Less is More: Federated Graph Learning with Alleviating Topology Heterogeneity from A Causal Perspective
Authors: Lele Fu
Bowen Deng
Sheng Huang
Tianchi Liao
Shirui Pan
Chuan Chen
Venue: ICML 2025 poster (2025)
Keywords: Federated learning
Graph neural networks
Topology heterogeneity
Primary area: deep_learning->graph_neural_networks

Abstract: Federated graph learning (FGL) aims to collaboratively train a global graph neural network (GNN) on multiple private graphs with preserving the local data privacy. Besides the common cases of data heterogeneity in conventional federated learning, FGL faces the unique challenge of topology heterogeneity. Most of existing FGL methods alleviate the negative impact of heterogeneity by introducing global signals.
However, the manners of creating increments might not be effective and significantly increase the computation amount. In light of this, we propose the FedATH, an FGL method with Alleviating Topology Heterogeneity from a causal perspective. Inspired by the causal theory, we argue that not all edges in a topology are necessary for the training objective, less topology information might make more sense.
With the aid of edge evaluator, the local graphs are divided into causal and biased subgraphs. A dual-GNN architecture is used to encode the two subgraphs into corresponding representations. Thus, the causal representations are drawn closer to the training objective while the biased representations are pulled away from it. Further, the Hilbert-Schmidt Independence Criterion is employed to strengthen the separability of the two subgraphs. Extensive experiments on six real-world graph datasets are conducted to demonstrate the superiority of the proposed FedATH over the compared approaches.

Less is More: Federated Graph Learning with Alleviating Topology Heterogeneity from A Causal Perspective
Lele Fu' Bowen Deng'
Sheng Huang'
Tianchi Liao Shirui Pan
Chuan Chen
Abstract
Federated graph learning (FGL) aims collabora- tively train global graph neural network (GNN) multiple private graphs with preserving the local data privacy. Besides the common cases of data heterogeneity in conventional federated learning, FGL faces the unique challenge of topol- ogy heterogeneity. Most of existing FGL methods alleviate the negative impact of heterogeneity by introducing global signals. However, the man- ners of creating increments might not be effec- tive and significantly increase the computation amount. In light of this, we propose the FedATH, an FGL method with Alleviating Topology Het- erogeneity from a causal perspective. Inspired by the causal theory, we argue that not all edges in topology are necessary for the training ob- jective, less topology information might make sense. With the aid of edge evaluator, the local graphs are divided into causal and biased subgraphs. A dual-GNN architecture is used to encode the two subgraphs into corresponding rep- resentations. Thus, the causal representations are drawn closer to the training objective while the biased representations are pulled away from it. Further, the Hilbert-Schmidt Independence Cri- terion is employed to strengthen the separability of the two subgraphs. Extensive experiments on six real-world graph datasets are conducted to demonstrate the superiority the proposed Fe- dATH over the compared approaches.
Server
Client 1
Client 2
Client
(a) Topology heterogeneity in FGL
Warserstein Distance
Fossi.
FacATH
u x On
Label Index
(b) Wasserstein distancc on WikiCS dataset.
Figure The illustration topology heterogeneity in FGL. (a) shows that the nodes of same classes in different local graphs may be connected to these diverse classes, resulting in the topology heterogeneity (the bolded node as example). (b) compares the average wasserstein distance between the embedding from different local graphs for the same classes. It can be that the proposed FedATH effectively mitigates the embedding divergence caused by topology heterogeneity for different local graphs, thus alleviating the biased training of local GNNs.
has attracted wide attention for ensuring private data is not compromised. In light of this, various FL algorithms have flourished and been applied in many scenarios. Notably, an important assumption behind them cannot ignored, that is, the samples on each client are independent and not correlated with each other, such as images and text. Meanwhile, graph data (Wu al., 2020; Liu et al., 2022a; Wang et al., 2025; Deng et al., 2025; Cai et al., 2024b) is nowadays prevalent and may also present distributed storage such as transaction networks of multiple banks. Specifically, graph data has both feature attributes and topology structure, each node in a graph is connected with other nodes via
1. Introduction
Federated learning (FL. (Yang et al., 2019; Ye al., 2023; Huang al., 2024; Liao al., 2024; Fu al., 2025a; Hu al., 2024) is distributed model training approach that
'Sun Yat-sen University 'Griffith University, Correspondence to: Chuan Chen <chenchuan& mail.syssuedu.co
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
the edges. This complicated data format raises significant challenges for Fl., which gives birth federated graph learning (FGL) (Liu et al., 2024; Tan et al., 2023; Xie et al., 2021; Meng et al., 2024; Cai et al., 2024a; Wan et al., 2024).
less similar. It can be seen that existing SOTA FGL meth- ods such as FGSSL (Huang et al., 2023a) and FedTAD (Zhu et al., 2024) cannot consistently guarantee that the embedding divergences from different clients are decreased, demonstrating that they fail to effectively address the prob- lem of topology heterogeneity. As a result, we reflect on whether conducting increments is really conducive to mit- igating topology heterogeneity. Are there better ways to accomplish this goal? Inspired by the causal theory, we believe that not all edges in a local topology are necessary for the training objective, and only a part of them plays deterministic role, while the rest is unimportant or even has a negative effect. Less local topology information may make more sense. The component consisting of determin- istic edges considered as the causal subgraph, while the rest is considered as the biased subgraph. When each client explores the causal subgraph, the node embedding is as rele- vant as possible to the training objective, thereby mitigating topology heterogeneity and preventing biased training of local GNNs.
As an emerging technique for distributed graph analysis, FGL aims to train a powerful global graph neural network (GNN) via incorporating multiple private graphs. Driven by practical requirements, many endeavors have been imple- mented. For example, (Zhang et al., 2021; Chen et al., 2024; Liu et al., 2022b; Tian et al., 2024) attempted to repair the neighbor nodes or global graph information, then allowing local GNNs to capture a wider scope of node information for enhancing the model training. The above approaches provide novel perspectives and achieve promising results. Unfortunately, FGL like traditional FL also suffers from the curse of data heterogeneity. The common manifestations of data heterogeneity are label shift and feature shift, both of which guide the local models to optimize in the direction of the locally optimal solutions, thereby weakening the ability of global model. In addition to the two kinds of data hetero- geneity, a particular heterogeneity form in FGL. needs to be emphasized, i.e., topology heterogeneity.
In this paper, we propose an FGL. method with Alleviating Topology Heterogeneity (FedATH) across multiple clients. Concretely, we adopt an edge evaluator to assess the im- portance of each edge. Based on the assessment results, the local graphs are divided into causal subgraphs and bi- ased subgraphs. Further, a dual-GNN architecture is used to encode the two kinds of subgraphs into corresponding representations. The cross entropy loss is used to reinforce the correlation of causal representation with the training objective while the negative entropy loss is used to disasso- ciate the biased representation from the training objective. To enhance the separability of them, the Hilbert-Schmidt Independence Criterion (HSIC) is introduced to maximize their independence. Notably, only the local causal GNNs are uploaded to the server for aggregation without increasing communication burden. Fig. 1(b) shows that the proposed FedATH significantly reduces the embedding divergence between varying local graphs, this is because the exploration of causal subgraphs effectively handles the topology het- erogeneity across multiple clients. Generally, the principal contributions of this paper are concluded as follows:
As presented in Fig. (a)) topology heterogeneity shows that the nodes of the same classes may be connected to other nodes of different classes on different clients. When local GNNs encode the node information, the embedding com- position of a node is not only determined by itself, but also influenced by its neighbor nodes. Then the topology hetero- geneity across clients directly induces the heterogeneity of local embedding even for the same categories, resulting in biased training of local GNNs. Therefore, how to alleviate the impact of topology heterogeneity is particular concern. Some efforts have been made for addressing this issue. For instance, Huang et al. (Huang et al., 2023a) used the global model to calibrate local embedding and structures. Zhu et al. (Zhu et al., 2024) generated a pseudo graph with the reliable knowledge from multiple clients, which served the dis- tillation data for training the global model. Xia et al. (Xia et al., 2024) augmented the local graph data via exploring the topological complementarity of various private graphs. In a nutshell, they attempt to conduct increments (e.g., addi- tional global graphs or global representations) to shrink the heterogeneity of local graphs, but the used techniques, such as knowledge distillation and contrastive learning, signifi- cantly increase the computation and communication volume. More importantly, they might not be effective in reducing the negative impact of topology heterogeneity.
We provide a novel perspective on the problem of topol- ogy heterogeneity in FGI., mitigating the negative im- pact of topology heterogeneity across different clients by diminishing superfluous information rather than creating new increments.
Inspired by the causal theory, we divide the local graphs into causal and biased subgraphs with the aid of the edge evaluator, and the HSIC is adopted to enforce their separability. Finally, only the local causal GNNs are shared for aggregation.
Fig. 1(b) compares the average wasserstein distance (WD) between the embedding from different local graphs for the same classes, where the WD is used measure the sim- ilarity between distributions. Smaller WD indicates that the two distributions are more similar, and the opposite is
large number of experiments are conducted on six
2

