$S^2$FGL: Spatial Spectral Federated Graph Learning
Authors: Zihan Tan
Suyuan Huang
Guancheng Wan
Wenke Huang
He Li
Mang Ye
Venue: ICML 2025 poster (2025)
Keywords: Federated Learning
Safety in Machine Learning
Primary area: social_aspects->safety

Abstract: Federated Graph Learning (FGL) combines the privacy-preserving capabilities of Federated Learning (FL) with the strong graph modeling capability of Graph Neural Networks (GNNs). Current research addresses subgraph-FL from the structural perspective, neglecting the propagation of graph signals on the spatial and spectral domains of the structure. From a spatial perspective, subgraph-FL introduces edge disconnections between clients, leading to disruptions in label signals and a degradation in the semantic knowledge of the global GNN. From a spectral perspective, spectral heterogeneity causes inconsistencies in signal frequencies across subgraphs, which makes local GNNs overfit the local signal propagation schemes. As a result, spectral client drift occurs, undermining global generalizability. To tackle the challenges, we propose a global knowledge repository to mitigate the challenge of poor semantic knowledge caused by label signal disruption. Furthermore, we design a frequency alignment to address spectral client drift. The combination of **S**patial and **S**pectral strategies forms our framework $S^2$FGL. Extensive experiments on multiple datasets demonstrate the superiority of $S^2$FGL. The code is available at https://github.com/Wonder7racer/S2FGL.git.

FLGL::
Spatial Spectral Federated Graph Learning
Zihan Tan Suyuan Huang" 1 Guancheng Wan
Wenke Huang
He Li
Mang Ye
Abstract
Structure Inertia Score
1%4 SIS
Federated Graph Learning (FGL) combines the
capabilities of Federated Learning (FL) with the strong graph modeling capability of Graph Neural Networks (GNNs). Current research addresses subgraph-FL from the structural perspective, neglecting the propagation of graph signals on the spatial and spectral do- mains of the structure. From spatial perspective, subgraph-FL. introduces edge disconnections be- tween clients, leading to disruptions in label sig- nals and degradation in the semantic knowledge of the global GNN. From spectral perspective, spectral heterogeneity causes inconsistencies in signal frequencies across subgraphs, which makes local GNNs overfit the local signal propagation schemes. As a result, spectral client drift occurs, undermining global generalizability. To tackle the challenges, we propose global knowledge repos- itory mitigate the challenge of poor semantic knowledge caused by label signal disruption. Fur- thermore, we design frequency alignment to address spectral client drift. The combination of Spatial and Spectral strategies forms our frame- work F2IL Extensive experiments on multiple datasets demonstrate the superiority of 5"FGL. The code is available at https: //ts:///g/ //github.
Subgraph
Cora
Scenarios
Client Scale
Pub
Cite
Spectral
Poor Semantic
Heterogeneity
Knowledge
Cora
Citeseer
Pubmed
Figure I. In the first place, compared with centralized GNN train- ing, subgraph- -L encountering label signal disruption chal- lenge, leading to decreased structure inertia score and poor se- mantic knowledge for GNNs. Moreover, we demonstrate the heat map of the Kullback-Leibler divergence of eigenvalue distributions across clients. Inconsistency subgraph signal frequency caused by spectral heterogeneity leads to spectral client drift.
fraud detection (Wang et al., 2019; Tang et al., 2022). How- ever, in real-world scenarios, graph data often generated at the edge devices rather than centralized systems (Zhang et al., 2021a). To address this. Federated Graph Learning (FGL) has emerged (Fu et al., 2022; Liu & Yu, 2022; Tan et al., 2025; 2024; Huang et al., 2022; Wan al., 2024b; 2025b), leveraging the data privacy-preservin capabilities of Federated Learning (FL) (Huang et al., 2024; 2023b;c;
1. Introduction
Graph Neural Networks (GNNs) have demonstrated remark- able efficacy in modeling graph-structured data (Wan al., 2025a; Fang al., 2025), thereby finding applications across various domains, such as social networks (Fan et al., 2020: Zhang et al., 2022b), epidemiology (Liiu et al., 2024), and
2022) to enable the efficient distributed training of GNNs (Huang et al., 2024). A prominent application of FGL subgraph-FL, in which each participant holds a subgraph derived from the same overall graph data.
Although numerous FGL methods have attempted to provide solutions based on structure enhance effectiveness, in- cluding identifying structurally similar collaborators (Baek et al., 2023; Xie et al., 2021; Li et al., 2024), enhancing struc- tural knowledge exchange (Tan et al., 2023; Huang et al., 2023a; Tan et al., 2025), and retrieving generic information under structural shifts (Wan et al., 2024a; Tan et al., 2024). Nevertheless, these approaches overlooked the propagation
Equal contribution National Engineering Research Center for Multimedia Software, School of Computer Science, Wuhan University, Wuhan, China. Correspondence Mang Ye <ye- mang @ whh.eddu.n>>.
Proceedings of the 4dnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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S2FGL: Spatial Spectral Federated Graph Learning
Centralized Learning
Federated Learning
Client i
Client j
Spectra
Spectra
Client
Client
Label Signal Disruption
0
2
2
Learn
Learn
Labeled Node
Spectral Drifts
Server
Signal Propagation
(a) Label Signal Disruption
(b) Spectral Client Drifts
Figure 2. Problem Illustration. (a) From the spatial perspective, nodes in subgraph-FL lose label signals from originally nearby labeled nodes due to edge loss, namely label signal disruption. Correspondingly, GNNs suffer from poor semantic knowledge, leading to a deteriorated global GNN. (b) From the spectral perspective, spectral heterogeneity induces inconsistencics in signal frequencies across subgraphs, leading to spectral client drift in the signal propagation paradigms of GNNs and degraded global generalizability.
of graph signals within the structure. Specifically, graph signal propagation can be analyzed from two perspectives: the spatial and the spectral domain. Specifically, the spatial domain governs the explicit transmission of signals among linked nodes, while the spectral domain characterizes signal diffusion across varying frequency spectra.
graphs and optimize in different spectral directions, leading to spectral client drift and degraded generalizability. Based on our analysis, we pose the question: II) How can we alle- viate spectral client drift under spectral heterogeneity?
To address the challenge of poor semantic knowledge under label signal disruption in Question D), we propose Node Label Information Reinforcement (NLIR). Specifically, our strategy leverages structurally and semantically representa- tive nodes to construct prototype-based global repository of semantic knowledge. During training. NLIR calculates the similarity distribution between all representative proto- types with node features, which provides multidimensional semantic localization of nodes. Consequently, our strategy injects semantic knowledge from the repository into the lo- cal GNN during training, effectively mitigating the issue of poor semantic knowledge under label signal disruption.
From the spatial perspective, due to edge loss, we hypoth- esize that nodes subgraph-FL. lose label signals from originally nearby labeled nodes. This degradation ham- pers the ability of GNNs learn comprehensive semantic knowledge, resulting in poor global performance and re- duced generalizability. Correspondingly, we define this phenomenon as label signal disruption, which naturally ex- ists in subgraph-FL. For verifying. inspired by graph active learning research (Han et al., 2023), we investigate how the Structure Inertia Score (SIS) varies in subgraph-FL. Specifi- cally, SIS evaluates the influence and significance labels on graphs. In Fig. SIS decreases in subgraph-FL compared with centralized training. Correspondingly, existing methods suffer from poor semantic knowledge. Based on our empirical analysis, we pose the question: 1) How can we address the challenge of poor semantic knowledge under label signal disruption?
