mathcal{D}}\\_k, \\tilde{\\mathcal{D}}\\right) & \\leq \\sum\\_{l \\neq k}^K\\left[d\\_{\\mathcal{H} \\Delta \\mathcal{H}}\\left(\\tilde{\\mathcal{D}}\\_k, \\tilde{\\mathcal{D}}\\_l\\right)+d\\_{\\mathcal{H} \\Delta \\mathcal{H}}\\left(\\tilde{\\mathcal{D}}\\_l, \\tilde{\\mathcal{D}}\\right)\\right] \\\\\n\\end{aligned}\n$$\nAssume $\\forall l \\in [K]$, $d\\_{\\mathcal{H}\\Delta\\mathcal{H}}(\\tilde{\\mathcal{D}}\\_{l},\\tilde{\\mathcal{D}})\\leq \\epsilon\\_{0}$, it can obtain\n$$\n(K-1) d\\_{\\mathcal{H} \\Delta \\mathcal{H}}\\left(\\tilde{\\mathcal{D}}\\_k, \\tilde{\\mathcal{D}}\\right)\n\\leq \\sum\\_{l \\neq k}^K\\left[d\\_{\\mathcal{H} \\Delta \\mathcal{H}}\\left(\\tilde{\\mathcal{D}}\\_k, \\tilde{\\mathcal{D}}\\_l\\right)+\\epsilon\\_0\\right]\n$$\n\n$$\n\\frac{1}{K} \\sum\\_{k \\in[K]} d\\_{\\mathcal{H} \\Delta \\mathcal{H}}\\left(\\tilde{\\mathcal{D}}\\_k, \\tilde{\\mathcal{D}}\\right)\n\\leq \\frac{1}{K(K-1)} \\sum\\_{k \\in[K]} \\sum\\_{l \\neq k}^K\\left[d\\_{\\mathcal{H} \\Delta \\mathcal{H}}\\left(\\tilde{\\mathcal{D}}\\_k, \\tilde{\\mathcal{D}}\\_l\\right)+\\epsilon\\_0\\right]\\\\\n$$\n\n$$\n\\leq \\frac{1}{K(K-1)}\\sum\\_{k \\in[K]} \\sum\\_{l \\neq k}^Kd\\_{\\mathcal{H} \\Delta \\mathcal{H}}\\left(\\tilde{\\mathcal{D}}\\_k, \\tilde{\\mathcal{D}}\\_l\\right) +\\underbrace{\\frac{1}{K(K-1)} \\sum\\_{k \\in[K]} \\sum\\_{l \\neq k}^K \\epsilon\\_0}\\_\\epsilon.\n$$\n\nThen, we have following **Theorem**\n\n>**Theorem 1**. Given a FGL system with global distribution ${\\mathcal{D}}$ and local distribution ${\\mathcal{D}}\\_{k}$, the generalization error for any hypothesis with the probability at least $1-\\delta$ ($0<\\delta\\leq 1$) is\n $$\n\\mathcal{R}\\_{{\\mathcal{D}}}(h) \\leq \\frac{1}{K} \\sum\\_{k \\in[K]} \\hat{\\mathcal{R}}\\_{\\tilde{\\mathcal{D}}\\_k}\\left(h\\_k\\right)\n+\\frac{1}{K(K-1)} \\sum\\_{k \\in[K]} \\sum\\_{l \\neq k}^K d\\_{\\mathcal{H} \\Delta \\mathcal{H}}\\left(\\tilde{\\mathcal{D}}\\_k, \\tilde{\\mathcal{D}}\\_l\\right)+\\epsilon+\\frac{1}{K} \\sum\\_{k \\in[K]} \\lambda\\_k+\\sqrt{\\frac{4}{m}\\left(d \\log \\frac{2 e m}{d}+\\log \\frac{4 K}{\\delta}\\right)}.\n$$\n\nAccording to the GNN's generalization analysis in [1],\nwe can obtain the upper bound of $d\\_{\\mathcal{H}\\Delta\\mathcal{H}}(\\tilde{\\mathcal{D}}\\_{k},\\tilde{\\mathcal{D}}\\_{l})$:\n$$\nd\\_{\\mathcal{H} \\Delta \\mathcal{H}}\\left(\\tilde{\\mathcal{D}}\\_k, \\tilde{\\mathcal{D}}\\_l\\right) \\\\\n$$\n\n$$\n\\leq 1+\\frac{B\\_{rank}}{m} \\sup \\_f\\left\\|f\\left(G\\_k\\right)-f\\left(G\\_l\\right)\\right\\|\\_F \\\\\n$$\n\n$$\n\\leq 1+\\sup \\_f B\\_W^2 B\\_X\\left(\\frac{D\\_{\\min }+m}{D\\_{\\min }^2 m}\\right)\\left\\|\\mathbf{A}\\_k-\\mathbf{A}\\_l\\right\\|\\_F+\\frac{B\\_W^2}{m}\\left\\|\\mathbf{X}\\_k-\\mathbf{X}\\_l\\right\\|\\_F\n$$\nwhere $B\\_{rank}$, $B_{X}$, $D_{min}$ denote relevant constants,\n$\\left\\|\\mathbf{A}_k-\\mathbf{A}_l\\right\\|_F$ and $\\left\\|\\mathbf{X}_k-\\mathbf{X}_l\\right\\|_F$ measure the divergences between different topologies and features, respectively. Then, we can see that when the topological and feature differences of different client subgraphs are reduced, the generalization ability of the global model will be enhanced. The proposed FedPAM uses FGW-OT to achieve the optimal transport between the global anchor graph and each subgraph, which objectively reduces the differences between different subgraphs. Therefore, the generalization ability of the proposed FedPAM will be improved.\n\n[1] Towards understanding generalization of graph neural networks.\n\n**Analysis for Computational Complexity:** The computation complexity mainly originates from two aspects: the FGW-OT for aligning representation spaces and the low-rank tensor optimization for adaptively learning global model parameters.\nFor the FGW-OT, the computation of $G$ (an intermediate variable) takes $\\mathcal{O}(N_{E}M+M^{2}N)$, where $N_{E}$ denotes the number of edges, $M$ denotes the number of anchors, and $N$ denotes the number of nodes.\nNotably, the adjacency matrices are often sparse, so the computational complexity can be significantly decreased.\nLikewise, the OT based on Sinkhorn algorithm takes $\\mathcal{O}(MN)$.\nThen, the FGW-OT takes $\\mathcal{O}(N_{E}M+M^{2}N+MN)$.\nFor the low-rank tensor optimization, the update of $\\boldsymbol{\\mathcal{G}}$ with $D_{l,1}\\times D_{l,2}\\times K$ occupies the dominant computational complexity.\nFirst, the FFT and the inverse FFT require $\\mathcal{O}(D_{l,1}^{2}D_{l,2}\\log(K))$, where $D_{l,1}$ and $D_{l,2}$ are the dimensions for the $l$-th layer's parameters, $K$ is the number of clients, and $D_{l,1}$ is the higher dimension.\nSecond, the T-SVD costs $\\mathcal{O}(D_{l,1}D_{l,2}^{2}K)$. Then, the low-rank tensor optimization takes $\\mathcal{O}(D_{l,1}^{2}D_{l,2}\\log(K)+D_{l,1}D_{l,2}^{2}K)$.\nOverall, the computational complexity for the proposed FedPAM is $\\mathcal{O}(N_{E}M+M^{2}N+MN+D_{l,1}^{2}D_{l,2}\\log(K)+D_{l,1}D_{l,2}^{2}K)$.\n\n**R2 to W2:** Thanks for this good comment. We have explained the computational complexity in \\textbf{R1}.\nFor the proposed FedPAM, its communication overhead is $2\\times(K|\\mathbf{W}\\_{k}|+KML)$, where $|\\mathbf{W}\\_{k}|$ denotes the model parameter volume, $L$ denotes the dimension of anchors.\nCompared to other FGL methods, FedPAM additionally transmit a set of anchors, which are essentially multiple feature vectors and do not take up much communication traffic.\nFor the scalability in practical federated environments, we test FedPAM with large-scale clients, e.g. 50 and 100 clients. The experimental results are reported in Tables 1 and 2. We can see that FedPAM still achieves superior performance.\n\n*Table 1: Performance comparison with 50 clients, where the SimCLR is employed in local training.*\n| Method   | PubMed (ACC) | PubMed (Fscore) | OA (ACC) | OA (Fscore) |\n|----------|------------|---------------|--------|-----------|\n| FedAvg   | 73.57      | 65.15         | 43.59  | 31.57     |\n| FedProx  | 70.42      | 59.40         | 40.83  | 28.98     |\n| MOON     | 70.44      | 59.45         | 42.78  | 30.44     |\n| FedX     | 70.43      | 59.43         | 47.81  | 34.83     |\n| FedU$^2$ | 72.65      | 63.60         | 44.07  | 30.62     |\n| **FedPAM** | **74.29** | **67.31**     | **49.85** | **38.34** |\n\n*Table 2: Performance comparison with 100 clients, where the SimCLR is employed in local training.