HYPERION: Fine-Grained Hypersphere Alignment for Robust Federated Graph Learning
Authors: Guancheng Wan
Xiaoran Shang
Yuxin Wu
Guibin Zhang
Jinhe Bi
Liangtao Zheng
Xin Lin
Yue Liu
Yanbiao Ma
Wenke Huang
Bo Du
Venue: NeurIPS 2025 spotlight ()
Keywords: Federated Learning
Graph Learning
Robustness
Geometric Learning
Primary area: machine_learning_for_sciences

Abstract: Robust Federated Graph Learning (FGL) provides an effective decentralized framework for training Graph Neural Networks (GNNs) in noisy-label environments. However, the subtlety of noise during training presents formidable obstacles for developing robust FGL systems. Previous robust FL approaches neither adequately constrain edge-mediated error propagation nor account for intra-class topological differences. At the client level, we innovatively demonstrate that hyperspherical embedding can effectively capture graph structures in a fine-grained manner. Correspondingly, our method effectively addresses the aforementioned issues through fine-grained hypersphere alignment. Moreover, we uncover undetected noise arising from localized perspective constraints and propose the geometric-aware hyperspherical purification module at the server level. Combining both level strategies, we present our robust FGL framework,**HYPERION**, which operates all components within a unified hyperspherical space. **HYPERION** demonstrates remarkable robustness across multiple datasets, for instance, achieving a 29.7\% $\uparrow$ F1-macro score with 50\%-pair noise on Cora. The code is available for anonymous access at \url{https://anonymous.4open.science/r/Hyperion-NeurIPS/}.