Considering the spectral client drift posed by spectral hetero- geneity in II), propose Frequency- aware Graph Model- ing Alignment (FGMA). Our method utilizes the similarity relationship of the node feature of the frozen global GNN and the local GNN to reconstruct spectra that incorporates GNNs adjacency awareness. FGMA then projects the high- frequency and low-frequency components of the features onto this spectrum. Subsequently, by aligning the local projections with the global one, we encourage the GNNs to learn globally generic frequency processing scheme, thereby mitigating spectral client drift.
empirically demonstrate that the
From the spectral perspective, inconsistencies in signal fre- quencies across clients caused by spectral heterogeneity in- duce spectral client drift in the signal transmission schemes of GNNs, thereby undermining the collaboration. To verify this phenomenon, we examine graph spectra across clients and demonstrate the heterogeneity in Fig. 1. It reveals in- consistent eigenvalue distribution across clients. As a result, GNNs learn distinct signal propagation schemes of sub-
In conclusion, our key contributions are:
First, we identify and empirically reveal the issue of
2

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S2FGL: Spatial Spectral Federated Graph Learning
poor semantic knowledge under label signal disruption. In addition, we reveal the spectral client drift under spectral heterogeneity in subgraph-FL.
perspective on local data for the not-true classes, FEDGEN (Zhu et al., 2021) ensembles user information in data-free manner to regulate local training, and SCAFFOLD (Karim- ireddy et al., 2020) uses variance reduction for the client drift phenomenon. Instead, strategies of personalized FL (pFL) aim to customize models that perform optimally for each client (Wu et al., 2023; Zhou & Konukoglu, 2023; Li et al., 2021: Zhang al., 2023b). Specifically. FedALA (Zhang et al., 2023c) proposed adaptive masks to achieve personalized aggregation, DBE (Zhang et al., 2023a) stores domain biases for elimination, and FedRoD (Chen & Chao, 2022) leverages two heads for global and personalized tasks.
We design our framework SaFGL including strat- egy Node Label Information Reinforcement and Frequency-aware Graph Modeling Alignment, effec- tively addressing the challenges of poor semantic knowledge and spectral client drift in subgraph-FI.
We conduct extensive experiments on various datasets, validating the superiority of our proposed 52FGL.
Graph Spectrum Being related closely to graph connec- tivity, signal propagation, and structure, graph spectra have proven essential in performing various tasks on graph- structured data. For instance, it plays an essential role in anatomy detection, (Gao et al., 2023; 'ang et al., 2022), graph condensation (Kreuzer et al., 2021; Liu et al., 2023), and graph contrastive learning (Bo et al., 2023a; Liu et al., 2022). Additionally, spectral GNNs (Wu et al., 2020) based on spectral filters are showing powerful ability in modeling graph data and attracting more attention. Specifically, exist- ing research either (He et al., 2021b; Defferrard et al., 2016; He et al., 2022; Wang & Zhang, 2023) leverages various orthogonal polynomials to approximate arbitrary filters, or utilizes neural networks to parameterize the filters (Liao et al., 2019; Bo et al., 2023b). Although the potential of graph spectrum has been explored in various scenarios and tasks, the spectral domain in generalizable subgraph- has remained unexplored. Consequently, current methods suf- fer from optimization diverging on spectra and are trapped in suboptimal learning. Instead, our approach remarkably mitigates the challenge by targeted alignment on spectra.
2. Related Work
Federated Graph Learning. Federated graph learning leverages the powerful graph modeling capabilities of GNNs along with the privacy-preserving attributes of federated learning, thus gaining increasing attention these days (He et al., 2021a; Fu et al., 2022; Liu & Yu, 2022; Wan al., 2025b). Current FGL research can generally be catego- rized into two types: intra-graphe grara FGL and inter-graph FGL. Intra-graph FGL research primarily focuses on subgraph-FL scenarios, where each client participates in the collabora- tion with a part of the whole graph (Zhang et al., 2021b). Correspondingly, the training targets include missing link prediction (Chen et al., 2021; Baek al., 2023), node clas- sification (Huang et al., 2023a; Li et al., 2024; Wan et al., 2024a; Zhu et al., 2024), and so on. On the other hand, clients in inter-graph FGL own independent local graph data, such as multiple graphs from different domains (Tan et al., 2023; Xie et al., 2021). In this paper, we focus on subgraph-FL scenarios of intra-graph FGL. Specifically, we are the first to empirically reveal and address the challenge of poor semantic knowledge under label signal disruption and client drift under spectral heterogeneity among sub- graphs, while existing methods inevitably fail spatially and spectrally due to the lack of targeted solutions.
Graph Signal Propagation: Graph signal propa- gation describes how node signals diffuse on graph structures. In the spatial domain, propagation occurs through explicit signal passing along edges. In the spectral domain, propagation is characterized by how signals distribute across different frequency compo- nents. Label Signal Disruption: As subgraphs experi- ence edge loss, nodes lose critical label signals con- taining class knowledge from their formerly adjacent labeled neighbors. Consequently, it limits the ability of GNNs to capture class distinctions accurately, lead- ing to poor semantic knowledge under label signal disruption. Spectral Client Drift: Inconsistencies in signal frequencies on graph spectra across subgraphs lead to spectral heterogeneity and diverging signal propagation schemes, causing spectral client drift and degrading the generalizability of the global model.
Federated Learning. Federated learning (Huang et al., 2023c; 2024; Yang et al., 2023; Wan et al., 2024a) has gained increasing attention in recent years it addresses the issue of data silos while ensuring data privacy. Several research directions have emerged from FL, including robust- ness (Xu et al., 2022; Hong et al., 2023; Zhu et al., 2023; Fang & Ye, 2022), fairness (Chen et al., 2024; Ezzeldin et al., 2023; Ray Chaudhury et al., 2022), and asynchronous federated learning (Xu et al., 2023; Zhang et al., 2023d). Generally, FL can be categorized into two main types by their optimization objective: traditional FL (tFL) and per- sonalized FL (Hu et al., 2024; Shang et al., 2022; Lv et al., 2024; Smith et al., 2017). Research of tFL aims at aggregat- ing a highly generalizable global model (McMahan et al., 2017; Li et al., 2020; Acar et al., 2021; Zhang et al., 2022a). For instance, FedNTD (Lee et al., 2022) preserves the global
3

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S2FGL: Spatial Spectral Federated Graph Learning
Feature h
Similarity
Low
High
Global Model
Global Repository
Local Model
Representative Node
For Node
h
Similarity Distribution
1 Eq. (11)
p( H)
Z
glow /high m
Client Objective:
FKD
H
CFGMA
LCE + LI + A2CMAMA
( H)
loo /hig
Similarity
Distribution
Eq. (11)
A SALC
Server
0
1 High
2
Feature h²
Similarity
Low
(a) Node Label Information Reinforcement
(b) Frequency-aware Graph Modeling Alignment
Figure 3. Framework Illustration. (a) Node Label Information Reinforcement (NLIR) leverages structurally and semantically representative global prototype repository. It provides multidimennsonal semantic localization of nodes through similarity distribution and allows LeRd inject the semantic knowledge during training. high and low spectral adjacency awareness with the global GNN for a generic signal propagation scheme, mitigating spectral drifts.
(b) Frequency-awarg Graph Modeling Alignment (FGMA) aligns local
3. Problem Statement
Definition 3.2. Structure Inertia Score (SIS): The SIS quantifies the cumulative influence of the training nodes on the entire graph and is defined as:
Notation. Let the graph data be represented as G (V, where V is the set of nodes with |V| N vertices, and E V x V denotes the set of edges connecting these nodes. The adjacency matrix is represented by A N where A.r = 1 indicates the presence of an edge eve E E, and Aue otherwise. Moreover, the feature vector matrix the graph G. The Laplacian matrix given by == D A. where D is the degree matrix. The unitary matrix U is composed of the eigenvectors of L. To distin- guish between local and global properties, we introduce the following notation: the symbol represents local properties or entities, whereas g denotes global properties or entities.