*\n| Method   | PubMed (ACC) | PubMed (Fscore) | OA (ACC) | OA (Fscore) |\n|----------|------------|---------------|--------|-----------|\n| FedAvg   | 68.21      | 56.76         | 37.90  | 26.01     |\n| FedProx  | 68.23      | 56.78         | 36.18  | 24.13     |\n| MOON     | 68.08      | 56.62         | 38.84  | 26.95     |\n| FedX     | 69.27      | 58.25         | 47.04  | 33.18     |\n| FedU$^2$ | 71.82      | 62.66         | 38.08  | 26.23     |\n| **FedPAM** | **72.36** | **65.22**     | **48.58** | **37.32** |\n\n**R3 to W3:** Thank you for this valuable suggestion. We added a new unsupervised FL algorithm, namely FLPD [1]. The experimental results are shown in Table 3. It can be seen that the proposed FedPAM achieves more superior performance.\nThe differences between FedPAM and the three works [2][3][4] are from two aspects.\nFirst, we propose a new framework for unsupervised federated subgraph learning. To the best of our knowledge, we are the first to study this task. Second, the two proposed modules, RSA and AGPL, are novel and can well address the challenges of unsupervised federated graphs.\n\n*Table3: Performance comparison with a new unsupervised FL method FLPD.*\n| SSL       | Method  | Cora ACC | Cora Fscore | CiteSeer ACC | CiteSeer Fscore |\n|-----------|---------|----------|-------------|--------------|-----------------|\n| **Simsiam** | FLPD    | 57.27    | 45.43       | 40.63        | 26.41           |\n|           | FedPAM  | **61.40**| **52.85**   | **49.53**    | **38.73**       |\n| **SimCLR**  | FLPD    | 56.95    | 43.22       | 38.91        | 22.87           |\n|           | FedPAM  | **58.62**| **45.21**   | **43.90**    | **31.61**       |\n| **BYOL**    | FLPD    | 65.37    | 54.01       | 44.82        | 32.34           |\n|           | FedPAM  | **66.01**| **55.23**   | **48.81**    | **38.69**       |\n\n\n[1] Prototype similarity distillation for communication-efficient federated unsupervised representation learning\n\n[2] Federated Contrastive Learning of Graph-Level Representations\n\n[3] Federated Self-Explaining GNNs with Anti-shortcut Augmentations\n\n[4] Subgraph Federated Unlearning"}, {"review_id": "RnCZaDma3S", "reviewer": "Reviewer_9QFW", "summary": "This paper proposes a new framework, FedPAM, to tackle unsupervised federated graph learning (FGL). The authors identify two major challenges in this setting: (1) the difficulty in aligning the representation spaces across clients due to lack of shared semantics, and (2) the limitations of conventional averaging in aggregating heterogeneous local models. To address these, FedPAM integrates two core components: (1) Representation Space Alignment (RSA): Uses learnable global anchor points and fused Gromov-Wasserstein Optimal Transport to align local client embeddings into a shared global space. (2) Adaptive Global Parameter Learning (AGPL): Stacks local model parameters as tensors and applies low-rank tensor optimization to adaptively fuse them, capturing higher-order correlations among clients. The framework is evaluated on eight graph datasets with three types of self-supervised losses. The results show consistent performance improvements over existing federated and unsupervised baselines.", "questions": "Please see Weaknesses", "limitations": "Yes", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- The paper focuses on a novel unsupervised federated graph learning setting, which is a real need for privacy-preserving, scalable graph learning.\n\n- Experiments are conducted on eight diverse graph datasets, using three SSL losses, and compared against a wide range of baselines. Ablation studies, parameter sensitivity, and visualizations are included to support the method’s effectiveness and robustness.\n\n- The paper is overall easy to follow and well-organized.\n\nWeaknesses:\n- Both FGW-OT and tensor decomposition with ADMM are computationally expensive. This limits the implementation of FedPAM to large-scale FGL systems.\n\n- The proposed challenges of “Inability to align the representation spaces of various clients” and “Unable to adaptively learn the optimal parameters of global model” fail to capture the unique characteristics of the unsupervised FGL problem. The experimental settings also follow the normal graph partitioning strategies in previous supervised FGL problems. It would be helpful if the authors could validate the challenges in the experiment section.\n\n- It seems that both the proposed alignment and aggregation techniques are independent with the unsupervised learning nature of the problem. Can we use them in normal supervised FGL systems?\n\n- It would be helpful to add experimental results of the complexity or time cost of FedPAM.\n\n- The results are reported as point estimates without error bars or statistical testing, which limits confidence in the observed gains.", "was_revised": true, "final_justification": "The additional experiments and clarifications helped address some of my concerns, particularly regarding scalability and the broader applicability of the proposed components. While I still find the motivation somewhat generic to the unsupervised FGL setting.", "rebuttal": "**R1 to W1:** Thank you for this valuable comment. We test the proposed FedPAM with more clients. The experimental results are reported in Tables 1 and 2.\nWe can see that FedPAM can still maintain its leading position in large-scale client scenarios. Although FGW-OT and low-rank tensor decomposition bring a large amount of computation, it is worth it.\n\n*Table 1: Performance comparison with 50 clients, where the SimCLR is employed in local training.*\n| Method   | PubMed (ACC) | PubMed (Fscore) | OA (ACC) | OA (Fscore) |\n|----------|------------|---------------|--------|-----------|\n| FedAvg  |73.57 | 65.15| 43.59  | 31.57 |\n| FedProx | 70.42| 59.40| 40.83  | 28.98 |\n| MOON | 70.44 | 59.45 | 42.78  | 30.44 |\n| FedX |70.43 | 59.43 | 47.81  | 34.83|\n| FedU$^2$ | 72.65| 63.60| 44.07  | 30.62     |\n| **FedPAM** | **74.29** | **67.31**| **49.85** | **38.34** |\n\n*Table 2: Performance comparison with 100 clients, where the SimCLR is employed in local training.