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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
real-world graph datasets, the experimental results demonstrate that the proposed FedATH is more supe- rior than the SOTA conventional FL and FGL methods.
Tan al. (Tan al., 2023) proposed share the topology encoding networks while maintaining the feature encoding networks specific. Pan et al. (Pan et al., 2024) devised an incentive mechanism to retain the fairness among multi- ple agents in federated graph system. For the node-level, each client stores a subgraph. Zhang et al. (Zhang et al., 2021) generated the missing neighbor nodes for each client with federated training, promoting the performance of local GNNs. Chen et al. (Chen et al., 2021) developed graph sampling strategy and federated graph convolutional oper- ation for distributed graph data. L, et al. (Li et al., 2023b) considered the impact of topology structure and proposed a topology-awer -wware federated aggregation manner. Kong et al. (Kong et al., 2024) adopted a federated fusion strategy for local anomalous neighbor embedding, enhancing the difference between anomalous nodes and neighbor nodes.
2. Related Work
2.1. Federated Learning
With the increased awareness of protection for private data, FL has flourished as a means of distributed model train- ing. As the pioneering algorithm, FedAvg (McMahan et al., 2017) has demonstrated the superiority of FI., but it is highly sensitive the heterogeneous data. Therefore, how to over- come the detrimental effects induced by heterogeneous data has always been central concern in FL. For the scenario of heterogeneous labels, (Li et al., 2020; Karimireddy et al., 2020; Li et al., 2021; Fu et al., 2025e; Huang et al., 2025) corrected the bias between the local models and global model to prevent the local models from falling into local optima. Knowledge distillation is an effective method for transferring information between different models, which is also introduced into FL to address the issue of catastrophic forgetting caused by heterogencous data. (Li & Wang, 2019; Zhu et al., 2021; Shao et al., 2024; Xie et al., 2024) passed the global knowledge to clients by leveraging knowledge dis- tillation, promoting the generalization of local models. For the scenario of heterogeneous features, (Hong et al., 2023; Wang et al., 2023; Zhang et al., 2024) adopted an adver- sarial training method, eliminating the divergence between various domains by fooling the discriminator. (Huang et al., 2023b; Li et al., 2023a; Yan et al., 2024; Qi et al., 2023: Meng et al., 2025; Fu et al., 2025h) explored the generalized global prototypes and aligned the representation spaces of different clients through contrastive learning. Despite the aforementioned methods achieve impressive results, they perform unsatisfactorily when the clients' private data is graph-structured, which is because graph data is far more complex than common image or text data. Accordingly, it is necessary to develop tailored FGL algorithms.
3. Preliminaries
Graph Neural Networks. Given a graph dataset G' (V, E, X). V denotes the node set, E denotes the edge set, and X E RNxd denotes the node feature matrix, where N and d are the number of nodes and feature dimension, respectively. For each node i2 E V, it has a feature attribute x; (the i-th row of X) with label yı E [C]. where C' is the number of categories. GNNs aim to aggregate the neighbor- hood information of nodes to improve the discriminability of their representations through a certain defined informa- tion propagation mechanism. Generally, the calculation of the l-layer GNN is formulated as h h1#1 5(h', (GG()),,,)) VV,
(1) where hy is the embedding of the I-layer for the i-th node, esy denotes the edge between the i-th and j-th nodes, AGG(-) is the defined aggregation operator of neighbor nodes, 8(.) denotes the activation function. Especially, hy = x; is the raw feature.
Federated Graph Learning. In federated graph system, a centralized server and K clients are included, each client stores private graph dataset G₁ (Vx, Eg, X²). Each node vg characterized as (xf, v (vi E [N&]), where N₂ is the number of the k-th local graph's 's nodes. The vanilla FGL is to directly transplant FedAvg into the federated scenario. Thus, the objective optimization of FGL is written as
2.2. Federated Graph Learning
FGL aims to train a decent GNN with distributed graph data. Unlike traditional FL strategics, FGL requires to addi- tionally consider the impact of topology structure on model training. Overall, FGL is categorized into two types based on the graph data format on the clients. The first type is the graph-level, where each graph is considered as a sample such as molecular graphs and protein graphs. The second type is the node-level, where each node in a graph serves as a sample such as citation network and social network. For the graph-level, each client has a set of graphs. Xie et al. (Xie et al., 2021) dynamically grouped clients into different clusters according to the gradients of local GNNs.
K (W K
min
(2)
W
where W is the parameter of the k-th local GNN, Ry denotes the k-th empirical risk and is defined as
Rs (W+) = E wi)) (FFa(WWlCer Wxl( vi))).
(3)
where (.) denotes the loss function for the k-th client. When each communication round completes, the updated
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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
Causal embedding
fc
Causal GNN
Blased embedding
Biased GNN
srver
Causal edge
f2
cC
Edge Evaluator
B Blased edge
CE
fi
H
Client 1
Client K
dxx
LDEp
LENT
fk
H
Figure 2. The overview of the proposed FedATH. The pink box presents the general FGL framework, the green box shows the local training process on each client, and the blue box shows the aggregation procedure on the server. In particular, the yellow box displays the meaning of used graphics.
global GNN is obtained by
In FGL, the nodes of same categories in different local graphs might be connected to ones of various categories due to the heterogeneous topology, contributing to the biased training of local GNNs. However, inspired by the above SCM, not all edges topology are essential for the node semantics, some are the biased factors and even play the negative roles. If each client adopts the common GNN en- coding manner, the representation is mixed with causal and biased variables, which is not conducive the homogeneity across clients. Hence, we expect to identify which edges determinant for the semantic and explore the key topology on each client. The part formed by keeping the important edges are regarded as causal subgraph, while the rest are re- garded as biased subgraph. When each client disentangles the local graph into causal and biased subgraphs, the labels are only associated with the causal subgraph and stripped from the biased subgraph, then the topology heterogencity of various local graphs can be alleviated.
Nk Wk: N
W
(4)
where - ΣK N₂ the total number of nodes. How- ever, simple aggregation inescapably causes the perfor-
mance degeneration due to the topology heterogeneity across multiple clients.
4. Proposed Method
Concretely, the proposed FedATH consists of two important modules: Subgraph Division via Edge Evaluation and Dis- entanglement of Causal and Biased Representations, Fig. 2 illustrates the framework of the proposed FedATH. The details are elaborated as follows.
4.1. Inspiration from Causal Theory
The proposed FedATH is inspired by the causal theory. To better understand it, the graph generation procedure from the structure causal model (SCM) (Schölkopf et al., 2021; Fan et al., 2022) has to be first elaborated. As shown Fig. 3(a), there are four kinds of causal relationships between various variables in the general cases. (1) C + G +- B. The observed graph is produced by the unobserved causal variable C and biased variable B. (2) C + Y. The causal variable C fundamentally determines the semantics Y. (3) Ce B. There may be redundant entanglement between C' and B. (4) G > R R Most GNNs directly map the raw graph G into latent representation R, then yields the semantic label Y.
R
Y
G
C
X
B
(a) Genera SCM
(b) SCM of FedATH
Figure 3. The SCMs common FGL methods and the proposed FedATH, where the grey circle denotes the observed variable and the white circle denotes the latent variable.

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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
4.2. Subgraph Division via Edge Evaluation
Concretely, a causal GNN f and biased GNN S& are equipped on each client, they encode the corresponding subgraphs into latent space
To divide the local graph into causal and biased subgraphs, we propose to train edge evaluator 4 each client for deciding the affiliation of edges. Specifically, for the k-th local graph Ga = (Vi, Ex, Xx), the contribution degree of each edge eij E Es between node U; E V& and node vy E Vi is evaluated by
H FE(GRIWE) H fl(G)/
where W and W are the parameters of f& and fg. respec- tively. For a concise denotation, the subscript k omitted in the following presentation. The causal representation H is considered to capture the information that is strongly rel- evant to the objective, the cross entropy is used to constrain its distance from the ground truth labels, which is written as
Cj = (xx((x{((x]]
(5)
where || denotes the concatenation of two vectors, the evalu- ator b can be specified as a Multi-Layer Perceptron (MLP).
Further, to enable Cay to be probabilisticlly meaningful, it mapped into [0, 1 by
Way sigmoid
(6)
LCE = (M 1y log (osfstaam) (8)
where way can be viewed as the probability that edge Eaj be- longs to the causal subgraph, while 1 wij is the probability that edge Eiy belongs to the biased subgraph. Frequently, the adjacency matrix A E X of local graph G is conducted based on the edge set Eg for an intuitive expres- sion. Likewise, the edge mask matrices of causal and biased subgraphs can be formulated 22 = [yy E R N.N,... N N. and 2 [1 - wiy] E N& respectively. Further, the adjacency matrix Ag is disentangled into causal adja- cency matrix A = A @ S2 and biased adjacency matrix A = Ax @ S2 where @ denotes the Hadamard prod- uct. When Ag and A are obtained, the causal subgraph G = (Ag, X&) and the biased subgraph G (A Xx) can be constructed.
where he is the i-th row of H By, denotes the one-hot vec- tor for the label y;. N1 denotes the number of training nodes. Due the absorption of unnecessary or even negative in- formation from neighbor nodes, the biased representation H' is regarded to be not discriminative, as a direct result of which its predicted label probability distribution appears smooth rather than sharp. Hence, the negative entropy loss is adopted to achieve the goal:
N log (og(soflmax(hh))
LENT = -EQUAIINEE NND Σ
(9)
where hy is the i-th row of HJ, N₂ is the number of nodes in the k-th local graph.
With the help of the edge evaluator, each client explores the causal and biased subgraphs, which further assist the GNNs to explore the corresponding latent representations. Notably, the edge evaluator is only trained on local clients and not shared, which elicits two advantages. First, feder- ated learning emphasizes efficient communication, while private evaluator does not incur additional communication burdens. Second, the topologies of different local graphs have different characteristics, a private edge evaluator cap- tures the personalized information, and a shared evaluator may interfere with the extraction of local causal and biased subgraphs. Moreover, the experimental comparison between shared and unshared edge evaluator is expanded in detail in the experimental section.
According to the causality presented in Fig. 3, the causal representation He and the biased representation H6 should be independent of each other. To measure the depen- dence between H and HJ, the Hilbert-Schmidt Indepen- dence Criterion (HSIC) is introduced. Give two variables P = [PI. P2, ..... PN)] and Q = (qI 92, qN). the map- ping functions vú and á map them into kernel spaces P and @: (p) E P. s(q) E Q. Then, the inner product for two vectors in kernel spaces can be written as K1 (1 (2) = (v(P))) (P2)), K2(q1, q2) = (ó(q1), ((q2)). Then, it has following definition.
Definition 4.1. Given a set of independent observed vari- ables X == [(pp. 91), -..; (P.v, (N)), ; an empirical estima- tor of HSIC(X, P, Q) is defined as
4.3. Disentanglement of Causal and Biased Representations
HSIC Q) (N - 1) CK, (10)
For causal subgraph G% and biased subgraph G how to guarantee their disengagement becomes the principal con- cern. In response to this problem, we propose to adopt a dual-GNN architecture to encode the two subgraphs into latent representations and achieve the disengagement of the two that is driven by the designed losses upon them.
where Tr(.) denotes the trace of a matrix. K, and K2 are the Gram matrices, whose each entry is computed by Kiij = Hn (P),P2), K2,ú = k2 q2) respectively. C is the centralized matrix for the Gram matrix and is defined as H = I 1/N, where I E R NNN is an identity matrix.
5