STS(P,t) = max (Pig -t))
(2)
Here, i is the PPR matrix, and {0,1) nm is a binary vector indicating the training nodes, where 1, node part of the training set, and tj 0 otherwise. The SIS aggregates the maximum personalized PageRank values from each node to any labeled node, effectively measuring the strongest influence each node in the graph receives from the training set. A higher SIS indicates greater structural in- ertia, suggesting that the labels have a significant inftuence over the network overall graph structure.
Definition 3.1. Personalized PageRank (PPR): The PPR matrix quantifies the influence each node has on every other node within the graph and is defined as:
4. Methodology
P= (I-
(1)
4.1. Motivation
Here, 10,1) is the teleportation probability, represent- ing the probubility of the random walk restarting from the source node. A typical value is 0.15, which makes the con- tinuation probability (1 (1-a)= = 0.85. I is the identity matrix, A is the adjacency matrix of the graph, and D is the degree matrix with Das denoting the degree of node i.
Signal propagation over graph structures fundamentally shapes the signal transmission paradigm of GNNs. There- fore, rather than focusing solely on challenges arising from static graph structures in subgraph-FL. is crucial to con- sider the dynamies of signal propagation. Correspondingly,

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S2FGL: Spatial Spectral Federated Graph Learning
we conduct our analysis from both the spatial and spectral domains from the perspective of the graph signal. Specif- ically, the spatial domain governs the explicit passing of signals between connected nodes, while the spectral do- main captures signal diffusion across different frequencies. Accordingly, we empirically validate the presence of two major challenges from the spatial and spectral perspectives: label signal disruption and spectral client drift. These phenomena respectively pose challenges of poor seman- tic knowledge and spectral client drift, which severely constrain the potential of subgraph-FI. collaboration. Foblrraph--F.collaaorrtooo.
client drift. Consequently, our strategy effectively enhances the global generalizability spectrally.
4.2. Node Label Information Reinforcement
First of all, we introduce the Structure-Aware Label Cen- trality (SALC) metric, denoted as ASACC for node u. It is defined the combination of the label influence centrality Ai and the structural prominence score A
SALC = Λ + Ale
(3)
Motivation of NLIR. Graph data is fragmented across clients in subgraph-FL, which inevitably disrupts seman- tic signals from labeled nodes across clients. Label signal disruption undermines key pathways for propagating se- mantic information. This results in biased local feature representations, which ultimately degrade the performance of GNNs. For validation, we investigate the relationship be- tween the decrease in SIS and the client scale. Specifically, the SIS exhibits a downward trend as the client scale in- creases, with notably lower scores in the range emphasized by mainstream subgraph-FL studies, thereby highlighting the label signal disruption phenomenon. We aim to miti- gate its negative impact by preserving valuable semantic knowledge across fragmented subgraphs. Correspondingly, we propose NLIR. By selecting nodes with both structural representativennss and rich semantic information for the construction of a global repository and injecting it during local training, NLIR reduces the information loss inherent in the subgraph scenarios. Subsequently, NLIR assesses the similarity distributions between node features and all representative prototypes for both local and global GNNs during local training, thereby enabling multidimensional se- mantic localization of nodes. By aligning the two similarity distributions, it effectively injects semantic knowledge and enhances feature modeling semantically.
where A, assesses the structural representativeness of node u, while AL, quantifies the influence propagation of labels.
A = mmax (Pu..
A =
Σ DEVL
P(L)
(4)
0,80
where Pa.e is the (u, v)-th element of the standard PPR matrix P. and Tn represents the prior importance score of node u, typically initialized to I for all nodes. The structural prominence score captures the maximum influence exerted by any node on node u, weighted by its importance. As clarified, the PPR matrix used for computing label influence centrality is denoted as P(2), and defined as:
((L) (I--(((-))D-
(5)
Here, V2 denotes the set of labeled nodes, and Fiu rep- resents the influence of node v on node u as captured by P(L), To accurately capture the influence of labeled nodes, the inclusion of self-loops in A' ensures that each labeled node's own label contributes its Ay.- Compared to the intu- itive approach of directly selecting labeled nodes, the SALC metric ASALC considers both structural representativenes: of the nodes and diffusion of label signals, thus avoiding biases caused by isolated labeled nodes. It is also capable of selecting unlabeled nodes that still possess rich label signals and structural advantages. This improves knowledge qual- ity, enriches the repository, and mitigates the label signal disruption problem. After computing the SALC scores, we rank the nodes based on their ASALC values and select the top K nodes, where the default value of K is 1/3 of the total number of nodes. Subsequently. for each class the local prototype H at each client is computed as the mean feature vector of the selected nodes belonging to class c:
P(L)
Motivation of FGMA. We reveal the challenge of spectral client drift in subgraph-FL in Fig. 1, where GNNs across different clients capture frequency information inconsis- tently due to graph spectral heterogeneity. This further leads to overfitting local signal propagation frequencies, which brings spectral conflicts during collaboration and compro- mises the generalizability of the global GNN. To address spectral client drift, we propose the Frequency-aware Graph Modeling Alignment. Specifically, FGMA reconstructs the local graph spectra with the GNN adjacency awareness by calculating the node similarity matrix. Subsequently, by projecting feature representations separately onto high- and Now-frequency components of the reconstructed spectra and aligning the local and global projections, FGMA promotes the learning of a generalizable spectral signal propagation paradigm across clients, thereby reducing frequency-base based discrepancies during collaboration and mitigating spectral
H =
Σ h DEV
(6)
DH
where Vi represents the set of selected nodes categorized as class on client i, and h2, is the feature vector of node u on client i. Once the local prototypes are computed, clients upload their prototypes to the server along with the node count. For each class e, the server aggregates the
5

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S2FGL: Spatial Spectral Federated Graph Learning
prototypes from a percent of the clients by weighting each local prototype according its sample size. Four global anchor prototypes are constructed for each class. Each global prototype H is computed as:
For L' let (um (owaa " J==1 be the eigenvectors corresponding to the smallest eigenvalues, which represents low-frquueccy, While {um I represents figh-frequency. Similarly, for I we obtain Rei ( m==1 and {um
Kug
(ight kuy
m==1 denotes the largest eigenvalues. which
The feature matrix h is then projected onto each of these eigenvectors. For instance:
high.9 keis
Im=11 "mm
H
H
H
EvENT DV
Σ EEA?
H
(7)
Z Low = (u,, u
lowT )h, Zhish = (um u
high highT") hh.