*\n| Method   | PubMed (ACC) | PubMed (Fscore) | OA (ACC) | OA (Fscore) |\n|----------|------------|---------------|--------|-----------|\n|FedAvg| 68.21| 56.76| 37.90 |26.01|\n|FedProx| 68.23| 56.78| 36.18 |24.13|\n|MOON| 68.08| 56.62| 38.84 |26.95|\n|FedX| 69.27| 58.25|47.04 |33.18|\n| FedU$^2$ | 71.82| 62.66|38.08| 26.23|\n|**FedPAM**| **72.36**| **65.22** |**48.58**| **37.32**|\n\n**R2 to W2:** Thank you for this insightful comment. \nFirst, due to the problem of data heterogeneity, the problem of representation space misalignment exists in both supervised and unsupervised scenarios. In the unsupervised scenario, this problem is more prominent. We centrally train a GNN model on the Cora dataset to output a representation space close to the ideal. Then we train a global model through FedAvg in the supervised and unsupervised scenarios respectively. Through wasserstein distance (WD) calculation, we found that the WD between the global representation output by the supervised FedAvg and the centrally trained representation is 3.67, while the WD between the global representation output by the unsupervised FedAbg and the centrally trained representation is 9.05. Therefore, in the unsupervised scenario, the representation space misalignment is much worse. At the same time, we also illustrate the problem of representation space misalignment in the unsupervised scenario in Figure 1(a).\nSecond, the inability to adaptively learn optimal global model parameters is a problem common in both supervised and unsupervised federated graph learning. The proposed AGPL is a plug-and-play component that can also be used in supervised federated graph scenarios. In the response to weakness 3, we conducted experiments demonstrating its effectiveness in supervised scenarios. In summary, the proposed FedPAM addresses both the specific challenges of unsupervised federated graphs and the general challenges of federated graph learning, making it a significant work.\n\n**R3 to W3:** Thanks for this interesting question.\nIn fact, both of the proposed Representation Space Alignment (RSA) and Adaptive Global Parameter Learning (AGPL) can be used in supervised FGL systems. We compared the supervised federated graph learning algorithms with the proposed FedPAM and the experimental results are reported in Table 3.\nIn the local training, the supervised CE loss is added into training loss.\nNotably, RSA is used to align the representation spaces of different clients, which is especially important in unsupervised scenarios. In supervised scenarios, since the label semantics of different clients are shared, this in itself is a strong alignment signal.\nTherefore, when only the RSA module is used (FedPAM-RSA), the effect is only slightly improved.\nAGPL aims to explore high-level correlations between clients and adaptively learn optimal global parameters, which is applicable in both unsupervised and supervised scenarios. Therefore, its performance is superior to FedPAM-RSA. When using the complete FedPAM, FedPAM achieves remarkable results, even surpassing SOTA unsupervised FGL on the CiteSeer dataset.\n\n*Table: 3 Performance comparison with various supervised FGL methods.*\n| Method | Cora (ACC) | Cora (Fscore) | CiteSeer (ACC) | CiteSeer (Fscore) | PubMed (ACC) | PubMed (Fscore) |\n|---------------|----------|-------------|--------------|-----------------|------------|---------------|\n|FedAvg|73.59|72.04| 68.85 |68.09|82.42|80.20|\n|FedProx| 74.29    | 72.89 | 69.65 | 68.75 | 82.43| 80.21|\n|MOON| 74.22    | 71.88| 68.78 | 68.18 | 82.49| 80.41|\n|FGSSL| 74.47    | 72.94| 70.02 | 69.19 | 82.38| 79.88 |\n|FedPUB| 75.35    | 73.08| 65.36| 62.66| 82.67 | 80.43  |\n|FedTAD| 74.29    | 71.57 | $\\underline{71.32}$ | $\\underline{70.43}$    | 82.72      | 80.26         |\n|FedATH| **77.90**| **76.93**   | 70.32  | 69.25  | **84.06**  | **82.71**     |\n|FedPAM-RSA|74.10    | 72.69       | 69.22 | 68.37           | 82.56      | 80.32         |\n|FedPAM-AGPL| $\\underline{75.83}$ | $\\underline{73.69}$ | 71.22    | 70.10           | 83.03      | 81.21         |\n|FedPAM|75.52|73.40|**71.63** |**70.52** |$\\underline{83.56}$  | $\\underline{81.32}$ |\n\n\n**R4 to W4:** Thank you for this valuable comment. The computation complexity mainly originates from two aspects: the FGW-OT for aligning representation spaces and the low-rank tensor optimization for adaptively learning global model parameters.\nFor the FGW-OT, the computation of $G$ (an intermediate variable) takes $\\mathcal{O}(N_{E}M+M^{2}N)$, where $N_{E}$ denotes the number of edges, $M$ denotes the number of anchors, and $N$ denotes the number of nodes.\nNotably, the adjacency matrices are often sparse, so the computational complexity can be significantly decreased.\nLikewise, the OT based on Sinkhorn algorithm takes $\\mathcal{O}(MN)$.\nThen, the FGW-OT takes $\\mathcal{O}(N_{E}M+M^{2}N+MN)$.\nFor the low-rank tensor optimization, the update of $\\boldsymbol{\\mathcal{G}}$ with $D_{l,1}\\times D_{l,2}\\times K$ occupies the dominant computational complexity.\nFirst, the FFT and the inverse FFT require $\\mathcal{O}(D_{l,1}^{2}D_{l,2}\\log(K))$, where $D_{l,1}$ and $D_{l,2}$ are the dimensions for the $l$-th layer's parameters, $K$ is the number of clients, and $D_{l,1}$ is the higher dimension.\nSecond, the T-SVD costs $\\mathcal{O}(D_{l,1}D_{l,2}^{2}K)$. Then, the low-rank tensor optimization takes $\\mathcal{O}(D_{l,1}^{2}D_{l,2}\\log(K)+D_{l,1}D_{l,2}^{2}K)$.\nOverall, the computational complexity for the proposed FedPAM is $\\mathcal{O}(N_{E}M+M^{2}N+MN+D_{l,1}^{2}D_{l,2}\\log(K)+D_{l,1}D_{l,2}^{2}K)$.\n\nFurthermore, we report the running time of compared methods on three datasets in Table 4. We can see that the proposed FedPAM does indeed have lower runtime efficiency than most comparison algorithms, but in order to achieve satisfactory results, a longer runtime is necessary. At the same time, it is worth noting that compared to FedU$^2$, FedPAM is still acceptable in terms of runtime.