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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
Therefore, the dependence loss between H and HD can be written
Corollary 5.2. Given an FGL system, the k-th and 1-th em- pirical local distributions are denoted Di and Di, its gen- eralization error follows Theorem 5.1. For dHAH Di), the following inequality holds.
(II) where the inner kernels are specified as K1 H"
- HSIC(H
Dmiin' + (D
1
K2 = H''² Through minimizing the objective ex- pressed as Eq. (1I), the disentanglement of causal and biased representations are enhanced. Here, the overall loss function is formulated as
5 + sup Bw BwBx Bx
D
1
B
(I|A + -1Allp)))+++
|
m
(14)
[ = LCe + CENT + ADDEP
(12)
where sup denotes the supremum, Bw and Bx denote the network parameters of GNN and the upper-bound constants with respect the data features X, respectively. D Dmin denotes the minimum degree.
where À is the trade-off parameter. When the local training completes, only the causal GNN is uploaded to the server for aggregation while the biased GNN and the edge evaluator remain private.
When the local causal subgraphs are explored, the edge weights in the adjacency matrices are reduced from I to [0, 1]. the Frobenious norms of ||Ax||F and || are de- creased, then the bound of generalization error with respect to the global hypothesis can be shrunk. Hence, the gener- alization ability of global model learned by the proposed FedATH is enhanced. The detailed proof process refers to the Appendix.
5. Theoretical Analysis
Here, we provide the generalization analysis for the pro- posed FedATH. Denote D and Dx. the global and local dis- tributions, respectively. Dj denotes the empirical local dis- tribution. h& is the local hypothesis learned on the local em- pirical distribution Dr and defined as fı : X + Vr mapping the data features into predicted labels. h = 1/KE Ek=1 hi denotes the global hypothesis integrated by local hypoth- esis. H denotes the hypothesis space of VC-dimension d. Moreover, without losing generality, it is specified that Ni = = NK = m.
6. Experiments
6.1. Datasets
The comparative experiments are conducted on six real- world graph datasets covering four categories. Cora, PubMed, and ogbn-arxiv (Yang et al., 2016) are three kinds of citation networks, depicting the citation relation- ships between various papers. Photo (Shchur et al., 2018) is a co-purchase network, recording items that are purchased together. WikiCS (Mernyei & Cangea, 2020) a Wiki-page network and constructed based on Wikipedia, recording the relationships between diverse computer science subjects based hyperlink. Roman-empire (Platonov et al., 2023) is an article syntax network via counting the Roman Empire. The details of above six datasets are reported in Table 7. To simulate the distributed graphs, the Louvain method (Blon- del et al., 2008) used to separate the complete graph to multiple clients, e.g., 10, 15, 20.
Theorem 5.1. Given an FGL system with global distribu- tion D and local distribution Da., with the the probability at least 8 (0 ō S. 1), the generalization error for any hypothesis h4p satisfies
RD(h) S
1 K
Σ
RDD (hx)
kE[K
K
Σ Σ
dHaH (Dx.D.) +
+
K(K 1)
kE[K] Lit
+
1
Σ
Σ
A + Vml
4
dlog
2em
+ log
4K
K
d
8
KE[R]
(13)
where R.D, (hx) denotes the empirical risk on Dr. (DA is the H-distance between Dr and Di. e denotes a upper-bound constant with respect to dH v) E [K], As= = ming (Ro(h) + RD,(() denotes the optimal risk on D and Dj-
6.2. Compared Methods
We compare the proposed FedATH with nine federated learning algorithms, including conventional and graph- oriented methods. FedAvg (McMahan al., 2017) is used as the baseline. FedProx (Li et al., 2020), MOON (Li et al., 2021), FedOPT (Reddi et al., 2021), and FedProto (Tan et al., 2022) are four conventional federated learning ap- proaches, effectively coping with the common distributed data. FedSage+ (Zhang et al., 2021), FGSSL (Huang et al., 2023a), FedPUB (Baek et al., 2023), FedTAD (Zhu et al.,
reveals that the generalization error of an FGL system depends mainly on two factors: the distribution divergence between local graph data dHAH D:) and the number of observed samples m. In particular, we further have following corollary for dRAH( (Dz, Di).
Theorem 5.1
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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
Cora
PubMed K =
ogbn-arxiv
Type
Method
= 10
K 15
K 10
K 20
K 10
15
K 20
FedAvg
73.59
69.52
62.44
82.42
81.51
80.80
35.75
34.74
33.57
FedProx
74.29
69.96
63.19
82.43
81.55
80.82
35.67
34.35
33.62
MOON
74.22
70.46
61.35
82.49
81.57
80.25
34.89
33.69
33.88
FL
FedOPT
74.56
71.31
63.71
81.52
80.45
80.92
36.42
34.41
35.53
FedProto
74.56
70.04
63.36
82.70
81.54
80.82
35.84
34.56
33.57
FedSage+
73.98
67.35
65.00
82.36
78.23
78.66
41.02
36.64
37.21
FGSSL
74.47
72.21
65.89
82.38
81.86
80.99
39.22
36.61
35.20
FGL
FedPUB
75.35
72.43
66.44
82.67
79.06
79.60
39.02
36.87
36.41
FedTAD
74.29
72.23
63.74
82.72
82.03
81.19
37.65
36.28
34.65
FGL
FedATH
77.90
73.42
67.97
84.06
83.61
83.03
42.21
38.46
39.54
Table Performance comparison (ACC %) on Cora, PubMed, and oghn-arxiv datasets for all compared methods, where BL denotes the baseline, the optimal results are bolded and the suboptimal results underlined.
Photo
WikiCS
Roman-empire
Type
Method
K = 10
== 15
K = 20
K 10
K = 15
K = 20
K 10
K 15
K 20
BL
FedAvg
87.19
86.04
84.35
69.56
66.33
67.54
34.41
33.83
32.23
FedProx
87.12
86.49
84.35
69.47
66.22
67.57
34.30
33.64
32.04
MOON
86.45
85.24
81.47
70.03
67.16
66.53
33.97
33.78
33.03
FL
FedOPT
87.73
86.10
84.20
69.39
68.11
66.46
34.35
33.44
32.10
FedProto
87.32
87.86
84.82
70.04
66.97
68.16
34.93
33.64
32.26
FedSage+
88.46
86.80
84.37
71.72
69.32
70.69
41.59
39.35
39.12
FGSSL
88.56
86.17
84.27
70.63
68.97
68.64
36.96
36.85
35.21
FGL
FedPUB
88.79
87.17
84.21
72.30
69.82
69.88
37.31
36.12
35.47
FedTAD
87.76
86.60
84.94
71.63
69.32
69.01
39.01
37.94
37.28
FGL
FedATH
90.33
88.61
85.50
75.22
72.16
71.38
48.18
47.19
45.26
Table 2. Performance comparison (ACC %) Photo, WikiCS, and Roman- cennree datasets for all compared methods, where BL denotes the baseline, optimal results are bolded and the suboptimal results are underlined.
2024) are four graph-oriented federated learning approaches, tailored for the distributed graph data.
gating model parameters is not effective against the topolog- ical heterogeneity. can also be seen that the conventional federated learning algorithms achieve minor gains versus FedAvg, and even less effective than FedAvg in many cases, showing that they cannot cope well with the topology heterogeneity in FGL. Conversely, the tailored FGL meth- ods achieve relatively good results, which thanks to their strategies for heterogeneity of graph data such as fixing the global graph or generating the global representations. How- ever, the proposed FedATH achieves the optimal results over the other compared approaches, demonstrating that using less key information is more conducive to more gains for the performance.
6.3. Implementation Details
For the backbone causal and biased GNNs, a 2-layer graph convolutional network is adopted, encoding the raw graph data into 64-dimensional embedding. The Adam is employed optimizer with the learning rate set 0.001. The numbers of communication rounds and local training epochs are fixed 100 and 3, respectively. Considering the node classification task, classification accuracy (ACC) and F1-score (F1) are employed as the evaluation met- rics. For the trade-off parameter X, it is tuned in range of (0.001, 7,10)
6.5. Ablation Study
6.4. Performance Comparison
Effects of Different Losses. In addition the tropy loss, two important losses are included in the proposed FedATH: LENT and LDgP. The tailored ablation experi- ments are designed to verify their effects Tables 3 and 4. can be seen that when either item is removed, the
The experimental results for all compared methods are re- ported in Tables and 2, where three cases of client num- ber considered, i.e., 10. 15, 20. Undoubtedly, FedAvg achieves inferior performance, indicating that simply aggre-

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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
performance of FedATH is inevitably weakened, suggest- ing that both of them play essential role in achieving the separation of causal and biased subgraphs. Only when both losses are present, the proposed FedATH reaches the optimal results.
Photo
WikiCS
Sharing
K= 10
= 15
K 20
K 10
K 15
K 20
CG+EE
86,96
85.20
84.78
74.55
71.27
67.45
CG+BG
86.87
85.58
84.86
74.21
70.86
69.48
CG+EE+BG
86.61
85.62
84,90
74.77
71.57
67.40
CG
90.33
88.61
85.50
75.22
72.16
71.38
Effeets of Sharing Different Components. Overall, three network modules are equipped in each client, including the edge evaluator, the causal GNN, and the biased GNN, We test the performance with sharing various components. As shown in Tables 5 and 6, the proposed FedATH reaches its optimality when only the local CGs are shared in general. When local EEs and BGs are involved sharing, the perfor- mance suffers from varying degrees of degradation. This is because they capture the biased information caused by local topology heterogeneity, which is not conducive model generalization.
Table 6. The performance comparison (ACC %) with various shared components on Photo and WikiCS datasets, where CC, BG, and EE denote the causal GNN, biased GNN, and edge eval- uator, respectively.
is set on WikiCS dataset. Different datasets have different statistical characteristics, and appropriate À is needed to achieve well separation of causal subgraph and biased subgraph. Hence, it is necessary to introduce A. Further- more, the impact of label ratio is verified in Fig. 5. The larger the label ratio, the higher the performance is obtained for all algorithms. Fortunately, the proposed FedATH still maintains leading performance.
Cora
PubMed
CEXT LDEP
K
10
= 15
K=
K=
K
20
x
73.59
69.52
62.44
82.42
81.51
80.80
/
76.40
72.92
66.29
83.78
83.29
82.44
x
75.09
72.12
63.01
83.95
83.36
82.42
Clsnsitcation ACC (8)
72.24
12.5
71:22
77.90
73.42
67.97
84.06
83.61
83.03
Classitatioon ACO
67.8
Table 3. Ablation results (ACC %) with respect to two principal losses on Cora and PubMed datasets.
65.0
62.5
ENT DDEP
Photo
WikiCS
000
0.07
00001
0.071
0.51
K
10
K 15
K 20
K =
10
K 15
K 20
(a) Cora
(b) WikICS
X
X
87.19
86.04
84.35
69.56
66.33
68.73
/
89.24
85.01
85.06
72.07
70.00
70.47
Figure 4. The performance comparison when the trade-off of parame- ter À is tuned, where the client number is set as 1.5.
x
89.81
86.66
85.24
72.92
69.86
71.15
90.33
88.61
85.50
75.22
72.16
71.38
Table 4. Ablation results (ACC %) with respect to two principal losses Photo and WikiCS datasets.
1o0000 conooon s00 (8)
Clorgifictio cation ACC (%5
Cora
PubMed
Sharing
K= 10
K= 15
K =
20
K
10
K= 15
K = 20
Frokey
Tadiig
fed
CG+EE
78.69
73.34
67.02
83.23
83.01
81.77
4001
Mo0s
CG+BG
76.59
73.17
66.45
83.16
82.94
82.53
Fechas halth
CG+EV+BG
78.25
73.94
67.19
83.18
83.13
81.74
Label Ramoo
Label Renio
(a) Cora
(b) WikiCS
CG
77.90
73.42
67.97
84.06
83.61
83.03
Figure 5. The performance comparison for all compared methods with different label ratios, where the client number is as 15.
Table 5. The performance comparison (ACC %) with various shared components on Cora and PubMed datasets, where CG, BC, and EE denote the causal GNN, biased GNN, and edge eval- uator, respectively.
7. Conclusion
In this paper, we propose new FGL method called FedATH to cope with the topology heterogeneity across different lo- cal graphs. We recognize that correcting model training by creating increments could tough. Instead, we mitigate the topology heterogeneity via reducing the superfluous infor- mation. Specifically, the local edge evaluators are developed
6.6. Hyperparameter Study
The trade-off parameter X balances the contribution of HSIC loss, its importance is validated by tuning the values in {0.0001, 0.001, ...; 10). From Fig. 4, we can observe that a larger is required on Cora dataset while a smaller one
8