(11)
Hgg
Applying these projections for each m E .......,ue (lie...,ksi)) we obtain several sets of projected feature matrices used in the loss computation. Specifically, local features h are pro- jected onto low/high-frequene cigenvectors of the local graph, yielding Ziloo and Z,m respectively. Similarly, frozen global inference features h are projected onto corre- sponding eigenvectors from L yielding /g. and Zighigh high Conequently, loss CFGMA is then defined as the sum of MSE over all eigenvector eigenvector-projected pairs:
where Hg. represents the k-th global prototype for class c, No is the set of clients randomly selected to contribute to the k-th global prototype for class c, and C' is the number of classes. The global repository H. contains all the global prototypes and will be broadcast back to clients. After the global knowledge repository is constructed, it is distributed to the local clients along with the model parameters. The global features hg used in the following loss formulation refer to the frozen inference features extracted locally us- ing the distributed global model. To regulate local training, we propose federated knowledge distillation loss func- tion aimed at harmonizing the semantic feature localization of the local GNN with its global counterpart, namely by aligning their similarity distributions for the representative prototypes stored in the global repository:
Avir
LEGMA
=
(.wow low (MSE(Z m
Zihhoo + MSR (Z
yg.low
Z high high m
(gghighh)
m=1
(12)
This loss addresses spectral heterogeneity by aligning client and global signal characteristies in both spectral domains. Combining strategies Node Label Information Reinforce- mentand Frequency-awart Graph Modeling Alignment, our framework FLL reinforces semantic knowledge during local modeling and mitigates spectral client drift. Ultimately, the loss for local training is:
LTKD=
Σ KI. (vp((, H)), a VEV.
MV
where KL(--)) represents the Kullback-Leibler divergence, and (-) is the softmax function applied to similarity scores computed by the function phh, H),, which returns vector of cosine similarities between the feature and all prototypes.
L= = + A + X2CrGMA,
(13)
where LEE denotes the standard cross-entropy loss for node classification, while \₁ and X2 are balancing hyperparame- ters for the proposed methods NLIR and FGMA.
4.3. Frequency-aware Graph Modeling Alignment
To emphasize the GNN spectral adjacency awareness and more accurately capture similarities between nodes, we leverage the feature matrix h to compute node similarity ma- trices. In the construction of the following similarity matrix, denotes operations applied both h and h², Specifically, for each node u, we identify its kai most similar neighbors based the cosine similarity of their feature vectors h.. We then construct a sparse self-similari matrix S' as:
5. Experiments
5.1. Experimental Setup
Datasets. We conducted experiments on various datasets to validate the superiority of our framework F2 The ho- mophilic graph datasets include Cora, Citeseer, and Pubmed, while the heterophilic graph datasets comprise Texas, Wis- consin, and Minesweeper. The following provides descrip- tion of each dataset. Cora (McCallum et al., 2000) dataset consists of 2708 scientific publications classified into one of seven classes. There are 5429 edges in the network of cita- tions. 1433 distinct words make up the dictionary. Citeseer (Giles et al., 1998) dataset consists of 3312 scientific publi- cations classified into one of six classes and 4732 edges. The dictionary contains 3703 unique words. Pubmed (Sen et al., 2008) dataset consists of 19717 scientific papers on diabetes that have been categorized into one of three categories in the PubMed database. The citation network has 44338 edges.
h h.
among the top Ksim
S' u,v
otherwise
Subsequently, we calculate the graph laplacian matrix L' based on this sparse similarity matrix S' as:
I' D'=D'-S"
(10)
where D' is the diagonal degree matrix of S' Moreover, we perform eigendecomposition on the laplacians Il* and I
6

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S2FGL: Spatial Spectral Federated Graph Learning
Table Performance Comparison with the sate-of-ine th-l--of-hh-arr the-art art methods on homophilic and heterophilic graph datasets. report the node classification accuracies with the performance improvement over FedAvg. The best results are highlighted in bold.
Methods
Cora
CiteSeer
PubMed
Texas
Wisconsin
Minesweeper
FedAvg [ASTAT17]
81.9 0.7
74.3 + 0.4
87.3 + 0.3
72.8 + 2.2
77.6 2.7
79.6 + 0.1
FedProx [arXiv18]
82.1 + 0.5 10.2
74.4 = 0.3 1
87.9 + 0.4 10.6
73.5 + 3.7 10.7
77.3 + 3.4 10.3
79.7 + 0.1 fo.1
FodNova [NcurIPS20]
81.6 + 1.2 L0.3
74.4 + 0.4 10.1
88.2 - 0.5 10.9
73.0 4.4 10.2
77.4 4.2 Jo.2
79.9 + 0.4 ro.3
FedFa [TCLR23]
82.7 + 0.5
10.8
74.9 + 0.6 10.6
87.8 + 0.5 +0.5
73.9 3.6 11.1
78. + 4.6 10.5
80.1 + 0.3 to.5
FedSage+ [NeurIPS 191
82.3 + 0.7
+0.4
75.2 + 0.3 10.9
88.2 + 0.7 10.9
73.7 = 4.0 10.9
79.0 + 3.3 f1.4
79.9 + 0.2 fo.3
FedStar [AAA123]
82.6 1 0.5
70.7
74.5 0.3 70.2
88.1 0.6 p0.8
74.3 2.7 +1.5
78.3 4.7 10.7
79.8 1 0.1 +u.2
FedPub [ICML23]
82.3 4 0.8
10.4
74.8 + 0.7 10.5
88.0 + 0.4 +0.7
73.4 + 3.5 1o.6
77.8 + 3.1 10.2
79.9 + 0.2 +0.3
FGSSL. [[[CA123]
82.6 + 0.4
10.7
74.9 0.2 10.6
87.6 0.7
10.3
73.6 4.6 10.8
77.8 3.8 10.2
79.9 + 0.2 t0.3
FedGTA [VLDB24]
82.4 4 0.8
10.5
75.1 + 0.5 10.8
87.7 + 0.9 po.4
72.6 + 4.2 40.2
77.8 + 4.1 10.2
80.2 + 0.3 f0.6
FGGP [AAA124]
82.5 1 0.4
10.6
74.7 0.5 10.4
87.5 0.4 70.2
73.6 2.8 10.8
78.2 3.4 no.6
80.4 1: 0.3 +0.8
S²FGL (ours)
83.4 1 0.2 11.5
76.0 + 0.3 11.7 88.6 0.2
11.3
74.8 + 2.3 12.0
79.0 + 1.0 11.4
80.5 + 0.1 p0.9
Texas and Wisconsin datasets are subsets of the WebKB dataset (Craven et al., 1998). The WebKB dataset was in- troduced 1998, comprising web pages from the computer science departments of various universities, including the University of Texas and the University of Wisconsin. The dataset is commonly used for tasks such as webpage clas- sification and link prediction, serving as a benchmark for evaluating machine learning models in graph-based learning scenarios. Minesweeper (Baranovskiy et al., 2023) dataset synthetic graph dataset inspired by the Minesweeper game. In this dataset, the graph is structured as a regular 100x 100 grid, where each node represents a cell connected to its neighboring nodes, except for edge nodes, which have fewer neighbors. The primary task is to predict which nodes contain mines. This dataset is commonly used to evaluate the performance of GNNs under heterophily.
algorithm to partition the graph into subgraphs assigned to different clients. For each dataset, we divide the nodes into training, validation, and testing sets with ratios of 60%, 20%, and 20%, respectively. Additionally, we simulate var- ious collaborative scenarios by configuring the number of clients to 10 for Cora, Citeseer, Pubmed, and Minesweeper datasets, and 3 for the Texas and Wisconsin datasets. The primary evaluation metric is the node classification accu- racy on the clients' test sets. We conduct each experiment five times and report the average accuracy from the last five communication epochs the final performance. We con- duct experiments with the ACM-GCN (Luan al., 2022), which achieves strong ability on both homophilic and heterophilic graph datasets.
5.2. Experiment Results
Evaluation Metric. Following mainstream FGL research experimantal practices, we utilize the accuracy of the node classification task the evaluation metric.