HYPERION: Fine-Grained Hypersphere Alignment for Robust Federated Graph Learning
Guancheng Wan Xiaoran Shang Yuxin Wu Guibin Zhang", Jinhe Bi Liangtao Zheng Xin Lin², Yue Liu2, Yanbiao Ma², Wenke Huang Bo Du liiiig-aaximiiinssnmmee
Wuhan University 2NUS
München RRemmin University of China SUCD
guanchengvan0vhu. edu. cn
Abstract
Robust Federated Graph Learning (FGL) provides an effective decentralized frame- work for training Graph Neural Networks (GNNs) in noisy-label environments. However, the subtlety of noise during training presents formidable obstacles for developing robust FGL systems. Previous robust Fl. approaches neither adequately constrain edge-mede mede-meddtted error propagation nor account for intra-class topological differences. At the client level, we innovatively demonstrate that hyperspheri- cal embedding can effectively capture graph structures in fine-graine manner. Correspondingly, our method effectively addresses the aforementioned issues through fine- mme-grained hypersphere alignment. Moreover, noise arising from localized perspective constraints and propose the geometric- aware hyperspherical purification module at the server level. Combining both level strategies, we present our robust FGL. framework, HYPERION, which oper- ates all components within unified hyperspherical space. HYPERION demon- strates remarkable robustness across multiple datasets, for instance, achieving a 29.7% t F1-macro score with 50%-pair noise Cora. The code is available https: /tttps//github... hwt.a//.....n.aaaacceennna/aaa/a
uncover undetected
Introduction
has recently emerged as key area in decentralized machine learning. FL enables multiple clients collaboratively train a shared global model while preserving data privacy [65] 20 To leverage graph-structured data from diverse participants. Graph Neural
Federated Learning (FL)
20 34
Networks (GNNs) 22
13
31 have been integrated into FL, giving rise to Federated Graph Learning
(FGL) 55 49 16
45
FGL combines two paradigms, effectively ensuring privacy [1
19 [50] while
enabling efficient distributed graph learning through neural message-passing mechanisms, which propagate node features and hidden representations in graph data.
also introduces new vulnerabilities. Prior studies demonstrate that even minor structural or semantic perturbations can lead misclassification GNNs 7 [57 68 72). These subtle differences may obscure critical information that defines node relationships and class boundaries. In FGL, coarse- -rarine representations of nodes within the same class can forcibly smooth out local topological differences, impeding the effective filtering of subtle noise and hindering the accurate capture of real semantic information and the underlying graph structure. Hence, we pose the following question: I) How can we learn class representations that are robust to noise while capturing subtle structural differences between
As shown in Figure I. although FGL offers numerous benefits
Equal Contribution. Corresponding Author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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similar nodes? Such noise is not only inherently difficult to detect but also pervasive in graph data. Studies show that existing datasets can easily contain over 30% label errors [3] 4T. Recent FL methods address label noise via label correction 47 60 and self-supervised learning [T0 58 8].
but these methods do not explicitly model the complex topological characteristics of graph data. Therefore, when dealing with graph data exhibiting complex topological structures, these approaches typically aggregate neighbor features indiscriminately, mixing noise with valid signals and degrading both alignment level and generalization performance. This leads to the question: II) How can we adaptively identify and select high-confidence, stable nodes in each client's 'ss noisy graph data?
Nevertheless. it is extremely difficult to completely remove noisy nodes solely relying on the client. The federated framework's 's privacy constraints limit each elient's view to its local subgraph, preventing a global perspective. As a result, client models learn only local semantics and limited topological context, causing certain abnormal nodes to appear "normal" locally and evade detection. The limitation of this local perspective indirectly damages the generalization ability of the global model. Therefore, we ask: III How can we robustly refine semantic and topological knowledge during global aggregation and transfer it efficiently to the global model?
To holistically address these challenges, we propose HYPERION: a Hyperspherical- Embedding- Framework for Robust Federated Graph Learning, where all compo- nents operate on a unified hyperspherical space. To address issue I). we introduce Topolog- ical Prototypes Hyperspherical Learning (TP-HSL) to fully capture the rich topological differences between nodes of the same class. Our method projects training node samples onto a hyperspherical embedding space. On the one hand, it maximizes the minimum spherical angle between different class prototype clusters, actively amplifying inter-class differences and enhancing the discriminability of decision boundaries. On the other hand, it minimizes the average spherical angle between nodes of the same class and their prototype centers to ensure tight intra-class clustering and strengthen structural correlations. Compared to conven- tional ome-class-one-proooyype
Prone hop
OIDLocal Perspective
Structure Similirsty
aand
Noda
Instry
Reeling
Soandant
Mode 407
Nadda 948
Denesten
Graphaenc
labell
(1) Wrrnng Neighhor Pooling
Wrong Server
Figure 1: Problem Illustration. We describe the chal- lenges FGL encounters under noisy labels: I) The coarse- grained representation method leads ineffective dif- ferentiation between similar nodes, resulting in the cou- pling of noise and valid information. II) The edges between nodes in graph facilitate the propagation of noise. IIII The restricted view of individual clients leads to missed detection of certain noise.
approaches 46 18 5T, TP-HSL provides finer structural modeling capabilities and heightened topological sensitivity. To address problem we propose Hyperspherical Consistency Noise Calibration (HS-CNC), which constructs a perturbed view of the graph. We retain only high-confiden nodes that consistently map to the same prototype cluster across views and filter out potential noisy nodes with unstable mappings. This process explicitly exposes the potential perturbation- sensitive areas through the "amplification of noise shifts" effect, ensuring information purity while effectively strengthening the correlation between nodes and the true topological structure. To solve issue III) we propose Geometric-Aware Hyperspherical Purification (GA-HSP). Driven by the Wasserstein distance, we distill, refine, and aggregate local prototype knowledge into robust global prototypes, constructing a well-defined global hypersphere. Then, by leveraging the covariance structure between the client hyperspheres and the global hypersphere, we apply the Mahalanobis distance to assess each node's 's outlier risk and climinate drifting nodes biased by noise. To summarize, we make the following key contributions:
Problem Identification. We study challenging problem: overcoming the label noise in FGL. Our focus is on mitigating the negative influence of the label noise while overcoming several key limitations of existing solutions, including reliance on coarse gaarsed representations, neglect of the graph data's 's topological structure, and the absence of re-correction from global perspective. Practical Solution. We introduce HYPERION, a novel and effective methodology that disentangles complex topological structures and mitigates malicious noise in FGL through hyperspherical representation. With the help of several technical innovations, HYPERION significantly enhances the model's 's ability to distinguish subtle structural differences while maintaining strong robustness.