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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
distinguish the local graphs into the causal subgraphs and biased subgraphs. A dual-GNN architecture maps the two subgraphs into latent representations. With the aid of the designed losses, the separabilityy of the two subgraphs is enhanced. The experimental results verify the advancement of FedATH over other compared methods. However, in real-world scenarios, node labels are often unavailable, and the lack of unified semantic information further exacerbates the difficulty of federation training. In future work, will explore how to achieve effective federated graph learning in unsupervised scenarios.
Federated graph learning framework with global self- supervision. Information Sciences, 657:1199700 2024.
Chen, F., Li, P., Miyazaki, T., and Wu, C. Fedgraph: Fed- erated graph learning with intelligent sampling. IEEE Transactions on Parallel and Distributed Systems, 33(8): 1775-1786, 2021.
Deng, B., Wang. T., Fu, L., Huang, S., Chen, C., and Zhang, T. Thesaurus: Contrastive graph clustering by swapping fused gromov-wassersten couplings. In Proceedings of the AAAI Conference on Artificial Intelligence, pp. 16199-
16207, 2025.
Acknowledgement
Fan, S., Wang, X., Mo, Y., Shi, C., and Tang. J. Debiasing graph neural networks via learning disentangled causal substructure. In Proceedings of the Advances in Neural Information Processing Systems, pp. 24934 2494-44446, 2022.
The research is supported by the National Key R&D Pro- gram of China (2023YFB2703700) the National Natural Science Foundation of China (62176269).
Fu, L., Huang, S., Lai, Y., Liao, T., Zhang, C., and Chen, C. Beyond federated prototype learning: Learnable semantic anchors with hyperspherical contrast for domain-skewed data. In Proceedings of the AAAI Conference on Artificial Intelligence, pp. 16648-16656. 2025a.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Fu, L., Huang, S., Lai, Y., Zhang, C., Dai, H.- N., Zheng, Z., and Chen, C. Federated domain- independent prototype learning with alignments of rep- resentation and parameter spaces for feature shift. IEKE Transactions on Mobile Computing, pp. 1-16, 2025b. doi=10.1 100.011099MMC.22255555000000
References
Back, J., Jeong, W., Jin, J., Yoon, J., and Hwang, S. J. Personalized subgraph federated learning. In Proceedings of the International Conference on Machine Learning, pp. 1396-1444 1415, 2023.
Fu, L., Li, Y., Huang. S., Chen, C., Zhang, C.,, and Zheng, Z. Parameter-oriented contrastive schema and multi-level knowledge distillation for heterogeneous federated learn- ing. Information Fusion, 121: 121:103123, 2025c.
Barnes, L. P., Dytso, A., and Poor, H. V. Improved informa- tion theoretic generalization bounds for distributed and federated learning. In Proceedings of the IEEE Interna- tional Symposium on Information Theory, pp. 1465-1470, 2022.
Hong, J., Wang, H., Wang, Z., and Zhou, J. Federated robustness propagation: Sharing adversarial robustness in heterogeneous federated learning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 37, pp. 7893-7901, 2023.
Blondel, V. D., Guillaume, J.-L., Lambiotte, R.. and Lefeb- vre, E. Fast unfolding of communities in large networks. Journal of statistical mechanics: theory and experiment, (008(10):P1000) 2008
Hu, M., Zhou, P., Yue, Z., Ling, Z., Huang, Y., Li, A., Liu, Y., Lian, X., and Chen, M. Fedcross: Towards accurate federated learning via multi-model cross rross-aggregation. In Proceedings of the IEEE International Conference on Data Engineering, pp. 2137-2150, 2024.
Cai, J., Zhang, Y., Fan, J., and Ng, S.-K. Lg-fgad: An effective federated graph anomaly detection framework. In Proceedings of the International Joint Conference on Artificial Intelligence. pp. 3760-3769, 2024a.
Huang, S., Fu, L., Li, Y., Chen, C., Zheng, Z., and Dai, H.-N. cross-client coordinator in federated learning framework for conquering heterogeneity. IEEF Transac- tions on Neural Networks and Learning Systems, 36(5): 8828 8842, 2025.
Cai, J., Zhang, Y., Lu, Z., Guo, W., and Ng. S.-K. To- wards effective federated graph anomaly detection via eelf-boosted knowledge distillation. In Proceedings of the ACM International Conference on Multimedia, pp. 5537-5546, 2024b.
Huang, W., Wan, G., Ye, M., and Du, B. Federated graph semantic and structural learning. In Proceedings of the International Joint Conference on Artificial Intelligence, pp. 3830-3838, 2023a.
Chen, C., Xu, Z.., Hu, W., Zheng, 7.,, and Zhang, J. Fedgl:

---

Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
Huang, W. Ye, M., Shi, Z., Li, H., and Du, B. Rethink- ing federated learning with domain shift: A prototype view. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition (CVPR), Pp. 16312-16322, 2023b.
Liu, Y., Jin, M., Pan, S., Zhou, C., Zheng, Y., Xia, F., and Philip, S. Y. Graph self-supervised learning: A survey. IEEE Transactions Knowledge and Data Engineering, 35(6):5879- 5900, 2022a.
Liu, Z., Yang, L., Fan, Z., Peng, H.,, and Yu, P. S. Federated social recommendation with graph neural network. ACM Transactions on Intelligent Systems and Technology, 13 (4):1-24, 2022b.
Huang, W., Ye, M., Shi, Z., Wan, G., Li, H., Du, B., and Yang, Q. Federated learning for generalization, robust- ness, fairness: A survey and benchmark. IEEE Transac- tions on Pattern Analysis and Machine Intelligence, 2024.
doi=10.1 11A.00011000TPMMM224444444530
McMahan, B., Moore, E., Ramage, D., Hampson, S., and y Arcas, B. A. Communication-effiiien learning of deep networks from decentralized data. In Proceedings of the International Conference on Artificial Intelligence and Statistics, pp. 1273-1282, 2017.
Karimireddy, S. P.. Kale, S., Mohri, M., Reddi, S., Stich, S., and Suresh, A. T. Scaffold: Stochastic controlled averaging for federated learning. In Proceedings of the International Conference on Machine Learning, pp. 5132- 5143, 2020.
Meng, L., Liang, K., Yu, H., Liu, Y., Zhou, S., Liu, M., and Liu, X. Fedean: Entity-awar aware adversarial negative sampling for federated knowledge graph reasoning. IEEE Transactions on Knowledge and Data Engineering, 36 (12):22068211) 2024.
Kong, X., Zhang, W., Wang. H., Hou, M., Chen, X., Yan, X., and Das, S. K. Federated graph anomaly detection via contrastive self-supervised learning. IEEE Transac- tions Neural Networks and Learning Systems, 2024. doi=10.1 (0..//TNNSS.022..44444
Meng, L.,, Qi, Z., Wu, L., Du, X., Li, Z., Cui, L., and Meng, X. Improving global generalization and local personalization for federated learning. IEEE Transactions on Neural Networks and Learning Systems, 36(1):76-87, 2025.
Li, D. and Wang, J. Fedmd: Heterogenous feder- ated learning via model distillation. arXiv preprint arXiv: 1ET::191000588/ 2019.
Li, J., Li, F., Zhu, L.., Cui, H., and Li, Prototype- guided knowledge transfer for federated unsupervised cross-modal hashing. In Proceedings of the 31st ACM International Conference on Multimedia, pp. 1013-1022, 2023a.
Mernyei, P. and Cangea, C. Wiki-cs: A wikipedia- based benchmark for graph neural networks. arXiv preprint M.Xiv:2007.02901 2020.
Pan, C., Xu, J., Yu, Y., Yang. Z., Wu, Q., Wang, C., Chen, L., and Yang, Y. Towards fair graph federated learn- ing via incentive mechanisms. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pp. 14499-14507, 2024.
Li, Q. He, B., and Song, D. contr-ccnt federated learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 10713- 10722, 2021.
Li, T., Sahu, A. K., Zaheer, M., Sanjabi, M., Talwalkar, A., and Smith, V. Federated optimization in heteroge- neous networks. In Proceedings of Machine Learning and Systems, pp. 429-450, 2020.
Platonov, O., Kuznedelev, D., Diskin, M., Babenko, A., and Prokhorenkova, L. A critical look at the evaluation of gnns under heterophily: Are we really making progress? arXiv preprint arXiv:2302. 16640, 2023.
Li, X., Wu, 7., Zhang, w., Zhu, Y., Li, R.-H., and Wang, G. Fedgta: Topology-aware averaging for federated graph learning. Proceedings of the VLDB Endowment, pp. 41 50, 2023b.
Qi, Z., Meng, L., Chen, Z., Hu, H., Lin, H., and Meng, X. Cross-silo prototypical calibration for federated learning with non-iid data. In Proceedings of the ACM Interna- tional Conference on Multimedia, pp. 3099-3107, 2023.
Liao, T., Fu, L., Chen, J., Wang. Z., Zheng. Z., and Chen, C. A swiss army knife for heterogeneous federated learning: Flexible coupling via trace norm. Advances in Neural In-
Reddi, S. J., Charles, Z., Zaheer, M., Garrett, Z., Rush, K., Konečnj,, J., Kumar, S., and McMahan, H. B. Adaptive federated optimization. In Proceedings of the Interna- tional Conference Learning Representations, 2021.
formation Processing Systems, 37: 139886-13999911 2024.
Liu, R., Xing, P., Deng, Z., Li, A., Guan, C., and Yu, H. Federated graph neural networks: Overview, techniques, and challenges. IEEE Transactions on Neural Networks and Learning Systems, 2024. doi: 10.1109/TNNLS,200 3360429.
Schölkopf, B., Locatello, F., Bauer, S., Ke, N. R., Kalch- brenner, N., Goyal, A., and Bengio, Y. Toward causal representation learning. Proceedings of the IEEE, 109(5): 612-634, 2021.
10