\n\n*Table 4: Comparison of running time (seconds) on three datasets.*\n| SSL      | Method    | Cora    | CiteSeer | PubMed   |\n|----------|-----------|---------|----------|----------|\n| **Simsiam** | FedAvg    | 47.72   | 49.23 | 48.32    |\n|          | FedProx   | 58.98   | 59.72    | 60.50|\n|          | MOON      | 67.28   | 67.39    | 66.84 |\n|          | FedU$^2$  | 2040.83 | 2041.00  | 2086.69  |\n|          | FedPAM    | 710.12  | 711.36   | 712.73   |\n| **SimCLR**  | FedAvg    | 41.77   | 42.00    | 41.43    |\n|          | FedProx   | 50.16   | 50.18    | 50.26    |\n|          | MOON      | 66.03   | 66.41    | 67.23    |\n|          | FedX      | 103.54  | 92.16    | 105.75   |\n|          | FedU$^2$  | 2092.78 | 2095.67  | 2102.21  |\n|          | FedPAM    | 582.62  | 602.36   | 605.89   |\n| **BYOL**    | FedAvg    | 62.14   | 62.02    | 62.37    |\n|          | FedU      | 62.10   | 63.78    | 62.70    |\n|          | FedEMA    | 63.33   | 62.45    | 63.00    |\n|          | Orchestra | 90.61   | 86.62    | 155.13   |\n|          | FedU$^2$  | 2137.52 | 2136.59  | 2207.17  |\n|          | FedPAM    | 828.67  | 826.81 | 830.01   |\n\n**R5 to W5:** Thanks for this good comment. In fact, we repeated each experiment five times and reported the mean values. To save space, we did not report the variances. Here, we supplemented the variances for the Cora in Table 5.\nIt can be seen that the variance of the FGL algorithm is relatively small, and these algorithms are generally stable.\n\n*Table 5: Performance comparison with standard deviations*\n| SSL       | Method          | Cora ACC          | Cora Fscore       |\n|-----------|-----------------|-------------------|-------------------|\n| **Simsiam** | FedAvg          | 54.38$\\pm$0.45    | 39.27$\\pm$0.37    |\n|           | FedProx         | 55.35$\\pm$0.42    | 40.83$\\pm$0.47    |\n|           | MOON            | 54.03$\\pm$0.68    | 39.81$\\pm$0.81    |\n|           | Fed$\\operatorname{U}^2$ | $\\underline{56.04\\pm0.82}$ | $\\underline{42.65\\pm0.79}$ |\n|           | FedPAM          | **61.40$\\pm$0.67** | **52.85$\\pm$0.61** |\n| **SimCLR**  | FedAvg          | 54.65$\\pm$0.51    | 39.95$\\pm$0.45    |\n|           | FedProx         | 55.26$\\pm$0.67    | 40.68$\\pm$0.66    |\n|           | MOON            | 54.56$\\pm$0.44    | 39.81$\\pm$0.47    |\n|           | FedX            | 56.73$\\pm$0.26    | 43.11$\\pm$0.28    |\n|           | Fed$\\operatorname{U}^2$ | $\\underline{57.09\\pm0.57}$ | $\\underline{43.45\\pm0.50}$ |\n|           | FedPAM          | **58.62$\\pm$0.66** | **45.21$\\pm$0.57** |\n| **BYOL**    | FedAvg          | 63.01$\\pm$0.77    | 50.51$\\pm$0.69    |\n|           | FedU            | 64.06$\\pm$0.81    | 51.92$\\pm$0.77    |\n|           | FedEMA          | 63.53$\\pm$0.68    | 51.23$\\pm$0.65    |\n|           | Orchestra       | 54.47$\\pm$1.11    | 39.63$\\pm$0.98    |\n|           | Fed$\\operatorname{U}^2$ | $\\underline{65.11\\pm0.62}$ | $\\underline{54.16\\pm0.58}$ |\n|           | FedPAM          | **66.01$\\pm$0.87** | **55.23$\\pm$0.85** |"}, {"review_id": "sIcQlF5ZM0", "reviewer": "Reviewer_pPE4", "summary": "This paper focuses on unsupervised federated graph learning. The authors propose a tailored framework named FedPAM consisting of two components: Representation Space Alignment (RSA) and Adaptive Global Parameter Learning (AGPL). Experiments on eight graph datasets to demonstrate the superiority of the proposed model compared with other baselines.", "questions": "Q1: What is the meaning of $\\mathbf{L}_k$ in Eq. (17)?\n\nQ2: Could the authors explain the intuition of Eq. (17)?\n\nQ3: How do we compute $\\mathcal{E}$?\n\nQ4: What is the value of $r$ in the experiments? How does it affect the performance of the proposed method?\n\nQ5: In Table 1 and Table 2, only some baselines are included. Could the authors explain why?", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "***Strengths***\n\nS1: The studied problem is important and promising.\n\nS2: The authors conduct experiments on eight graph datasets to evaluate the performance of the proposed method.\n\n***Weaknesses***\n\nW1: GNN formulation. The computation of a GNN in Eq. (1) looks questionable. The authors may check how previous studies formulate the forward pass of GNNs. \n\nW2: Using the product $\\mathbf{P}_k \\mathbf{Z}^T_k$ to measure the similarity between the anchors and nodes seems problematic. The product will depend on $\\mathbf{P}_k$’s and $\\mathbf{Z}_k$'s norms as well. A better metric is cosine similarities.\n\nW3: Notations. A letter should not denote different variables in the paper. For example, $\\lambda$ in Eq. (8) and Eq. (16); $\\mathbf{A}$ to denote the adjacency matrix, the relationships between anchors, and the optimal global GNN parameters.\n\nW4: Self-supervised learning strategies. The authors adopt Simsiam, SimCLR, and BYOL. It is quite weird not to use graph self-supervised learning methods.", "was_revised": true, "final_justification": "After the rebuttal, my concerns are addressed by the authors' response. The authors are expected to modify the submission accordingly in the revised version of their paper.", "rebuttal": "**R1 to W1:** Thank you for this rigorous advice. We found that the current form of writing is somewhat unclear. After checking previous work, we changed the forward propagation formula to the following form:\n>$\\mathbf{z}_i^{l+1} =\\sigma(\\mathbf{z}_i^l, AGG(\\mathbf{z}\\_{j}^l; j\\in\\mathcal{N}_i)), \\forall l\\in[L]$,\n\nwhere $\\sigma(\\cdot)$ denotes the activation function, $A G G(\\cdot)$ denotes the aggregation method, $\\mathcal{N}_{i}$ denotes  the neighborhood node set for the $i$-th node.\n\n**R2 to W2:** Thanks for this good question. Using cosine similarity is a more common method. In Eq. (11), we also use cosine similarity to calculate the similarity between $\\mathbf{P\\}_{k}$ and $\\mathbf{Z}\\_{k}$, and then compare the experimental results of different calculation methods in Table 1.