In this section, we comprehensively evaluate the proposed FGL by addressing the following questions:
Baselines. We compare FFLGL with several state-of-the- art approaches, including traditional federated learning methods such as FedAvg (McMahan et al., 2017), Fed- Prox (Li et al., 2020), FedNova (Wang et al., 2020), and FedFa (Zhou& & Konukoglu, 2023); federated graph learn- ing approaches including FGSSL (Huang et al., 2023a) and FGGP (Wan et al., 2024a); well as personalized fed- erated graph learning methods such as FedSage+ (Zhang et al., 2021b), FedStar (Tan et al., 2023), FedPub (Back et al., 2023), and FedGTA (Li et al., 2024). This compre- hensive set of baseline methods spans various FL and FGL paradigms, allowing us to evaluate the performance and ad- vantages of our proposed 52FGL across diverse scenarios.
Q1: How does S'FGL perform compared exist-
ing FL and FGL methods in subgraph-FL?
Q2: What is the impact of each component of FGL on the overall performance?
Q3: Does F²L maintain consistent performance across different hyperparameter and client scales?
Q4: Do NLIR and FGMA mitigate the effects of label signal disruption and spectral heterogeneity?
Q1: How does FGL perform compared to existing FL and FGL methods in subgraph-FL.
Implement Details. Following prevalent methodologies in FGL research, we employ the Louvain community detection
We present the results of node classification tasks across various FGL scenarios using multiple graph datasets, and
7

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S2FGL: Spatial Spectral Federated Graph Learning
summarize the final average test accuracy in Tab. 1. It demonstrates that our proposed FCL consistently outper- forms all baseline approaches across all six datasets. This superiority highlights the effectiveness of F²GL
80
Accurecy
00
75-
Q2: What is the impact of each component of FPG on the overall performance?
10
20
60
Number of Clients
Number of Clients
(a) Cora
(b) Citeseer
To evaluate the individual contributions of NLIR and FGMA strategies within the FGL framework, conducted ab- lation experiments on the Cora and Citeseer datasets, In this study, we removed each component to assess its im- pact on the overall performance. The results are presented in Tab. 2, which demonstrate that both NLIR and FGMA independently contribute to the overall performance.
Figure 5. Analysis of performance under different client scales.
and adaptability across varying hyperparameter settings and client partition configurations on both the Cora and Cite- seer datasets. These findings confirm that FGL not only sustains effectiveness under diverse conditions but also adapts seamlessly varying client scales, demonstrating its suitability for real- world subgraph-FL. scenarios.
Dataset
NLIR
FGMA
Cora
Citeseer
X
X
81.9 + 0.7
74.3 + 0.4
Q4: Do NLIR and FGMA mitigate the effects of label signal disruption and spectral heterogeneity?
/
83.2 + 0.4
75.6 + 0.3
x
V
82.6 + 0.3
75.0 + 0.2
In Fig. I, our experiments demonstrate that the SIS score in subgraph scenarios declines compared to centralized train- ing. while spectral heterogeneity exists among clients. Here, we further verify the targeted effectiveness of NILR and FGMA with respect to these two issues, respectively. First, in Fig. 6 (a), we investigate the relationship between the per- formance gain brought by NILR to FedAvg and the change in the SIS score. The results show that this method achieves higher performance when semantic signals are more limited, thereby confirming its effectiveness and targeted nature. Sec- ond, in Fig. (b), we examine the relationship between the performance improvement of FGMA for FedAvg and spec- tral heterogeneity, measured by the average KL divergence between the eigenvalue distributions of different clients. Ex- perimental results show that greater spectral heterogeneity corresponds to larger performance gains, confirming the effectiveness of our method. Specifically, green indicates the performance gain of the proposed method relative to FedAvg, blue denotes variations in SIS, and orange corre- sponds to spectral heterogeneity.
83.4 = 0.2
76.0 0.3
Table 2. Ablation study on key components of FGG
Q3: Does F'GLL maintain consistent performance across different hyperparameter and client scales?
We evaluated the stability and adaptability of our proposed framework FLL under varying hyperparameter configu- rations of and different client scales.
Varying Hyperparameters of NLIR and FGMA. For NLIR, we test hyperparameter sets of 100, 50, 10, and For FGMA, the settings are 0.01, 0.05, 0.5, and The re- sults in Fig. 4 indicate that our method maintains consistent performance across varying hyperparameter configurations.
Accuracy Groweh
Accuracy Growth
10.020
0.000
0.000
100
0.01
100
0.05
0.01
20
10
0.5
NLR
FGMA
Growti be
(a) Cora
(b) Citeseer
19099
Figure 4. Analysis of the performance growth between FCG and FedAvg under different hyperparameters of NLIR and FGMA
Clicut Seate"
Clint Client Scale
Varying Client Scales. We assessed performance with dif- ferent client scales: 5, 10 and 20. Specifically, we compared SºFGL with other FL and FGL basclines, including FedAvg, FedProx, and FGSSL. The results in Fig. demonstrate that S consistently delivers reliable results regardless of the client scales. Overall, F2G exhibits strong stability
(a) NILR
(b) FGMA
Figure 6. Analysis of the targeted effecctivnness of NILR and FGMA. (a) The performance gain from NILR inereases the semantic signal becomes limited. (b) The performance improve- ment from FGMA grows with higher spectral heterogeneity.
8

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S2FGL: Spatial Spectral Federated Graph Learning
6. Conclusion
ing with stable and scalable spectral encoding. NeurIPS, 36:45516-45533 2023a.
In this paper, we identify and empirically demonstrate two phenomena in subgraph-FI FL from both spatial and spectral perspectives of graph signal propagation: label signal disrup- tion and spectral heterogeneityy These phenomena pose chal- lenges of poor semantic knowledge and spectral client drift. To address these challenges, we propose two key strategies: NLIR and FGMA. NLIR selects structurally and semanti- cally representative nodes and constructs a global repository accordingly. By injecting semantic information from the repository into local training, it alleviates the poor semantic knowledge caused by label signal disruption. In addition, FGMA aligns and feature projections in both the high- and low-frequency reconstructed graph spectra, thereby promot- ing a generic signal propagation paradigm and mitigating client drifts under spectral heterogeneity. By integrating these strategies, FGI. effectively tackles both spatial and spectral challenges in subgraph-FL. Extensive experiments on multiple datasets demonstrate that FLFGL significantly enhances global generalizability.
Bo, D., Shi, C., Wang, L., and Liao, R. Specformer: Spectral graph neural networks meet transformers. arXiv preprint arXiv:2303.0100 2023b.
Chen, H.-Y. and Chao, W.-L. On bridging generic and personalized federated learning for image classification. In ICLR, 2022.
Chen, M., Zhang, W., Yuan, Z., Jia, Y., and Chen, H. Fede: Embedding knowledge graphs in federated setting. In IJCKG, pp. 80-88, 2021.
Chen, Y., Huang, W., and Ye, M. Fair federated learning under domain skew with local consistency and domain diversity. In CVPR, pp. 12077-12086, 2024.
Craven, M., DiPasquo, D., Freitag, D., McCallum, A.. Mitchell, T., Nigam, K., and Slattery, S. Learning to extract symbolic knowledge from the world wide web.
AAAI/AAI,, pp. 2, 1998.
Acknowledgement
Defferrard, M., Bresson, X., and Vandergheynst, Con- volutional neural networks on graphs with fast localized spectral filtering. In NeurlPS, 2016.
This work is supported by the National Key Research and Development Program of China (2024YFC3308400) and National Natural Science Foundation of China under Grant (62361 166629, 62176188, 62225113, 13, 623B2080), the Wuhan University Undergraduate Innovation Research Fund Project. The supercomputing system at the Supercomput- ing Center of Wuhan University supported the numerical calculations in this paper.