---

Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
Shao, J., Wu, F., and Zhang, J. Selective knowledge sharing for privacy-preeerving federated distillation without good teacher. Nature Communications, 15(1):349, 2024.
Xie, H., Ma, J., Xiong, L.., and Yang, C. Federated graph classification over non-iid graphs. In Advances in Neural Information Processing Systems, pp. 18839-18852, 2021.
Shchur, O., Mumme, M., Bojehevski, A., and Günnemann, S. Pitfalls of graph neural network evaluation. arXiv preprint arviv: /8/ 1.05868, 2018.
Yan, B., Zhang, H., Xu, M., Yu, D., and Cheng. X. Fe- drfq: Prototype-basee federated learning with reduced redundancy, minimal failure, and enhanced quality. IEEE Transactions on Computers, 73(4): (044)):086-0998 2024.
Tan, Y., Long, G., Liu, I.., Zhou, T., Lu, Q., Jiang, J., and Zhang, C. Fedproto: Federated prototype learning across heterogeneous clients. In Proceedings the AAAI Confer- ence on Artificial Intelligence, volume 36, pp. 8432-8440, 2022.
Yang, Q., Liu, Y., Chen, T., and Tong, Federated machine learning: Concept and applications. ACM Transactions on Intelligent Systems and Technology, 10(2):1-19, 2019,
Yang, Z., Cohen, W., and Salakhudinov, R. Revisiting semi-supervised learning with graph embeddings. In Proceedings of the International conference on machine learning, pp. 40-48, 2016
Tan, Y., Liu, Y., Long, G., Jiang, J., Lu, Q., and Zhang, C. Federated learning non-iid graphs via structural knowl- edge sharing. In Proceedings of the AAAI conference artificial intelligence, volume 37, pp. 9953-99661 2023.
Ye, M., Fang, X., Du, B., Yuen, P. C., and Tao, D. Hetero- geneous federated learning: State-of-the-art and research challenges. ACM Computing Surveys, 56(3): 2023.
Tian, C., Xie, Y., Chen, X., Li, Y., and Zhao, X. Privacy- preserving cross-domain recommendation with federated graph learning. ACM Transactions on Information Sys- tems, 42(5): 1-2, 2024.
Zhang, J., Zhao, L., Yu, K., Min, G., Al-Dubai, A. Y., and
Zomaya, A. Y. A novel federated learning scheme for generative adversarial networks. IEEE Transactions on Mobile Computing. 23(5):3633 3649, 2024.
Wan, G., Huang, W., and Ye, M. Federated graph learning under domain shift with generalizable prototypes. In Pro- ceedings of the AAAI conference on artificial intelligence, pp. 13429-15437, 2024.
Zhang, K., Yang, C., Li, x., Sun, L., and Yiu, S. M. Sub- graph federated learning with missing neighbor genera- tion. Proceedings of the Advances in Neural Information Processing Systems, pp.
Wang, H., Li, Y., Xu, W., Li, R., Zhan, Y., and Zeng, Z. Dafkd: Domain-aware federated knowledge distillation. In Proceedings of the IEEF/CVF Conference Com- puter Vision and Pattern Recognition, pp. 20412-20421, 2023.
2021.
Zhu, Y., X., Wu, Z., Wu, D., Hu, M., and Li, R.-H. Fedtad: Topology-awan data-free knowledge distillation for subgraph federated learning. In Proceedings of the International Joint Conference on Artificial Intelligence, 2024.
Wang, Y., Liu, Y., Shen, X., Li, C., Ding, K., Miao, R., Wang, Y., Pan, S., and Wang. X. Unifying unsupervised graph-level anomaly detection and out-of-distribution de- tection: A benchmark. In Proceedings of International Conference on Learning Representations, 2025.
Zhu, Z., Hong, J., and Zhou, J. Data-free knowledge distilla- tion for heterogeneous federated learning. In Proceedings of the International Conference on Machine Learning, pp- 12878-12889, 2021.
Wu, Z., Pan, S., Chen, F., Long, G., Zhang, C., and Philip, S. Y. A comprehensive survey on graph neural networks. IEEE Transactions on Neural Networks and Learning Systems, 32(1):4422)) 24, 2020.
Xia, Z., Zhang, X., Liang, L., Li, Y., and Gong, Y. Feder- ated graph augmentation for semisupervised node clas- sification. IEEE Transactions on Computational Social Systems, 11(3):3232-3244 2024.
Xie, C.,, Huang, D.-.., Chu, W., Xu, D., Xiao, C., Li, B., and Anandkumar, A. Perada: Parameter-efficient federated learning personalization with generalization guarantees. In Proceedings of the IEFE/CVF Conference on Com- puter Vision and Pattern Recognition, pp. 23838-23848 2024.

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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
A. Summary of Appendix
In the appendix, the following contents are included.
Summaries of the proposed FedATH in Algorithm and the used datasets.
Performance comparison with Fl metric.
Proof of Theorem 5.1 and Corollary 5.2.
Performance with increasing client number.
Convergence verification of the proposed FedATH.
B. Summaries of The Propose FedATH and The Used Datasets
We summarize the main steps of the proposed FedATH in Algorithm I and the main statistics of used graph datasets in Table 7.
Algorithm The flow FedATH
Input: Number of clients K, local training epochs E, communication rounds T, learning rate 7), trade-off parameter X.
local graph data G (Va, Ex, Xx), causal GNN fÊ. and biased GNN Λ2.
Output: Global causal GNN f.
1: Client Side:
2: for = 1: in parallel
for epoch e= E do
4: Calculate the casual and biased edge mask matrices S2g and 225 via Eqs. (5) and (6); Calculate the casual and biased representations Hi and H. via Eq. (7);
5:
6: Calculate LCE < (1 y.: hf) via Eq. (8);
7:
Calculate LENT + (h via Eq. (9);
8:
Calculate LDEP f H HD) via Eq. (11);
9: Update f jk
17 (CCE + LDEP);
10: Update f Jk
b. IL + LDEP);
11: end for
12: Upload the local causal GNN f to the server;
13:
end for
Server Side:
:o do
16:
Aggregate the parameter of global causal GNN via W ΣK k=
N N/WW
17:
Distribute the global causal GNN We, to clients;
18: end for
Dataset
Nodes
Features
Edges
Classes
Train / Val / Test
Cora
2,708
1,433
5,429
7
20% 40% / 40%
PubMed
19,717
500
44.338
3
20% /40 % / 40%
ogbn-arxiv
169,343
128
231,559
40
60% 20% / 20%
Photo
7,487
745
119,043
8
20% 40% / 40%
WikiCS
11,701
300
216,123
10
50%
20% / 30%
Roman-empire
22,662
300
32,927
18
50%
20% / 30%
Table 7. Descriptions of six graph datasets.
12

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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
Cora
PubMed
ogbn-arxiv
Type
Method
= 10
K 15
20
K 10
K
= 20
K 10
= 15
K 20
FedAvg
72.04
66.22
$8.80
80.20
79.00
78.87
26.05
26.17
25.07
FedProx
72.89
66.68
59.65
80.21
79.05
78.88
26.07
25.93
25.12
MOON
71.88
67.95
58.03
80.41
79.10
78.10
25.82
24.73
25.56
FL
FedOPT
67.73
63.64
59.75
78.10
75.33
76.51
24.06
24.23
26.35
FedProto
72.99
67.01
59.38
80.44
79.03
78.88
26.09
25.99
24.99
FedSage+
71.96
62.07
59.85
82.18
78.03
76.18
27.45
28.82
30,23
FGSSL
72.94
69.33
61.85
79.88
79.32
78.82
30.41
27.93
26.59
FGL
FedPUB
73.08
66.76
60.01
80.43
75.44
79.63
29.55
24.83
24.37
FedTAD
71.57
70.38
60.34
80.26
78.84
78.57
27.20
26.74
25.10
FGL
FedATH
76.93
72.16
65.80
82.71
81.82
81.31
34.90
30.94
31.97
Table Performance comparison (FI %) on Cora, PubMed, and ogbn-arxiv datasets for all compared methods, where BL denotes the baseline, the optimal results are bolded and the suboptimal results underlined.
Photo
WikiCS
Roman-empire
Type
Method
= 10
K 15
K = 20
K 10
K 15
K 20
K 10
K 15
K 20
BL.
FedAvg
84.85
83.06
81.37
63.65
61.66
62.38
30.34
29.74
27.80
FedProx
84.78
83.70
81.37
63.55
61.50
62.34
30.22
29.52
27.60
MOON
83.64
82.02
79.43
63.99
62.15
60.90
29.86
29.60
28.55
FL
FedOPT
84.92
83.02
80.26
60.53
60.37
56.86
30.50
29.58
28.08
FedProto
84.94
85.07
81.70
64.04
61.95
62.25
30.01
28.82
27.32
FedSage+
86.13
82.00
82.94
61.47
62.17
61.15
31.88
29.39
29.60
FGSSL.
85.99
82.62
80.24
64.84
64.11
62.91
33.59
33.08
31.52
FGL
FedPUB
85.86
83.53
79.32
63.79
59.18
59.71
30.15
29.22
28.71
FedTAD
85.24
83.45
81.20
65.31
63.69
63.10
35.08
33.80
33.21
FGL
FedATH
88.80
86.52
82.88
71.10
68.15
67.37
44.98
44.27
42.08
Table Performance comparison (F1 %) Photo, WikiCS, and Roman-empire datasets for compared methods, where BL. denotes the baseline, the optimal results are bolded and the suboptimal results are underlined.
C. Performance Comparison with F1
To avoid the biased evaluation for experimental results, we employ an additional widely used classification metric F1-score (FI), which can overcome the evaluation distortion caused by the unbalanced data. From Tables and 9, can be seen that the proposed FedTAH consistently achieves the optimal results just like ACC, showing that FedATH solidly enhances the ability of global model. Nevertheless, some algorithms achieve suboptimality on ACC but not FI, such as FGSSL and FedTAD, demonstrating that they vulnerable to unbalanced data.
D. Proof of Theorem 1 and Corollary 1
For Theorem 5.1, we provide the detailed proof process. Denote D and Dj the global and local distributions, respectively. Dr denotes the empirical local distribution. bą local hypothesis learned on the local empirical distribution D, and defined as h4 X y. mapping the data features into predicted labels. 1/K E h& denotes the global hypothesis integrated by local hypothesis. Ht denotes the hypothesis space of VC-dimension d. Moreover, without losing generality, it specified that N₂ = Nx
Theorem D.I. Given an FGL system with global distribution 'D and local distribution DE. with the the probability at least
13

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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
1 80 < < 1), the generalization error for any hypothesis by satisfies
Ro(h) -
1
Σ
Ro (hr)
K
kE[K]
1
Σ dHAH (Dx,
K(K
-
(15)
kE|K] Li
Σ kE[K]
A +
4
2em [dlog
+ loo 4K
m
d
where R-p, denotes the empirical risk on DRs dHA Di) is the H-distance between Di and DD, denotes upper-bound constant with respect to (D. E [K], A (Ro(( + RD. denotes the optimal risk on 'D and kj--
Corollary D.2. Given FGL system, the k-th and 1-th empirical local distributions are denoted as Dr. and Di, its generalization error follows Theorem 5.1. For duau (Dx, Di), the following inequality holds.
(1
1
(HA,(DN, D.) 1 sup By Bi Bx
Dmin +
D
(16)
By
(l|ANIF + (1)1))
| XillF:
where sup denotes the supremum, Bw and Bx denote the network parameters of GNN and the upper-hound constants with respect to the data features X, respectively. Dmin denotes the minimum degree.
When the local causal subgraphs are explored, the edge weights in the adjacency matrices are reduced from [0, 1], the Frobenious norms of ||A&|| and |||||e are decreased, then the bound of generalization error with respect the global hypothesis can be shrunk. Hence, the generalization ability of global model learned by the proposed FedATH is enhanced.
Proof. According to the generalization error of FL (Barnes et al., 2022; Zhu et al., 2021), we first introduce following Lemma D.3.
Lemma D.3. Given an FL system with global distribution D and local distribution Da. the generalization error for any hypothesis with the probability at least 1 (0 < 5 1)
1 RD(h) - ERD, (hx) +
1 Σ
(dHAN (Dx, Xx +V
2em 4 doog
4K
+ log
(17)
K
d
KE[K]
KE[K]
For the H-divergence dHAH DD. D), we further have
duam (Dx:D)<<m sduan (Dr. tduan (D..D)
(18)
(K 11)dHH (Dx: s >>> duan (De: D.) ddHAH (Di, (D)
14