\nThe difference in results between using the product and using cosine similarity is not significant. This is because entropy regularization smooths the solution space in the OT calculation, which has a certain normalization effect. Therefore, using the product calculation can still guarantee satisfactory results.\n\n*Table 1: Performance conparison with different similarity calculation methods.*\n| SSL       | Method          | Cora (ACC) | Cora (Fscore) | CiteSeer (ACC) | CiteSeer (Fscore) |\n|-----------|-----------------|----------|-------------|--------------|-----------------|\n| **Simsiam**  | FedPAM-Product  | 61.40    | 52.85       | 49.53        | 38.73           |\n|           | FedPAM-Cosine   | 61.85    | 53.12       | 49.89        | 39.31           |\n| **SimCLR**   | FedPAM-Product | 58.62    | 45.21       | 43.90        | 31.61           |\n|           | FedPAM-Cosine   | 58.93    | 45.83       | 43.90        | 31.61           |\n\n**R3 to W3:** Thanks for this rigorous comment. We will further improve the notations in the following revisions.\n\n**R4 to W4:** Thank you for this good comment. It is worth noting that Simsiam, SimCLR, and BYOL are the three most widely used self-supervised learning methods, applicable in almost all fields. In the field of graphs, the augmented view is achieved by perturbing the edges and masking some feature elements, while the contrast loss is no different, such as work [1] and work [2]. Of course, there are some more complex graph self-supervised learning methods, however, our focus is on how to achieve unsupervised joint modeling in distributed graph scenarios, and local contrast loss is not the focus of research.\n\n[1] Deep Graph Contrastive Representation Learning\n\n[2] Deep Graph Infomax\n\n**R1 to Q1:** Thanks for this rigorous comment. $\\mathbf{L}_{k}$ is the $k$-th slice of low-rank tensor $\\mathcal{L}$ corresponding to the parameters of the kth client.\n\n**R2 to Q2:** Thanks for this good comment. The original intention behind the optimization objective of Eq. (17) comes from two aspects:\nThe original intention behind the optimization objective of Eq. (17) stems from two aspects: First, matrix-based aggregation methods cannot capture higher-order correlations between different clients, and the complementary information between clients cannot be fully utilized. Therefore, we stack the parameters of different clients into a third-order tensor and use low-rank tensor decomposition in high-dimensional space to explore the complementarity between clients.\nSecond, traditional federated aggregation performs weighted aggregation based on sample size, which cannot adaptively learn optimal parameters. Therefore, we aim to learn a set of adaptive client weights while adaptively aggregating model parameters from different clients to obtain optimal global model parameters. Based on these two aspects, we establish a joint optimization loss, as shown in Eq. (17).\n\n**R3 to Q3:** Thanks for the good question. When $\\mathcal{E}$ is updated, the other variables are viewed as constants, the subproblem is formulated as\n$$\n\\min_{\\boldsymbol{\\mathcal{E}}}\\beta||\\boldsymbol{\\mathcal{E}}||_{1} + \\langle\\boldsymbol{\\mathcal{Y}},\\boldsymbol{\\mathcal{W}}-\\boldsymbol{\\mathcal{L}}-\\boldsymbol{\\mathcal{E}}\\rangle + \\frac{\\mu}{2}||\\boldsymbol{\\mathcal{W}}-\\boldsymbol{\\mathcal{L}}-\\boldsymbol{\\mathcal{E}}||\\_{F}^{2} \n$$\n\n$$\n= \\min\\_{\\boldsymbol{\\mathcal{E}}}\\beta||\\boldsymbol{\\mathcal{E}}||_{1} + \\frac{\\mu}{2}||\\boldsymbol{\\mathcal{E}}-(\\boldsymbol{\\mathcal{W}}-\\boldsymbol{\\mathcal{L}}+\\boldsymbol{\\mathcal{Y}}/\\mu)||\\_{F}^{2}\n$$\n\nFor the $l_1$-norm subproblem, we adopt the solution proposed in the work [3], then obtain the update rule for $\\mathcal{E}$, the details can be referred in [3].\n\n[3] The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices\n\n**R4 to Q4:** Thanks for this valuable comment. $r$ is a constant to smooth the distribution of client weights, the larger the value of $r$, the smoother the client weights, otherwise the sharper they are. it is set to 2 on all datasets.\nWe test the FedPAM's performance with differen $r$ in Table 2, it can be seen that most of the experimental results are best when $r=2$, which shows that a slightly smooth client weight distribution is conducive to achieving considerable results. A smooth client weight distribution is conducive to balancing the information of multiple clients, while a sharp client weight distribution will amplify the influence of some clients, resulting in biased fusion.\n\n*Table 2: Performance compasiron (ACC \\%) with different values of $r$.*\n| SSL     | Dataset  | r=0.5 | r=1     | r=2     | r=5  | r=10 |\n|---------|----------|-------|---------|---------|------|------|\n| **Simsiam** | Cora     | 58.82 | **61.56** | 61.40  | 60.21| 57.89|\n|         | CiteSeer | 46.91 | 48.52   | **49.53**| 47.21| 47.10|\n| **SimCLR**  | Cora     | 57.92 | 58.35   | **58.62**| 58.02| 58.02|\n|         | CiteSeer | 41.92 | **44.05** | 43.90  | 42.12| 41.56|\n| **BYOL**    | Cora     | 64.92 | 65.56   | **66.01**| 65.21| 64.23|\n|         | CiteSeer | 47.10 | 48.32   | **48.81**| 47.95| 46.92|\n\n**R5 to Q5:** Thanks for this valuable comment.\nFirst, unsupervised federated learning is still an unexplored field, with limited works and few available comparison methods. Second, we refer to the baseline setting method of unsupervised federated learning work [5] and conduct three types of comparisons based on model architecture. When using different SSL losses, the comparison methods used are also different, which is determined by the model design. For example, FedU, FedEMA, and Orchestra are based on the architecture with online network and target network.\nTherefore, when the SSL loss is Simsiam or SimCLR, these methods are not applicable.\nMOON uses the output of the global model to calibrate the output of the local model, and is also not applicable to the online network and target network architecture.\nIn summary, when using different SSL losses, only some baselines are available.