Ezzeldin, Y. H.,, Yan, S., He, C., Ferrara, E., and Avestimehr, A. S. Fairfed: Enabling group fairness in federated learn- ing. In AAAI, 2023.
Fan, W., Ma, Y., Li, Q. Wang, J., Cai, G., Tang, J., and Yin. D. A graph neural network framework for social recommendations. TKDE, 2020.
Impact Statement
Fang, X. and Ye, M. Robust federated learning with noisy and heterogeneous clients. In CVPR, pp. 10072-10081, 2022.
This paper presents work whose goal is advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
Fang, X., Easwaran, A., Genest, B., and Suganthan, P. N. Adaptive hierarchical graph cut for multi-granularity out- of-distribution detection. JEEE TAI, 2025.
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12


Reviews:
[{"review_id": "DRlQqZjOWm", "reviewer": "Reviewer_svqb", "summary": "The paper presents a novel framework called S2FGL (Spatial Spectral Federated Graph Learning) to address two key challenges in subgraph federated learning (FGL): Label Signal Disruption (LSD) and spectral client drifts. LSD occurs when subgraphs lose critical label signals due to edge losses between clients, which hampers the ability of Graph Neural Networks (GNNs) to learn class knowledge. Spectral client drifts arise from inconsistencies in signal frequencies across subgraphs, leading to degraded global generalizability. To solve these issues, the authors propose two strategies: Node Label Information Reinforcement (NLIR), which creates a global repository of class knowledge to restore label signals, and Frequency-aware Graph Modeling Alignment (FGMA), which aligns high- and low-frequency spectral components across clients to mitigate spectral drifts. Extensive experiments on various datasets demonstrate the effectiveness of S2FGL, outperforming existing methods in terms of global generalizability.", "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": "Dear Reviewer svqb: \n\nWe sincerely appreciate your time and effort and hope that our responses will address your concerns and lead to an updated score.\n\n## Experimental Designs Or Analyses\n**There are only experiments with ACM-GCN.**\n\nA1: $S^2FGL$ is a backbone-agnostic framework designed to address label signal disruption and spectral client drifts. In light of your thoughtful comment, we have incorporated results using GCN in Table 1, showing that $S^2FGL$ consistently outperforms current methods. The results will be included in the final version.\n\n*Table 1: Performance comparison using **GCN.***\n|Method|Cora|Citeseer|PubMed|Texas|Wisconsin|Minesweeper|\n|-|-|-|-|-|-|-|\n|FedAvg|80.2|69.5|84.9|63.4|62.1|77.9|\n|FedProx|80.3|69.2|85.2|64.0|62.2|78.0|\n|FedFA|80.8|70.1|85.4|64.7|63.4|78.3|\n|FedSSL|82.6|70.9|85.8|64.2|64.5|78.6|\n|FGGP|82.8|70.6|85.5|65.5|63.6|78.7|\n|$\\bf{S^2FGL}$|**83.7**|**72.2**|**86.8**|**66.8**|**65.5**|**79.1**|\n\n## Weakness\n**W1: Why NLIR is novel.**\n\nA2: For the target problem, we innovatively focus on label signal disruption (LSD), which brings significant semantic degradation. Though existing methods mitigate edge loss structurally, they ignore the semantic degradation under LSD and fail to offer targeted solutions. Therefore, current FGL methods inevitably suffer from limited GNN semantic recognition ability.\n\nMethodologically, existing prototype-based approaches exhibit two critical limitations under LSD: insufficient semantic richness and semantic deviation under structural biases. Correspondingly, we propose a novel metric $\\Lambda^\\text{SALC}$ (Eq. 3 and 4, Page 4), which jointly accounts for label influence and structural representativeness to effectively mitigate LSD. \n\n**W2: The similarity matrix requires second-order complexity. The projection requires doing EVD on a dense Laplacian.**\n\nA3: First, the computation of the feature similarity matrix is widely accepted in FGL. Specifically, FGSSL leverages it to align structural consensus, FGGP utilizes it for contrastive learning, and FedGL employs it for structure updates. Distinctively, we align the reconstructed spectrum to enable a highly expressive signal propagation scheme against spectral drifts, achieving the best performance.\n\nSecond, our method **does not** require an EVD on a dense Laplacian. Since only partial eigenvectors are needed, we leverage SciPy's sparse solver eigsh with $O(r\\cdot nnz\\cdot I)$. $r$ denotes the number of eigenvectors needed, $nnz$ is the non-zero element, and $I$ is required iterations.\n\n**W3: The authors need to verify the existence of the spectral challenge and prove the method can mitigate it.**\n\nA4: We kindly note that we have demonstrated the spectral challenge in Fig. 1(b). For further validation, we compute the Pearson correlation between spectral shift (KL divergence of a client's eigenvalue distribution from the global) and local accuracy of the global GNN under FedAvg. After runs under five random seeds, they are -0.39 (Cora), -0.27 (Citeseer), and -0.34 (Pubmed), indicating that greater spectral shift leads to higher inconsistency with global optimization and confirming spectral drifts.\n\nFGMA facilitates a generic signal propagation paradigm across clients to mitigate spectral drifts. For validation, Table 2 shows that FGMA is **more** effective under **higher** spectral heterogeneity, providing strong evidence for its specificity and effectiveness. Spectral heterogeneity here is the average KL divergence among clients' eigenvalue distributions.\n\n*Table 2: Correlation between **Spectal Heterogeneity** and **FGMA Improvement** on FedAvg.*\n|Client Scale|11|13|15|17|19|21|23|\n|-|-|-|-|-|-|-|-|\n|Spectral Heterogeneity|0.63|0.88|0.85|2.09|2.23|3.30|3.75|\n|FGMA Improvement (%)|0.73|0.97|0.92|1.27|1.29|1.51|1.57|\n\n**W4: Lack of complexity analysis.**\n\nA5: For a $k$-layer GNN with batch size $b$ and feature dimension $f$, the propagated features $X^{(k)}$ have a space complexity of $O((b+k)f)$, while linear regression has $O(f²)$. Key parameters include $n$, $m$, $c$ (nodes, edges, classes), $s$ (augmented nodes), $g$ (complemented neighbors), $p$ (trainable prototype matrix dimension), $Q$ (query set size for contrastive learning), and $N$ (selected clients per round). Analysis in Table 3 shows that $S^2FGL$ achieves top performance with reasonable overhead.\n\n*Table 3: Complexity analysis. Best in bold and second with underline.*\n|Method|Client Mem|Server Mem|Client Time|Server Time|\n|-|-|-|-|-|\n|FedStar|$\\bf{O(2((b+k)f+f^2))}$|$\\underline{O(N f^2)}$|$\\bf{O(2(kmf+nf^2))}$|$\\bf{O(Nf)}$|\n|FGSSL|$O(Q(b+k)f+f^2+n^2)$|$\\underline{O(Nf^2)}$|$O(Qkmf+Qnf^2+n^2f)$|$\\bf{O(Nf)}$|\n|FGGP|$O((n+sg)f+f^2+Qcp+n^2)$|$\\bf{O(Ncp)}$|$O((m+sg)f+(n+sg)f^2+Qcp^2)$|$\\underline{O(N^2(\\log(N)+c^2p^2)+Ncp)}$|\n|$S^2FGL$|$\\underline{O((b+k)f+f^2+n^2)}$|$\\underline{O(Nf^2)}$|$\\underline{O(kmf+nf^2+n^2f)}$|$\\bf{O(Nf)}$|\n\nThank you for your valuable feedback and hope that our rebuttal adequately addresses your concerns!", "claims_and_evidence": "The label signal disruption in subgraph-FL has already been recogonized and can not be taken as one contribution of this work.", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "There are no theoretical proofs.", "experimental_designs_or_analyses": "There are only experiments with one strong GNN backbone, ACM-GCN. But many graph federated learning works use other more naive but popular GNNs, e.g., GCN and GPRGNN. The authors should include the experiments with other GNNs used in prior works.", "relation_to_broader_scientific_literature": "This article addresses a common intra-graph federated learning task, identifying two challenges it faces: 1) Label Signal Disruption caused by the absence of cross-client links, and 2) Spectral Client Drift, a distribution shift resulting from inconsistent spectral domain distributions of graph signals across different clients. The first challenge is widely recognized, while the second has received little attention. Overally, the proposed method itself is a combination of existing techniques.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "**Strengths:**\n\n- This paper is well-organized and easy to follow.\n    \n- Although the spectral client drift problem is quite intuitive to conceive, this paper appears to be the first to explicitly pinpoint this issue.