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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
Assume AA E [K], duan D) S co, it can obtain
K dHAH
(K 1)HHAH (Dr.D) D)
Di) + En]
L
K
Σ [E[K]
dHAH (Dr,
1
Σ
NHSH (Dz, + sa
K
K(K
-
kE(K]
K
(19)
1
Σ
Σ
dHaH
(Dis Di)
K(K -
KE 17
E
Σ
K(K 1)
E0
kE[K] Ezst
Then, we have
RD(h) S
1 Σ
RDD (ha)
E
Σ dUAH (Dx.Di) +
K
K(K -1)
1)
kE[K]
KE[R] LEE
(20)
Σ
A +
4
dllo d
2em
+ log
4K
)
m
kE[K]
The proof completes.
Proof. Next, we estimate the upper bound of (HA, Di) in the context of FGL. nodes with their topology from the
distributions of the k-th and 1-th clients are sampled, respectively: G₁ = (Ax; X₁) ~ Dr. Gi- = (A,,X) ~ Dj. Taking a two-layer GCN as an example, it has fA, X) = (PXW,, W1 and W₂ denote the parameters first and second layers. Further, Suppose | S Bx, ||ill S Bw, and
where P = [D -=--II-""AA-DDD-->>>>
||2ll S Bw; then we have
If (G+) - f
sigmoid (o W, W2) - sigmoid (Pia (P,X,W,) Wa)l
PAO (PxX; W, W₂ - Pio (,,),W,) W2) llr
(21)
< W2llF || (1|x, -Xill= + | - Poll [|,l)p)
< By || - -x xxll + B2 |P -PilF
< BlyBx Bx ||Px Pill Biy |X Xill
For (AHAA(Drr D,), its supremum be written
dHAH (Dey DI)
= 2 sup
Pro, [ZA (z())) + (#w) - [=:: hu (21) tu (z:)
(22)
=2 sup NEHAH
PR [2: h (2k) = 1] - Pre, [ h (2)) = 11]
=2 sup HERAH
EED, h (2x) E., h (z4)
15

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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
According to the inequality 0.5]][((=]]
[[](=)] 5.5 (.)] it can be further derived
dHAN (Dx, Di)
$ 2 sup 0.5 p, (2k) 0.5 (21)
(23)
$ + sup ED (zx) - E. (z1))
Further, the empirical H-divergence is upper bounded as
dHAN (De. Di)
m
1 + sup
/ (ZR,))
m Σ / (#L.))
f
1
1
sup
f (ZR,)) (2(,)
m f
(24)
1
< 11 + - sup
If (EN,K) (z)a)
f
1
+ m
sup Is (Gk) - (Gi))l
$ 1 Brank
sup If (Gx) - (Gi)ll
m
Notably, the entrywise 1-norm
|A]] 1,ete adheres to
|||| S ||All1,dt $ (rank(A)]AAI 1r--
(25)
Brank (rank(f(G=)-
- f(Gi) is constant.
From Eq. (24). we can see that the upper bound of du (D,, D.) depends on sup, IIf(Gx) - f(Gi)lle. Recall Eq. (21),
supy (Gx) - [( bounded by ||Px P. We further have
Pk- - Pi
D 1/2 1/2
D
11/2 1/2 k
D, 1/2 A,D, 1/2
D 1/21 D 1/2 D 1/2 1112 A,D 11/2
(26)
+D 1/2 D,D 1/2 D 1/2 A, 1/2
D 1/2 (A D 1/2
(A
+D 1/2 A, 1/2 (P----1/AA,) 1/2
T₂
Then, it can obtain
P - P S |Tiler + T22|e-
(27)
For |Tile and Talle, we estimate their upper bound. Denote Dmin the minimum degree of D and D,, then it has
T F
D
1/2 (A
A1) D. 1/2
T
11
Σ
1/2
(As At)
1/2
Ass
(28)
( (-1122) 1/2
(A - Ai) (D min
||A
Aill?
min
1.j
A Arl
D
min
16

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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
T2 = x
1/2
A, 1/2
A, 1/2 1/2 +D; 1/2 A, D 1/2 -D; 1/2
k
(29)
[ 1/2
D
1/2
A,D; k
Denote & the (t, f)-th element of D
1/2
Di" 1/2
With the Taylor extension, we have
6
D Diu
1/2
D
D/1//
1/2
D
-322/2 Rsc
3/2
(D
Res
D(a)
D
Dia
1
D
3/2 kii
Σ
A Atg
(30)
2
Thus, it can be derived
|0:| S
Σ
(Akyy Ald)
(31)
2D
min
For the (i, element of T2 we can obtain
T2yl S |A
D
1/2 + D
1/2 DT,11/2 AAyy
A 15,1
(32)
Since Ar., $ and the entries of D
1/2 and D- 1/2 are bounded by D D-1/2 the following inequality holds:
T241 S 0il Atul
D
1/2
D 1/2 1122
|Ais .559
Kys
(Ak -taa)
11DD1112 + +D .1
Σ
(Agza Als.))
2D
3/2 min
2 2D 3/2
(33)
min
k
(ARja AGee) 1
(Ara
Aoe
Σ k
2 min
Then, follows that
T2 2
T21
igj
2
( Rick
Alad)
+
Σ (ARta Alsa))
2D2 min
8.J
(34)
(AR
Ala))
+ Σ (Aize - Alga)
2D2
min
2
21
D2
2
Σ
(A Ald)
m+
(Ak
A1na)
m
min
Actually,
=
|1b is the square of degree divergence between A₁ and At. Note that
A Ail = -E (AR - A
1
(35)
Σ 1pl < m max (w:)² $ m
According to Cauchy-Schwarc inequality, we have
72 Aen)
Σ (Auu - Alc)) =
O(AR (ARu
5m [AAE - Ain)
(36)
k
17

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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
Then, has
Σ Σ
Ars Alee)
Σ
(Aki
Arre)
(37)
S
(Aki - Alre2
;
R
m A| AIl
Hence, Eq. (34) turns out to be
2
22 2 1
m |A - Aill.
D2
min
(38)
T22
1221 S
m7
|| AillF
D
min
Thus, Eq. (27) is derived into
P2 Pille S ||Till + 1|T2ll
m
|| - Ailpt +
|| - AilF
"Dmin
D min
(39)
(Poing + m
|AA -All
D2 Puin
So far, we can obtain the upper bound of duaH Di):
dHau (De:D.) D.)
1+ Brank
sup |ff - -
m
Brank
< 1 +
sup Bly Bx |P P |X -XillF
m
Duin + m Da m
|| A + Bly | 1Xil
(40)
< 1 + sup Bl Bx
m
Bly
=1+ sup Bly Bx
||A+
Arlle +
| XxIlF
D Duin
D2
min
m
B
< 1 + sup Bly Bx
(1AAll + Adllp)
X Xills
Duin
D2
f
can be seen that duam (D,, 2 D:) is bounded by ||A|||r and ||A|||r. When the local causal subgraphs are explored, the edge weights in the adjacency matrices are reduced from [0, 1], the Frobenious norms of || and ||A||| decreased, then the bound of generalization error with respect the global hypothesis can be shrunk. Hence, the generalization ability of global model learned by the proposed FedATH is enhanced. The proof completes.
E. Performance with Increasing Client Number
We verify the performance of different methods the number of clients increases in Fig. 6. First, the performance of all algorithms degrades as the client number increases, this is because too many clients create fragmentation of information and disturbance to federated aggregation. Notably, the proposed FedATH consistently maintains the optimal. Second, for different graph datasets, the performance of the algorithms degrades to varying degrees as the client number increases. The reason is that different graph datasets have different levels of importance for the connectivity information, and the loss of connectivity information affects model performance to different degrees.
18

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Less is More: Federated Graph Learning with Alleviating Topology Heterogencity from A Causal Perspective
Ciacoiication ACC (NQ
Clarsif ACI
Number of Cherts
Number of Chenis (b) WikiCs
(a) Cora
Figure 6. The performance for all compared methods the client number increases.
F. Convergence Verification
We verify the convergence property the proposed FedATH compared other federated learning algorithms in Fig. can be observed that the performance of FedATH keeps steadily increasing faster rate and takes the lead after certain number of communication rounds. Notably, the convergence of some FGL methods is inferior due the complex computation process, e.g., FGSSL and FedPUB on PubMed dataset, while the proposed FedATH does not suffer from this case, proving its superiority of convergence.
Clarattattoo (S3)
Roaiion ACC
1aggar ACC
Fechno
techiss
wotn fefteee
Hoow
techers
fessage
Pobli
TedSor
NCON
Facttp
PedSFT
6 6. www.6666600000000000000000
0000000
Communication Round
Communicaion Round
Communcation Round
Communication Round
(a) PubMed
(b) ogbnnarn aixi-arxiv
(c) WikiCS
(d) Koman-empire
Figure 7. The classification ACC curves as the increasing communication round for all compared algorithms, where the client number is set
19