\n\n[5] Rethinking the representation in federated unsupervised learning with non-iid data"}, {"review_id": "HXn0tHCzuG", "reviewer": "Reviewer_Rhdr", "summary": "This paper addresses unsupervised settings in graph federated learning and defines two main challenges: 1) the misalignment of local representations due to the lack of a common signal (i.e., a supervised label), and 2) the difficulty of addressing heterogeneity across clients when aggregating local parameters on the server. The authors propose a global anchor set to align the representation space across clients, and a low-rank-based weighting mechanism for parameter aggregation. The authors provide a theoretically solid method supported by well-designed experiments.", "questions": "See Weaknesses and Questions above.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths.**\n\nS1. The paper is well-written and easy to follow, with the authors presenting their ideas clearly.   \nS2. The paper correctly identifies representation space misalignment as a critical challenge in unsupervised graph federated learning. The proposed solution for this, which leverages a global anchor set, is intuitive and theoretically sound.  \nS3. The experimental design is well-organized.  \n\n**Weaknesses.**  \n \nW1. The computational cost of the proposed method is a potential concern, given its reliance on both optimal transport and tensor decomposition, which are known to be computationally intensive operations.  \nW2. Since $\\mathcal{E}$ can be considered as a non-common semantics across other clients' parameter tensor set, it would be more informative if the authors address the potential privacy leak issues from this point.   \nW3. It would be more helpful to address the potential privacy risks that arise when a client's weight becomes extremely low. Since a low weight signifies that the client is a distinct outlier, this information could potentially be used to infer specific properties about that client.  \n\n**Questions.** \n  \nQ1. The parameter sensitivity study in Figure 5 shows a strong positive correlation between performance and the hyperparameter $\\lambda$, which weights the alignment loss $\\mathcal{L}_ {CE}$. This steep trend raises an interesting question about the trade-off between the representation learning from $\\mathcal{L}_ {SSL}$ and the representation alignment from $\\mathcal{L}_ {CE}$. While $\\mathcal{L}_ {SSL}$ is fundamentally necessary to prevent representation collapse in an unsupervised setting, the results suggest its relative contribution can be small as long as a strong alignment signal exists. It would be insightful if the authors could provide further analysis on this trade-off. For example, is there a performance plateau or even a decrease at much higher $\\lambda$ values? More importantly, what is the minimum relative weight of the SSL loss required to maintain stable and non-trivial representations?\n\nQ2. The AGPL module is presented as a novel method for adaptive parameter aggregation that tackles the shortcomings of standard federated averaging. To better isolate and demonstrate its effectiveness as a generalizable component, it would be highly compelling to see an experiment where AGPL is applied as an \"add-on\" to other baseline methods, such as FedAvg. Such an experiment would provide clearer evidence of AGPL's standalone contribution, independent of the RSA module, and strengthen the paper's claims about its utility.", "was_revised": true, "final_justification": "The rebuttal has addressed my concerns well. My primary concern was the need to demonstrate the isolated effectiveness of the AGPL module, and the authors have successfully shown this in their response, particularly in relation to Table 3. This clarification was crucial. As my concerns are now resolved, I lean towards accepting this paper.", "rebuttal": "**R1 to W1:** Thank you for this good comment. The computation complexity mainly originates from two aspects: the FGW-OT for aligning representation spaces and the low-rank tensor optimization for adaptively learning global model parameters.\nFor the FGW-OT, the computation of $G$ (an intermediate variable) takes $\\mathcal{O}(N_{E}M+M^{2}N)$, where $N_{E}$ denotes the number of edges, $M$ denotes the number of anchors, and $N$ denotes the number of nodes.\nNotably, the adjacency matrices are often sparse, so the computational complexity can be significantly decreased.\nLikewise, the OT based on Sinkhorn algorithm takes $\\mathcal{O}(MN)$.\nThen, the FGW-OT takes $\\mathcal{O}(N_{E}M+M^{2}N+MN)$.\nFor the low-rank tensor optimization, the update of $\\boldsymbol{\\mathcal{G}}$ with $D_{l,1}\\times D_{l,2}\\times K$ occupies the dominant computational complexity.\nFirst, the FFT and the inverse FFT require $\\mathcal{O}(D_{l,1}^{2}D_{l,2}\\log(K))$, where $D_{l,1}$ and $D_{l,2}$ are the dimensions for the $l$-th layer's parameters, $K$ is the number of clients, and $D_{l,1}$ is the higher dimension.\nSecond, the T-SVD costs $\\mathcal{O}(D_{l,1}D_{l,2}^{2}K)$. Then, the low-rank tensor optimization takes $\\mathcal{O}(D_{l,1}^{2}D_{l,2}\\log(K)+D_{l,1}D_{l,2}^{2}K)$.\nOverall, the computational complexity for the proposed FedPAM is $\\mathcal{O}(N_{E}M+M^{2}N+MN+D_{l,1}^{2}D_{l,2}\\log(K)+D_{l,1}D_{l,2}^{2}K)$.\n\nIn addition, we also report the running time on Cora, CiteSeer, and PubMed in Table 1. We can see that the proposed FedPAM does indeed have lower runtime efficiency than most comparison algorithms, but in order to achieve satisfactory results, a longer runtime is necessary. At the same time, it is worth noting that compared to FedU$^2$, FedPAM is still acceptable in terms of runtime.\n\n*Table 1: Comparison of running time (seconds) on three datasets.*\n| SSL      | Method    | Cora    | CiteSeer | PubMed   |\n|----------|-----------|---------|----------|----------|\n| **Simsiam** | FedAvg    | 47.72   | 49.23    | 48.32    |\n|          | FedProx   | 58.98   | 59.72    | 60.50    |\n|          | MOON      | 67.28   | 67.39    | 66.84    |\n|          | FedU$^2$  | 2040.83 | 2041.00  | 2086.69  |\n|          | FedPAM    | 710.12  | 711.36   | 712.73   |\n| **SimCLR**  | FedAvg    | 41.77   | 42.00    | 41.43    |\n|          | FedProx   | 50.16   | 50.18    | 50.26    |\n|          | MOON      | 66.03   | 66.41    | 67.23    |\n|          | FedX      | 103.54  | 92.16    | 105.75   |\n|          | FedU$^2$  | 2092.78 | 2095.67  | 2102.21  |\n|          | FedPAM    | 582.62  | 602.36   | 605.89   |\n| **BYOL**    | FedAvg    | 62.14   | 62.02    | 62.37    |\n|          | FedU      | 62.10   | 63.78    | 62.70    |\n|          | FedEMA    | 63.33   | 62.45    | 63.00    |\n|          | Orchestra | 90.61   | 86.62    | 155.13   |\n|          | FedU$^2$  | 2137.52 | 2136.59  | 2207.17  |\n|          | FedPAM    | 828.67  | 826.81   | 830.01   |\n\n**R2 to W2:** Thank you for this interesting suggestion.