\n    \n\n**Weakness:**\n\n- The problem NLIR aims to address and NLIR itself are not new things in federated learning since there are many works discussing the loss of egdes and prototype-based federeated learning. The authors should emphasize the innovative aspects of their method in addressing Challenge , compared with prior works.\n    \n- The cosine similarity matrix requires second-order complexity. And the projection step on p. 6 requires doing EVD on a dense Laplacian.\n    \n- The Spectral Client Drift challenge is novel. But the authors do not experimentally verify the existence of the second challenge or whether their proposed method can mitigate the spectral domain distribution shift of graph signals. It would be beneficial for the authors to provide experimental validation for Challenge 2, rather than relying on mere conjecture.\n    \n- Complexity analysis is missing.", "other_comments_or_suggestions": "see Above", "questions_for_authors": "see Above", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "TCSo1ggGRg", "reviewer": "Reviewer_yM48", "summary": "This paper investigates graph signal propagation in federated graph learning through both the spatial and spectral domains, highlighting the issues of label signal disruption and spectral client drift. In response, it proposes two methods: Node Label Information Reinforcement and Frequency-aware Graph Modeling Alignment, which address these identified challenges. Comprehensive experiments are performed to demonstrate the effectiveness of these methods.", "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": "Dear Reviewer yM48:\n\nWe sincerely appreciate the time and effort you have invested in reviewing our paper, as well as your favorable assessment of the motivation and design of our method. We hope that our rebuttal has effectively addressed your concerns.\n\n\n## Weakness\n\n**W1: The authors should consider conducting experiments with a larger number of clients to ensure that NLIR remains effective in configurations with more clients**\n\nA1: We kindly emphasize that our experimental setup adheres to standards in existing FGL research, incorporating suitably configured numbers of clients. Nonetheless, to address your concerns, we conducted additional experiments with a substantially larger number of clients on the large-scale graph dataset arxiv-year. The results in Table 1 confirm the robustness and effectiveness of our method in large-scale client settings. \n\n\n*Table 1: NLIR performance under large client scale.*\n\n| Client Scale | 50 | 100 | 150 |\n|:-:|:-:|:-:|:-:|\n| FedAvg  | 32.1  | 34.8 | 35.5 |\n| NLIR  | 32.9  | 35.7 | 36.2 |\n \n\n**W2: The ablation study should incorporate error bars.**\n\nA2: In response to your concern, we incorporate error bars in Table 2 as follows. We will incorporate your suggestions in the final version.\n\n\n*Table 2: Ablation study of $S^2FGL$ with error bars*\n\n| NLIR | FGMA | Cora | Citeseer |\n|:-:|:-:|:-:|:-:|\n| ✗ | ✗ | 81.9 ± 0.7     | 74.3 ± 0.4     |\n| ✓  | ✗  | 83.2 ± 0.4     | 75.6 ± 0.3     |\n| ✗  | ✓ | 82.6 ± 0.3     | 75.0 ± 0.2     |\n| ✓ | ✓  | **83.4 ± 0.5** | **76.0 ± 0.3** |\n\n## Other Comments Or Suggestions\n\nA3: We are grateful for your feedback regarding the typo. We will implement the revision in the final version.\n\n\nWe sincerely appreciate your thoughtful review comments and hope that our rebuttal has addressed the concerns you raised!", "claims_and_evidence": "Claims are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "The proposed methods are well-designed and appropriately address the identified challenges. The NLIR method creates a global category knowledge repository by enabling clients to upload multiple easily-learnable prototypes as reference points. By evaluating the similarity between each node and these reference points, it achieves precise semantic localization of nodes, thereby integrating globally accessible category information into local GNNs. The FGMA method utilizes the similarity matrix obtained from GNN inference to perform spectral reconstruction, aligning local and global features by projecting them onto the low-frequency and high-frequency spectral domains. The reconstruction allows FGMA to synchronize local and global message passing paradigms, effectively alleviating spectral drift.", "theoretical_claims": "The definitions are clear and correct, and there is no theoretical proof provided.", "experimental_designs_or_analyses": "This paper presents extensive experimental evidence, encompassing performance comparisons, ablation studies, and hyperparameter studies, among others. Overall, the experimental framework is well-constructed. Notably, in the Q4 of the experiments, the correlation between the NLIR method and the structure inertia score is examined, directly demonstrating the method's relevance. However, this raises a concern: under larger partition sizes, SIS does not significantly decrease. In such instances, will NLIR lose its original effectiveness? Consequently, the performance of NLIR with a substantial number of clients still needs to be validated, ideally with the SIS reduction level indicated. Furthermore, the ablation study should incorporate error bars to provide a more comprehensive analysis of the results.", "relation_to_broader_scientific_literature": "This paper innovatively identifies the issue of label signal disruption , with a strong motivation that is empirically validated. Moreover, the investigation into spectral drift presents a novel perspective in the context of subgraph-FL, encouraging further exploration of graph spectral domain in FGL.", "essential_references_not_discussed": "All essential references have been included, especially those related to the latest methods in subgraph-FL.", "other_strengths_and_weaknesses": "Strengths:\n1. The motivation is both strong and innovative, with exploration of the issues of label signal disruption and spectral client drift.\n2. The proposed solutions are well-suited and specifically targeted, effectively addressing the identified problems in the spatial and spectral domains.\n3. The experimental design is generally rigorous and effective. Moreover, the experiment in Q4, which examines the relationship between the performance of the NLIR method and SIS, successfully validates the motivation..\n\nWeaknesses:\n1. In the experiment Q4, as the partition size increases, SIS rises, leading to a decline in the performance of the NLIR. The authors should consider conducting experiments with a larger number of clients to ensure that NLIR remains effective in configurations with more clients.\n2. The ablation study should incorporate error bars for enhanced clarity and precision in the results.", "other_comments_or_suggestions": "Under the Notations section, there is a typo in line 122 where a word is repeated.", "questions_for_authors": "Please refer to the weaknesses, as no other questions have been posed.", "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": "EkuNVBKvHB", "reviewer": "Reviewer_yo13", "summary": "The authors propose S2FGL, a framework that simultaneously addresses the spatial and spectral challenges in federated graph learning. Instead of focusing on static graph structures, it provides solutions through the lens of graph signal propagation.", "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": "Dear Reviewer yo13:\n\nThank you for your thoughtful review and for acknowledging the value of our work. We hope the responses provided below will clarify your concerns and contribute to a more favorable evaluation.\n\n## Weakness\n\n**W1: The authors should conduct a hyperparameter study for the NLIR and include the performance comparison with FedTAD.**\n\nA1: Thank you for your thoughtful suggestions. We have added experiments regarding the prototype count hyperparameters in NLIR to Table 1 and included comparison experiments with FedTAD in Table 2. The results in Table 1 demonstrate the stability of the NLIR method under variations in prototype counts. The results in Table 2 show that our method consistently outperforms FedTAD, validating its efficiency. We will include these results in the final version.