Reviews:
[{"review_id": "797T0tl1rL", "reviewer": "Reviewer_AFBr", "summary": "The paper proposes to divide the local graph into causal subgraph and biased subgraph  for alleviating the topology heterogeneity, the causal subgraph possesses the key information for the downstream task, and the biased subgraph possesses the confusing information. Thus, only the causal graph neural networks are shared in the server. The generalization bound for the proposed FedATH is derived, and the experimental results also support the authors’ argument.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your valuable comments, we response your concerns as follows.\n\n**Response to Weakness1:** We test the performance of several federated learning methods when all the edges are removed on Cora dataset. It can be seen that although topological heterogeneity causes a decline in federated learning, the topology still favors feature learning.\nTherefore, it is not the case that removing the topology guarantees that optimal performance will be achieved. We give an upper bound on the generalization error in the theorem. However, for practical purposes, a learning system has a lower bound on its generalization error, but the analysis of the lower bound is difficult to infer.\n\n| Method    | Cora (K=10) | Cora (K=15) | Cora (K=20) |\n|-----------|-------------|-------------|-------------|\n| FedAvg    | 62.57       | 64.99       | 61.26       |\n| FedProx   | 64.22       | 65.43       | 62.47       |\n| MOON      | 63.97       | 65.34       | 61.60       |\n| FedOPT    | 62.75       | 64.61       | 62.17       |\n| FedProto  | 60.72       | 62.21       | 60.17       |\n\n**Response to Weakness2:** Validating the adaptability of federated learning algorithms in large-scale client scenarios is necessary. We conduct the experiments with 50, 80, and 100 clients on Photo dataset. It can been seen that the proposed FedATH still outperforms other methods in the scenario of large-scale clients, demonstrating that it has good adaptability in face of more clients.\n\n| Method    | Photo (K=50) | Photo (K=80) | Photo (K=100) |\n|-----------|--------------|--------------|---------------|\n| FedAvg    | 84.20        | 83.11        | 82.38         |\n| FedProx   | 84.23        | 83.11        | 82.41         |\n| MOON      | 84.79        | 82.99        | 82.37         |\n| FedOPT    | 80.56        | 78.78        | 79.16         |\n| FedProto  | 75.04        | 74.16        | 72.20         |\n| FedSage+  | 84.88        | 83.44        | 82.98         |\n| FGSSL     | 85.71        | 83.45        | 83.38         |\n| FedPUB    | 83.69        | 82.18        | 79.75         |\n| FedTAD    | 84.97        | 83.72        | 83.23         |\n| FedATH    | **86.10**        | **84.16**        | **83.43**         |\n\n**Response to Weakness3:** We have noticed that a review of causal learning is necessary, a review about causal learning is written as follows.\n\nCausal learning [1,2] aims to uncover causal factors that affects the data distributions and has gained widespread attention in machine learning.\nMining causal factors is often used in invariant representation learning. For example, Lu et al. [3] aimed to discover the independent variables in the data under the nonlinear setting. \n[4] further disentangled the causal assumptions into invariant factors and nonstable factors.\n[5] proposed a causal variational autoencoder that captured the causal structures in the data.\nIn recent years, to learn invariant representations in graph learning, some studies have also absorbed the idea of ​​causal learning. \nFor instance, [6] explored a key subgraph from the initial graph to capture the most important information.\n[7] proposed a graph contrastive invariant learning to enhance the performance of graph representation.\n[8] introduced the graph autoencoder to learn the invariant graph representations across multiple domains.\nInspired by the causal learning, we believe that the local graph consists of causal subgraph and biased subgraph, and sharing the causal information in them help enhance the generalization of the global model.\n\n[1] Pearl, J. Causality. 2009.\n\n[2] Causality for trustworthy artificial intelligence: status, challenges and perspectives. ACM Computing Surveys,  2025.\n\n[3] Invariant causal representation learning for out-of-distribution generalization. ICLR, 2022.\n\n[4] Invariant and transportable representations for anti-causal domain shifts. NIPS, 2022.\n\n[5] Causalvae: Disentangled representation learning via neural structural causal models. CVPR,  2021.\n\n[6] Learning causally invariant representations for out-of-distribution generalization on graphs. NIPS, 2022.\n\n[7]  Graph contrastive invariant learning from the causal perspective. AAAI, 2024.\n\n[8] Learning causal representations based on a gae embedded autoencoder. IEEE TKDE, 2025.\n\n**Response to Question1:**  We have released the code at https://anonymous.4open.science/r/FedATH-9517.\n\n**Response to Question2:** Label heterogeneity emphasizes that the label distribution for independent samples creates a bias. Topological heterogeneity, on the other hand, emphasizes more on the connecting edge relationship between nodes to produce variance. In other words, label heterogeneity can lead to topological heterogeneity, while topological heterogeneity may not be caused by label heterogeneity. \n\n**Response to Question3:** The purpose of Eq. 9 is to smooth the predictive distribution output by the biased network, not just for labeled nodes, but for all nodes.", "claims_and_evidence": "The authors claim that less edges are more conducive to handling the topology heterogeneity, then the generalization analysis and experimental results support their claims.", "methods_and_evaluation_criteria": "The proposed FedATH explores the local causal subgraph and improves the performance on the test datasets, making sensing of the domain of federated graph learning.", "theoretical_claims": "I checked the generalization analysis in Section 5 and the proof process in Appendix, I’m leaning towards them being right.", "experimental_designs_or_analyses": "Overall, the experimental section is ok.", "relation_to_broader_scientific_literature": "The representation division [1][2] has been applied into some scenarios, it aims at capturing an invariant representation from the coupled representation. The proposed FedATH is also based on this theory.\n\n[1] On learning invariant representations for domain adaptation. In ICML 2019.\n[2] Self-supervised learning of pretext-invariant representations. In CVPR 2020.", "essential_references_not_discussed": "No, the current literature discussion is adequate.", "other_strengths_and_weaknesses": "Strengths:\n\n1. In Figure 1, the authors visualize what topology heterogeneity is and the shortcomings of existing methods, showing that the proposed FedaATH is well motivated.\n\n2. The authors provide a rigorous derivation of the generalizability analysis for the proposed FedATH, and the conclusion shows that mining causal subgraphs and abandoning biased subgraphs is beneficial to enhance the generalizability of the model.\n\nWeaknesses:\n\n1. Corollary 5.2 illustrates that mining the causal subgraphs are beneficial for reducing the topology heterogeneity. Consider an extreme case when all edges are removed, whether the performance is optimal.\n\n2. In the experiments, the FL algorithms are tested with up to 20 clients. However, more clients will be involved in the practical application. So, it’s necessary to conduct the experiments with more clients, at least 100 clients.\n\n3. In related work, only FL and FGL are introduced. The causal learning is a key technique, a review about it is indispensable.", "other_comments_or_suggestions": "See above weakness.", "questions_for_authors": "1. The code of the proposed FedATH is unavailable, to validate its reproducibility, it’s suggested to release the code.\n\n2. Whether the so-called topological heterogeneity is just a form of label heterogeneity on a graph, and what is the essential difference between the two.\n\n3. In Eq. (9), is it reasonable to seek for an expectation for all nodes? Is it more appropriate to seek for an expectation for nodes that are only labeled?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tevlb1PrVP", "reviewer": "Reviewer_oUdB", "summary": "This paper proposed a reduced-edge based federated graph learning method that aims to mitigate the effects of topological heterogeneity on federated learning. Specifically, the proposed FedATH assesses the importance of each edge via an edge evaluator. Thus, the local subgraph is divided into causal and biased subgraphs. The causal subgraph is deemed to capture the essential information and perform the classification.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your professional advices, the detailed responses are written as follows.\n\n**Response to Weakness1:** The difference between causal and biased graphs is whether they contain critical edge information. We use an edge evaluator to assess the importance of each edge. Subgraphs that contain information about important edges are called causal subgraphs, and subgraphs that contain information about the remaining edges are called biased subgraphs.\n\n**Response to Weakness2:** In traditional federated learning, the samples are isolated from each other and do not affect each other in model training, the model is only affected by differences in data label distribution or feature distribution. In contrast, for federated graph learning, the samples have connecting edges between them, and the sample representations are affected by connecting edges between different nodes. For example, the same class of nodes may connect different classes of nodes in different clients, then the representations minght be heterogeneous even for same classes of nodes in various clients. Therefore, topological heterogeneity is different from traditional label heterogeneity or feature heterogeneity.          \n\n**Response to Weakness3:** First, to verify that sharing only causal networks is sufficient, we tested the algorithm performance when sharing different network components. In most scenarios, sharing only the causal network can achieve optimal results, which indicates that the causal factor is a key component for affecting the performance of the model. It should also be noted that sharing edge evaluators may also be beneficial to improve the performance, due to the fact that the fusion of edge information from different clients facilitates the quality of the evaluators. Second, the experimental results on Photo dataset with large-scale clients in the following table. We can see that the proposed FedATH still takes the leading position in the scenario of large-scale clients.\n\n| Method    | Photo (K=50) | Photo (K=80) | Photo (K=100) |\n|-----------|--------------|--------------|---------------|\n| FedAvg    | 84.20        | 83.11        | 82.38         |\n| FedProx   | 84.23        | 83.11        | 82.41         |\n| MOON      | 84.79        | 82.99        | 82.37         |\n| FedOPT    | 80.56        | 78.78        | 79.16         |\n| FedProto  | 75.04        | 74.16        | 72.20         |\n| FedSage+  | 84.88        | 83.44        | 82.98         |\n| FGSSL     | 85.71        | 83.45        | 83.38         |\n| FedPUB    | 83.69        | 82.18        | 79.75         |\n| FedTAD    | 84.97        | 83.72        | 83.23         |\n| FedATH    | **86.10**        | **84.16**        | **83.43**         |", "claims_and_evidence": "The claims of this paper are clear, and the experimental results support the claims.", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation criteria make sense to the federated graph learning field.", "theoretical_claims": "I check the Theorem 5.1 and Corollary 5.2, I believe that they are correct.", "experimental_designs_or_analyses": "Overall, the experiment parts are complete, but I still have some concerns:\n\n1.\tMore analysis should be explained about the effects of sharing different components.\n\n2.\tLarge-scale clients are an important challenge for federated learning, and I would like to know how the proposed algorithm performs in the face of large-scale clients.", "relation_to_broader_scientific_literature": "This paper follows the problem setting of literature [1] and [2] and aims to address the difficulties posed by topological heterogeneity for federated learning. However, unlike them, this work mitigates the problem of topological heterogeneity by reducing data information rather than increasing it. Furthermore, the proposed method adopts the HSIC criteria [3].\n\n  [1] Ke Zhang, Carl Yang, Xiaoxiao Li, Lichao Sun, and SiuMing Yiu. Subgraph federated learning with missing neighbor generation.\n\n  [2] Yinlin Zhu, Xunkai Li, Zhengyu Wu, Di Wu, Miao Hu, and Rong-Hua Li. Fedtad: Topology-aware data-free knowledge distillation for subgraph federated learning.\n\n  [3] Greenfeld, Daniel and Shalit, Uri. Robust learning with the hilbert-schmidt independence criterion.", "essential_references_not_discussed": "I'm not aware of any additional literature that needs to be discussed.", "other_strengths_and_weaknesses": "Strengths:\n\n1.\tIn terms of topological heterogeneity in federated graph learning, the authors provide new ideas for solving the problem by reducing data information rather than introducing new information.\n\n2.\tIt is remarkable that this paper theoretically guarantees that the proposed method can effectively solve the topological heterogeneity problem.\n\n3.\tThis paper is well-written and the presentation is excellent.\n\nWeaknesses:\n\n1.\tThe authors mention dividing this map into causal and biased subgraphs, however, their exact definitions are not made clear.\n\n2.\tIn fact, I don't fully understand what the authors mean by topological heterogeneity and how it differs from the common data heterogeneity of federal learning.\n\n3.\tThe questions about the experiments refer to the above ‘Experimental Designs and Analysis’.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "In addition to the above weaknesses, I still have one more concern:\n\nTo decouple the causal graph and the biased graph, the authors encoded the two graphs locally using two GCNs. This process seems to bring higher computational cost, whether the proposed method consumes a much higher runtime than the comparison method?