\nPrivacy protection is an important issue in federated learning, where $\\mathcal{E}$ contains client-specific information. Once the centralized server is attacked, this may cause client privacy leakage. The current tensor decomposition method has an explicit bias tensor $\\mathcal{E}$. Therefore, in order to protect the client's privacy information from being exposed, we can use the tensor Tucker decomposition to decompose a core (low-rank) tensor and a set of factor (bias) matrix. The mathematic form is written as\n>$\\mathcal{W}=\\mathcal{L} \\times_1 U_1 \\times_2 U_2 \\cdots \\times_M U_M$,\n\nwhere $\\mathcal{W}$, $\\mathcal{L}$, and ${U_{i}}\\_{i=1}^{M}$ denote the original tensor, the low-rank tensor, and a set of factor matrix. Since the factor matrices integrate the information of multiple clients, they will not expose client-specific information.\nLimited by the rebuttal time, we will further investigate a novel method for unsupervised FGL with strict privacy protection in the future.\n\n**R3 to W3:** Thanks for this insightful comment. \nIn federated learning, low client weights can lead to privacy leaks (for example, by inferring client data distribution or sample information through reverse engineering). To avoid this problem, low-weight clients can be clustered with similar clients based on their data distribution or model parameter similarity to form virtual \"super clients.\" Parameters are first aggregated within the group, and then the group participates in global aggregation, masking individual contributions. Clients within the group share the aggregated weights, preventing individual clients from having too low a weight. This is a very interesting research topic that we will further explore in future work.\n\n**R1 to Q1:** Thank you for this interesting question. \nWe continue to increase the value of $\\lambda$ to test whether FedPAM will encounter a performance bottleneck. The experimental results are reported in Table 2. We can see that when $\\lambda$ rises to 500 or 600, FedPAM encounters a performance bottleneck. However, it is worth noting that a larger value of $\\lambda$ is conducive to achieving satisfactory performance, thanks to the global anchor-guided representation space alignment. From the experimental results, to achieve satisfactory results, the weight of $\\lambda$ relative to SSL weight should be at least 10.\n\n*Table 2: Performance comparison with different values of $r$.*\n| Dataset   | $\\lambda=100$ | $\\lambda=200$ | $\\lambda=300$ | $\\lambda=400$ | $\\lambda=500$ | $\\lambda=600$ | $\\lambda=700$ |\n|-----------|--------------:|--------------:|--------------:|--------------:|--------------:|--------------:|--------------:|\n| Cora      |         62.83 |         63.29 |         65.92 |         64.96 |         67.65 |         67.56 |         37.10 |\n| CiteSeer  |         43.90 |         46.59 |         50.64 |         52.65 |         53.24 |         36.30 |         36.52 |\n\n**R2 to Q2:** Thank you for this good advice. In essence, the proposed AGPL module is a plug-and-play component. To verify its effectiveness, we stitch the AGPL module with the existing federated learning algorithm.\nNotably, we focus on testing the federated learning algorithm without special aggregation methods on the server side.\nThe experimental results are reported in Table 3.\nWe can see that when the existing federated learning algorithm is combined with the proposed AGPL module, the performance will be improved, which illustrates the effectiveness of the AGPL module.\n\n*Table 3: Performance comparison when the proposed AGPL is combined with different FL methods.*\n| SSL      | Method         | Cora ACC | Cora Fscore | CiteSeer ACC | CiteSeer Fscore | Ogbn-Arxiv ACC | Ogbn-Arxiv Fscore |\n|----------|----------------|----------|-------------|--------------|-----------------|----------------|-------------------|\n| **Simsiam** | FedAvg+AGPL   | 57.38    | 47.21       | 42.27        | 30.42           | 35.58          | 21.32             |\n|          | FedProx+AGPL   | 58.25    | 48.02       | 42.75        | 30.68           | 35.22          | 21.01             |\n|          | MOON+AGPL      | 56.70    | 46.56       | 41.02        | 28.83           | 35.58          | 21.33             |\n|          | FedPAM         | **61.40**    | **52.85**       | **49.53**        | **38.73**           | **35.97**          | **23.07**             |\n| **SimCLR**  | FedAvg+AGPL   | 56.65    | 44.18       | 41.35        | 27.80           | 47.65          | 35.45             |\n|          | FedProx+AGPL   | 56.86    | 44.43       | 41.58        | 27.93           | 46.97          | 35.86             |\n|          | MOON+AGPL      | 55.81    | 43.83       | 42.10        | 28.12           | 47.80          | 36.67             |\n|          | FedX+AGPL      | 57.53    | 44.96       | 42.86        | 30.23           | 48.01          | 36.95             |\n|          | FedPAM         | **58.62**    | **45.21**       | **43.90**       | **31.61**           |**48.19**          | **37.25**            |"}, {"review_id": "4XM7JNlPpv", "reviewer": "Reviewer_z9JY", "summary": "In this paper, the authors proposed a novel method for unsupervised federated graph learning. Specifically, the proposed FedPAM uses a set of anchors to explore the global rep resentation space, then use them to align the local representation spaces. To capture the high-order correlation among multiple clients, the low-rank tensor decomposition is adopted to enhance the consistency between multiple clients. Finally, extensive experiments are conducted to verify the effectiveness for the proposed FedPAM.", "questions": "1. The authors should explain why use the low-rank tensor decomposition for obtaining the global model weights.