\n\n*Table 1: Hyper-parameter study of prototype count in NLIR*\n| Prototype Count | Cora | Citeseer | Pubmed |\n|:-:|:-:|:-:|:-:|\n| 4  | 83.4 | 76.0 | 88.6 |\n| 8  | 83.6 | 76.3 | 88.5 |\n| 16 | 83.5 | 76.4 | 89.0 |\n\n*Table 2: Performance comparison with FedTAD*\n|Datasets|Cora|Citeseer|PubMed|Texas|Wisconsin|Minesweeper|\n|-|-|-|-|-|-|-|\n|FedAvg|81.9|74.3|87.3|72.8|77.6|79.6|\n|FedTAD|82.5|75.3|87.7|73.4|77.7|79.9|\n|$\\bf{S^2FGL}$|**83.4**|**76.0**|**88.6**|**74.7**|**78.4**|**80.5**|\n\n\n**W2: The authors should clarify the key advantages of the spectral solutions over spatial solutions in mitigating spectral client drifts.**\n\nA2: Methods that rely on the spatial domain fail to capture signal propagation patterns across different frequencies in the graph spectra. As a result, client drifts caused by GNNs overfitting to local frequency signal propagation paradigms are challenging to resolve for them. Instead, our method performs spectral reconstruction based on the adjacency awareness of different GNNs and promotes the alignment of local spectral signal propagation patterns with the globe. This approach enables the formation of a generalizable and strong spectral signal propagation paradigm, effectively mitigating spectral client drifts.\n\n\nThank you again for your efforts in reviewing our work. We hope our rebuttal has clarified and addressed your concerns!", "claims_and_evidence": "Given the inherent interconnection between the spatial and spectral domains, methods that perform well in the spatial domain are expected to positively impact the spectral domain. However, the authors do not adequately explain the key benefits of utilizing the spectral domain to address spectral client drifts.", "methods_and_evaluation_criteria": "The solutions provided by S2FGL are targeted, and the datasets used are commonly employed in federated graph learning research.", "theoretical_claims": "The definitions are rigorous", "experimental_designs_or_analyses": "The experiments are comprehensive. However, the authors should consider the impact of the number of representative prototypes for each class in the NLIR on performance and conduct a hyperparameter study. Furthermore, the performance comparison experiment should include FedTAD [1].\n\n[1] FedTAD: topology-aware data-free knowledge distillation for subgraph federated learning. IJCAI 2024", "relation_to_broader_scientific_literature": "The problems explored in this study are meaningful. First, the authors shift the focus from the typical issue of heterogeneity and examine the shortcomings of FGL in comparison to centralized graph learning from a spatial perspective. Second, the spectral domain has received insufficient attention in existing FGL studies, and this paper contributes to filling that gap.", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "The primary strengths and weaknesses have been discussed above.", "other_comments_or_suggestions": "There is a missing punctuation mark after the equation (9).", "questions_for_authors": "1. The authors should conduct a hyperparameter study for the NLIR and include a performance comparison with FedTAD.\n2. The authors should clarify the key advantages of the spectral solutions over spatial solutions in mitigating spectral client drifts.", "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": "MHQDqSvRp2", "reviewer": "Reviewer_uVfb", "summary": "This paper identifies and defines two significant limitations in FGL: label signal disruption and spectral client drifts. The proposed framework, S2S^2S2FGL, addresses these issues by consolidating globally accessible semantic knowledge and aligning the high- and low-frequency spectral domains. Abundant experiments validate its effectiveness.", "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": "Dear Reviewer uVfb:\n\nThank you for your encouraging comments on our work. We hope that our responses below will address your concerns and reinforce your positive evaluation.\n\n## Weakness\n\n**W1: A more detailed analysis of existing methods and their limitations \nis required, particularly for the FGL baselines.**\n\nA1: In response to your concerns, we further elaborate on the deficiencies of the FGL baseline methods evaluated in our study. Specifically, FedSage+ seeks to reconstruct subgraphs by locally training a neighborhood generator. FedGTA determines aggregation weights using mixed moments of neighbor features and local smoothing confidence. FGSSL aligns similarity matrices between local and global GNNs while employing node-level contrastive learning. Similarly, FGGP applies contrastive learning, integrating clustering prototypes to enhance prediction accuracy. FedPub utilizes similarity-based clustering on random graphs at the client side and employs local masks to incorporate the global model. FedStar introduces a specialized channel for processing graph structure, sharing only this channel’s parameters. In summary, existing approaches fail to address two critical issues: spatial label signal disruptions and spectral client drifts. Label signal disruptions weaken the semantic recognition ability of GNNs in FGL, thereby reducing its overall effectiveness. Additionally, spectral client drifts hinder collaboration among clients, making it difficult to establish an efficient spectral signal propagation paradigm. Our work aims to address this gap by focusing on these two underexplored aspects.\n\n**W2: Why does the second method align only the high-frequency and low-frequency spectral features, rather than other components?**\n\nA2: Both low-frequency and high-frequency areas of the graph spectrum are critical and highly representative. Low-frequency components typically capture the global structural properties of the graph, for instance, the second smallest eigenvalue indicates graph connectivity. Conversely, high-frequency eigenvectors contain detailed local information. Directly matching all eigenvectors is computationally prohibitive. Instead, selecting those associated with the smallest and largest eigenvalues reduces computational complexity while preserving the graph's essential structural characteristics.\n\n\nWe are grateful for your encouraging feedback and hope that our response has effectively addressed your concerns!", "claims_and_evidence": "The claims in this paper do not exhibit any significant flaws.", "methods_and_evaluation_criteria": "The two methods effectively address the issues outlined in Figures 1 and 2. Additionally, the Louvain partition and the datasets used in the experiments are standard, making them appropriate for evaluating the effectiveness of these methods.", "theoretical_claims": "NA", "experimental_designs_or_analyses": "The experimental designs presented in this paper are reasonable. Specifically, the authors employ the Louvain partition, a widely used method for simulating real-world subgraph-FL scenarios, and utilize a standard node classification task as the evaluation criterion.", "relation_to_broader_scientific_literature": "The two issues identified in this paper are highly relevant and provide new insights for future FGL research.", "essential_references_not_discussed": "None.", "other_strengths_and_weaknesses": "Strengths：\nThe problems explored in this paper are intriguing. Drawing inspiration from graph active learning, the authors introduce the structure inertia score to measure the significance of semantic signal strength, highlighting a key challenge in subgraph-FL. This work also investigates the heterogeneity of spectral signal propagation across clients. Furthermore, the paper comprehensively addresses these issues in both domains and proposes reasonable solutions.\n\nWeaknesses：\n1. The discussion of current FGL research from lines 26 to 30 is too general. A more detailed analysis of existing methods and their limitations is required, particularly for the FGL baselines.\n2. For the second method proposed in this paper, why does it align only the high-frequency and low-frequency spectral features, rather than other components? A more rational explanation is needed.", "other_comments_or_suggestions": "I have no additional comments or suggestions.", "questions_for_authors": "Please see 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: The reviewers highlighted this paper's novel setting, good writing, and convincing experiments. The authors also provided reasonable answers to the doubts about the details of the proposed dataset and method. The reviewers’ concerns were well-addressed. Therefore, the decision is to recommend Acceptance.
Decision: Accept (poster)