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "woUlBuVZx9", "reviewer": "Reviewer_eaWv", "summary": "To address the topology heterogeneity of FGL, the authors proposed an interesting idea, namely, Less is More. Concretely, the unnecessary edges are discarded while the necessary edges are maintained. The CE loss and NE loss are separately used to train the corresponding GNNs. The HSIC loss is adopted to enforce the training of the dual-GNN. From both theorical and experimental perspectives, the authors verify the superiority of the proposed FedATH.\n\n\n## update after rebuttal\nAfter the Reviewer-author discussion phase, I maintain my score and explicitly support acceptance.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your valuable comments, the detailed responses are provided as follows.\n\n**Response to Weakness1:** We have collected two new federated graph learning methods FGGP [1], FedGTA [2], and compared them with the proposed FedATH. The experimental results are reported in the following table, it can be observed that the proposed FedATH is more superior than the new FGL methods.\n\n| Method   | Cora (K=10) | Cora (K=15) | Cora (K=20) | Photo (K=10) | Photo (K=15) | Photo (K=20) |\n|----------|-------------|-------------|-------------|--------------|--------------|--------------|\n| FGGP [1] | 73.96       | 71.75       | 65.84       | 88.56        | 86.27        | 84.25        |\n| FedGTA [2] | 75.36     | 71.43       | 65.87       | 88.28        | 87.02        | 85.17        |\n| FedATH   | **77.90**       | **73.42**       | **67.97**       | **90.33**        | **88.61**        | **85.50**        |\n\n[1] Wan, G., Huang W., Ye M. Federated graph learning under domain shift with generalizable prototypes. AAAI, pp. 15429-15437, 2024.\n\n[2] Li, X., Wu, Z., Zhang W., etc. FedGTA: topology-aware averaging for federated graph learning. VLDB, pp. 41-50, 2024.\n\n**Response to Weakness2:** In fact, when we divide the subgraphs to multiple clients, each client has its own training set and test set. Thus, after federated training, the global model is tested on each client and eventually the average performance of all clients is reported.\n\n**Response to Weakness3:** We summarize the meanings of main symbols in the following table.\n| Symbol | Meaning |                                                                                                       \n|----------------------|------------------------------------------------------------------------------------------------------------------|\n| $G=(V, E, X)$ | A graph dataset, node set, edge set, feature matrix |\n| $G_k^c$, $G_k^b$ | The causal subgraph, the biased subgraph |\n| $A_k$, $H_k$ | The adjacency matrix, the latent embedding for the k-th client |\n| $C$, $K$ | The number of classes, the number of clients |\n| $N_k$, $N$ | The number of nodes for the k-th client, the number of total nodes |\n| $c_{ij}$, $w_{ij}$ | The importance score for edge $e_{ij}$, the normalized score |\n| $\\Omega_k^c$, $\\Omega_k^b$ | The causal edge mask, the biased edge mask for the k-th client |\n| $f_k^c$, $f_k^b$, $\\Phi_k$ | The causal GCN, the biased GCN, the edge evaluator for the k-th client |\n| $\\lambda$ | The trade-off parameter |\n\n**Response to Minor Weaknesses:** Thanks for the meticulous advice. We will improve the presentation in a revision version.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes, the traditional FGL methods tend to create an extra global information for correct the local training, while the presented idea goes the other way and advocate discarding the unnecessary edges.", "theoretical_claims": "Theorem 5.1 and Corollary 5.2 give the generalization bound for FGL, the proof is complete, which well supports the soundness of the proposed FedATH.", "experimental_designs_or_analyses": "In order to verify the effectiveness of the proposed method, the authors conducted a large number of experiments, and the experimental design and analysis are fully considered.", "relation_to_broader_scientific_literature": "The proposed FedATH aims to reduce the topology heterogeneity of FGL [1,2,3], discarding the unnecessary edges while maintaining the necessary edges. Compared to the existing literature, the proposed FedATH is well motivated and can make sense of the FGL field.\n[1] Chen F, Li P, Miyazaki T, et al. Fedgraph: Federated graph learning with intelligent sampling. IEEE Transactions on Parallel and Distributed Systems, 2021, 33(8): 1775-1786.\n[2] Xie H, Ma J, Xiong L, et al. Federated graph classification over non-iid graphs. Advances in neural information processing systems, 2021, 34: 18839-18852.\n[3] Tan Y, Liu Y, Long G, et al. Federated learning on non-iid graphs via structural knowledge sharing, Proceedings of the AAAI conference on artificial intelligence. 2023, 37(8): 9953-9961.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strength:\n    First, this paper proposes a new idea that discarding unimportant edges is more conducive to alleviating topological heterogeneity in federated graphs, which is different from the existing works. Second, many experiments are conducted on real-word graph datasets, the results demonstrate that the proposed FedATH outperforms other methods.\n\nMajor Weaknesses:\n    1) The proposed FedATH is compared with nine FL algorithms, only four of which are tailored to the distributed graphs. At present, there are many FGL studies. Therefore, it’s suggested to compare FedATH with more FGL methods.\n    2) The authors split a complete graph into multiple subgraphs via Louvain algorithm. When the training of FL finishes, how do the author test the algorithms? Are they performed on the original graph? The details of test period should be explained.\n\n Minor Weaknesses:\t\n1)\tThere are too many symbols in the paper, a Table is needed for summarizing them.\n2)\tSome typos and grammatical errors still appear, the authors should carefully check the paper.\n3)\tThe format for several figures is required to be changed, especially for the text size.", "other_comments_or_suggestions": "None.", "questions_for_authors": "None.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OYnH1EzxRB", "reviewer": "Reviewer_eW6P", "summary": "This work proposed a causal subgraph learning method for graph federated learning. The work consists of three critical components. First, the edge evaluator separates the local subgraph into causal subgraph and biased subgraph. Second, the dual-GNN is developed to encode the corresponding subgraphs. Third, the separability of causal and biased subgraphs is enhanced by HSIC. The experimental results on six graph datasets demonstrate the proposed method’s superiority.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Many thanks for your valuable feedback, we carefully reply your concerns as follows.\n\n**Response to Weakness(a):** The topology of a graph has a significant influence on node embedding. However, inspired by causal learning, we argue that only the critical topological information is a direct determinant for the class of node representations, which is called the causal factor, while the remaining topological information is called the biased factor.\n\n**Response to Weakness(b):** Admittedly, the general form of the algorithm's generalizability analysis has its roots in domain generalization theory. Indeed, most studies on domain adaptation and domain generalization further analyze the distribution discrepancy $d_{\\mathcal{H}\\Delta \\mathcal{H}}(\\tilde{\\mathcal{D}}_s,\\tilde{\\mathcal{D}}_t)$. \nSimilarly, we analyze the term for the upper bound, but for graph data, it is necessary to consider not only differences in feature distributions, but also differences between topological distributions. Therefore, we perform a customized upper bound analysis, which is not available in the existing work on federated graph learning.\n\n**Response to Weakness(c):** The computational complexity of FedATH mainly comes from the computation process on the clients and consists of three components. Let us take the computation of a client as an example. First, the forward propagation of dual-GCN costs $O(|E|L)$, where $|E|$ denotes the number of edge and $L$ denotes the dimension of embedding. Second, the computation of edge evaluator costs $O(|E|d)$, where $d$ denotes the feature dimension of nodes. Third, the calculation of HSIC criterion takes $O(N^3)$. Overall, the computational complexity of the proposed FedATH is $O(|E|L+|E|d+ N^3)$.\n\n**Response to Weakness(d):** We use Python and PyTorch to conduct the simulation environment, and perform the experiments of FedATH and compared experiments on a server equipped with an Intel(R) Core(TM) i9-10980XE processor, an RTX 3090 GPU, and 128 GB of RAM. Adam is selected as the optimizer, and the learning rate is set as 0.001. The numbers of communication rounds and local training epochs are fixed as 100 and 3, respectively. \n\n**Response to Weakness(e):** We will optimize the figures in the following revision for a better presentation.\n\n**Response to Essential References Not Discussed:**\nStudy [1] proposes to decouple graph data into features of multiple potential subspaces through a new neighborhood routing mechanism, which mainly aims to mine the decoupling factors behind the data and enhance the interpretability and generalization of the model. However, this mechanism may not be suitable for solving topological heterogeneity problems in federated graph learning, since the separation of multiple decoupling factors is computationally and communication-unfriendly, and it is difficult to judge which factor is really needed to be shared by all clients. In contrast, the proposed FedATH decouples the node representations from the topology level and only shares the causal networks, which is computationally and communicatively acceptable.\nStudy [2] proposes a kernel-based optimization method to reduce the discrepancy between the source and target domains and proves it theoretically. However, as described in Response to W(b), for federated graph learning, the effect of topology on the generalization error needs to be taken into account, and thus we tailor an analytical approach that is not found in the exsiting studies.\n\n[1] Ma, J., Cui, P., Kuang, K., Wang, X., Zhu, W. Disentangled Graph Convolutional Networks. ICML, pp. 4212-4221, 2019.\n\n[2] Muandet, K., Balduzzi, D., Schölkopf, B. Domain generalization via invariant feature representation. ICML, pp. 10-18, 2013.", "claims_and_evidence": "Yes, the authors claimed that mining a few of critical causal factors is more effective than introducing new global information, which is reasonable, and experimental results also demonstrated the validity of this argument.", "methods_and_evaluation_criteria": "Yes. This work adheres to the perspective that mining causal subgraphs is more conducive to reducing heterogeneity in graph federated learning, which is a novel viewpoint for this field. Furthermore, the experimental setup and evaluation criteria follow the classic works in graph federated learning, which is reasonable.", "theoretical_claims": "The proposed Theorem 1 shows the upper bound for a graph federated learning system. However, similar generalization analyses have appeared in many domain adaptation and domain generalization efforts, so it is difficult to see a difference between them.", "experimental_designs_or_analyses": "Yes. I have checked the experimental designs and analyses. The authors adopted the Louvain method to split the graph into multiple private subgraphs, and compared the proposed methods with nine typical algorithms. The designs are well thought out. However, the analysis about experimental results needs to be improved, such as hyperparameter study.", "relation_to_broader_scientific_literature": "This work proposed a decoupled graph representation learning method in the scenario of federated learning, which might be inspired by literature [1]. The proposed Theorem 1 in this work is drawn from literature [2]. \n\n[1] Ma J, Cui P, Kuang K, et al. Disentangled graph convolutional networks, ICML, 2019.\n[2] Muandet K, Balduzzi D, Schölkopf B. Domain generalization via invariant feature representation, ICML, 2013.", "essential_references_not_discussed": "Refer to ‘Relation to Broader Scientific Literature’", "other_strengths_and_weaknesses": "Strengths:\n\n   (a) This work proposed a graph decoupling method for graph federated learning, which is significantly different from the existing works. The idea is inspiring, and the writing is good.\n\n   (b) This work demonstrates the effectiveness of the proposed method from two aspects. First, the work conducts a theoretical framework to show the effectiveness. Second, extensive experiments are conducted to prove the claims.\n\n   Weaknesses:\n\n   (a) In Fig.3, the authors claim to capture the causal and biased factors of local subgraphs, but do not explain clearly what a causal factor is and what a biased factor is for local subgraphs.\n\n   (b) The work proposes a generalization analysis for graph federate learning, which has frequently presented in domain adaption and domain generalization. Hence, the differences between them must be explained.\n\n   (c) The computation complexity of the proposed methods must be analyzed.\n\n   (d) The simulation environment should be further introduced in details.\n\n   (e) The font of some figures is too small, it’s suggested to adjust them.", "other_comments_or_suggestions": "See the weaknesses.", "questions_for_authors": "See the weaknesses.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: In this submission, the authors propose a new federated graph learning method, which aims to tackle the challenge of topology heterogeneity, different from the traditional data heterogeneity. The proposed method is designed from a causal perspective, and extensive experiments on several tasks also confirm its benefits and great potentials.

During the rebuttal, all the reviewers appreciate this work. Please incorporate the new important discussion and experiments into the final version. Hope the authors find the discussions with reviewers useful and make this submission a better one.
Decision: Accept (poster)