\n2. Fig. 4 shows that the proposed FedPAM will degrade when the number of anchors increase. However, according to my understanding, more anchors, more accurate for the global space. Hence, the reason should be elaborated.\n3. For an algorithm, the complexity analysis is essential. Hence, the complexity analysis should be provided.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The idea is interesting, using a set of learnable anchors to locate the global space, then adopting them to align local representation spaces, which is appealing.\n2. The motivation is well demonstrated by Fig. 1, which verify that the local representation spaces are not aligned and the conventional federated aggregation is ineffective.\n3. The paper is well written and the readers can easily follow.\n\nWeaknesses:\n1. The motivation of using low-rank tensor decomposition is not clarified.\n2. Additionally transfer the anchors can increase the communication burden.\n3. The analysis for the experimental results are insufficient.", "was_revised": true, "final_justification": "This paper proposed a novel method for unsupervised FGL, the proposed RSA and AGPL modules guaranteed the effective of FedPAM by a large number of experiments. The added generalization and complexity analysis provided a comprehensive understanding for the proposed FedPAM. Likewise, the added experiments further verify the superiority and flexibility for FedPAM. Hence, it's suggested to be accepted.", "rebuttal": "**R1 to Q1:** Thank you for this valuable comment.\nThere are two main reasons for using low-rank tensor decomposition. First, most federated aggregation methods perform parameter aggregation at the matrix level and cannot capture high-order correlations between different clients. The so-called high-order correlation refers to the consistency and complementary information across clients, and low-rank tensor decomposition is used to enhance information fusion between clients. Second, the client weight calculation method based on the number of samples often cannot truly reflect the actual degree of client contribution. For example, although some clients have a large number of nodes, they might be extremely heterogeneous and cannot play a positive role in the optimization direction of the global model. Therefore, an adaptive weight learning method is needed.\n\n**R2 to Q2:** Thanks for this insightful comment. We expect to learn a set of representative anchors and reduce the running cost as much as possible, so the number of anchors should be small. Since we do not impose any constraints between anchors, for example, different anchors should be as dissimilar as possible. Therefore, too many anchors may cause the collapse of the anchor space, which is not conducive to the alignment of the representation space across clients.\n\n**R3 to Q3:** Thanks for this meaningful comment. The computation complexity mainly originates from two aspects: the FGW-OT for aligning representation spaces and the low-rank tensor optimization for adaptively learning global model parameters.\nFor the FGW-OT, the computation of $G$ (an intermediate variable) takes $\\mathcal{O}(N_{E}M+M^{2}N)$, where $N_{E}$ denotes the number of edges, $M$ denotes the number of anchors, and $N$ denotes the number of nodes.\nNotably, the adjacency matrices are often sparse, so the computational complexity can be significantly decreased.\nLikewise, the OT based on Sinkhorn algorithm takes $\\mathcal{O}(MN)$.\nThen, the FGW-OT takes $\\mathcal{O}(N_{E}M+M^{2}N+MN)$.\nFor the low-rank tensor optimization, the update of $\\boldsymbol{\\mathcal{G}}$ with $D_{l,1}\\times D_{l,2}\\times K$ occupies the dominant computational complexity.\nFirst, the FFT and the inverse FFT require $\\mathcal{O}(D_{l,1}^{2}D_{l,2}\\log(K))$, where $D_{l,1}$ and $D_{l,2}$ are the dimensions for the $l$-th layer's parameters, $K$ is the number of clients, and $D_{l,1}$ is the higher dimension.\nSecond, the T-SVD costs $\\mathcal{O}(D_{l,1}D_{l,2}^{2}K)$. Then, the low-rank tensor optimization takes $\\mathcal{O}(D_{l,1}^{2}D_{l,2}\\log(K)+D_{l,1}D_{l,2}^{2}K)$.\nOverall, the computational complexity for the proposed FedPAM is $\\mathcal{O}(N_{E}M+M^{2}N+MN+D_{l,1}^{2}D_{l,2}\\log(K)+D_{l,1}D_{l,2}^{2}K)$."}]

Author rebuttal:

Meta-review: The work proposed a federated graph learning method via incorporating FGW and tensor decomposition. The experiments showed the effectiveness in comparison to a few baselines. I summarize the strengths and weaknesses below.

**Strengths**
1. The idea of using FGW and tensor decomposition for graph federated learning is novel.
2. The paper is easy to follow and well-organized.
3. The baselines and datasets in the experiments are sufficient. The results demonstrated the superiority of the proposed method over a few baselines.

**Weakness**
* The theoretical analysis is limited, though the authors tried to analyze the generalization bounds during the rebuttal.

During the rebuttal, the authors addressed most concerns of the reviewers, such as the computational complexity, running time, and more baselines. During the discussion, most reviewers supported accepting the paper.

In addition to the reviews given by the reviewers, I have the following comment. During the rebuttal, the authors tried to use the theoretical results of domain adaptation to derive a generalization bound for the proposed model. However, the theory usually requires an independence assumption, which does not hold for the nodes on a graph, because they are connected. In addition, using $|A_k-A_l|$ to measure the topological divergence makes no sense, since they may have different numbers of nodes and are permutation invariant. So, if the paper is accepted, the authors should be careful about the added theoretical results from the rebuttal.

Given the novel idea, the impressive numerical performance, the recognition of the reviewers, and the fact that the theoretical result is not compulsory for the current work, I recommend accepting the paper.
Decision: Accept (poster)