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Experimental Validation. We conducted extensive experiments on five mainstream datasets under different noise types and ratios. The results demonstrate that our approach outperforms the state-of- the-art methods in multiple FGL. environments For instance, under the 50% -0irr noise setting on the Cora dataset, our method achieves an impressive Fl-macro score of 51.15%, outperforming the second-best method of 39.41 % by significant margin.
2 Preliminaries
Following the typical FGL. framework, M participants (indexed by m) collabora- tively train a shared global model using their private graph data. Participant m holds a graph 9m (Vm,Amm Am), where Vn = {e]ken Nm is the node set containing [mm Am E (0. (0.1) is the adjacency matrix with AG = 1 if there is an edge between nodes 2/ and U/ (and 0 otherwise), and Xm {x,) ) 24 E R' is the node feature set of dimension d. Moreover, Vm E {0, N xC is the label matrix, where each label y 10, is one-hot vector over O classes. See Appendix[A A detailed notation.
Notations.
Nn. nodes,
Problem Formulation. We focus on the semi-supervised node classification problem. Only a small set of nodes V is labeled for training, denoted as V% Vn \ V where N the number of labeled nodes. The remaining nodes are unlabeled and denoted as Vi Given X'm and Am the goal of node classification is to train classifier fe... (.mmAnm - Di where the model parameters optimized by minimizing the following objective:
min L(Jom. (Ni, Am). Om
(1)
where C is loss function that measures the discrepancy between predictions and ground-truth labels. In this way, according to the Empirical Risk Minimization (ERM) principle, the well-trained classifier se... can generalize effectively unseen nodes VM!
However, in real-world scenarios, the available labels y may be corrupted, which degrades the generalization ability of the m-th client's classifier fe... We denote these noisy labels V (Ji,..., Jr where )% represents their corresponding ground-truth labels. To realistically model label noise multi-source data, consider two common types of label noise, defined follows:
This noise model assumes that the true label has probability E (0, 1) of being uniformly flipped to any of the other classes with equal probability, Formally, for all +
Uniform noise 44
pVy w/ i) d d-11 1
(2)
Pair noise 63]
This noise model assumes that the true label can only be flipped to specific paired class with a fixed probability e, while remaining unchanged with probability 1 e.
The optimization objective to learn a generalizable global model through the federated learning process that performs well under noisy conditions while maintaining strong robustness.
Methodology
3.1 Framework Overview
Inspired by our observations in Sec. that FGL is sensitive to label noise, we propose HYPERION finely enhance the model's ability capture subtle structural differences among similar nodes and thereby improve robustness to noise. HYPERION comprises three key components: I) on each client, we extract local graph knowledge in a hyperspherical embedding space, where multiple class-specife prototype clusters capture fine-grained structural patterns; II) we select nodes whose embeddings remain consistently stable relative to their prototype clusters under perturbed views to ensure reliability; III) after the aggregation, we employ Wasserstein-driver prototype distillation and Mahalanobis-guit guided node purification at the server to refine and transfer complex structural knowledge. The detailed description of HYPERION is illustrated in Figure 2

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(0) Mooosntttiic-Anne Amare Hyperspherical Purdfication
Hyperspherica Modeling with Topological Protetypes
Haveertein--wee Server Diviliation
(b) 1ly perspherical Consistency
Topelogical Protetypes
M
Fand and LAY
Noose Callbration
Hyperspherical Learning
DO
Mahalonotas- Baved
Esrs Reea Hevey
Projection
Naighorthed Sparetiicnins
Masuaasabe Resod
AA
Eyfoed
Architecture illustration of HYPERION. HYPERION comprises Topological Prototypes Hyperspherical Learning (TP-HSL), Hyperspherical Consistency Noise Calibration (HS-CNC) and Geometric-Aware Hyper- spherical Purification (GA-HSP). Best viewed in color and zoom for details.
Figure 2:
3.2 Topological Prototypes Hyperspherical Learning (TP-HSL)
Existing methods suffer from noise amplification due to their reliance on coarse Euclidean representations that fail to capture multiple structural variations among intra-class nodes, We address this limitation by introducing prototype clusters on the hypersphere, which reveal and mitigate noise propagation through fine-gran fine-grained topological modeling.
Motivation.
Hyperspherical Modeling with Topological Prototypes. Each client projects its node features Xm,, and adjacency Am, into a unit hypersphere via an independent feature projector 0%
Zm, 6% (Xmey Am.); Zm;
Zyp
(3)
|>
The hyperspherical embedding representation 2m; can be modeled using the von Mises-Fisher (vMF) distribution [35] 55). The vmF distribution well-suited for accurately measuring the angular differences between embeddings, especially in high- high-dimensional spaces. In graph data, node embeddings are typically nigh-dimensional, and the vmF distribution, through its concentration parameter, effectively quantifies the similarity and dissimilarity between node embeddings [55] 21). This geometric property makes the model more sensitive to meaningful semantic relationships while being less susceptible noise and feature magnitude variations. Compared traditional Euclidean spaces, the hyperspherical projection offers superior noise robustness by encoding semantic similarity in angular relationships rather than absolute positions, which is particularly erucial for handling label noise and structural variations in decentralized graph data. This approach allows the model to better capture fine-grained topological differences while maintaining strong generalization across clients.
However, prototype- prototype-based based modeling with a single center suffers from inherently limited expressive- ness [32). which fails to capture all the underlying topologies adequately. To solve the issue, allocate a prototype cluster of shape [M, C, K). defined as P = {p E [C], k E [K], m e [M]]. which partitions the entire hyperspherical space into multiple topology-awan subspaces, each centered around a different prototype:
K
viewsp( (Zm.) w P R) Σ c. (n) exp(mp(.* o.F. =m;))
(4)
km
where p denotes the centroid prototype in the k-th topological subspace for class on the m-th client, week denotes the corresponding prototype weight, and k the concentration parameter. This approach ensures that the node embeddings within each class are better aligned with the corresponding prototypes, thereby improving both intra-class cohesion and inter-class distinction. By doing so, 24, we further compute its prediction probability as follows:
finely capture the structural differences within each class. For the input vector
Σ wiick exp(xpll... 2m,)) .
ply y ye Zmyi (wi, P (
(5)
Σe, Ek winh exp(xpmn* c.k 2m.)

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Topological Prototypes Regularization Learning. To strengthen inter-class separability and intra-class cohesion, we introduce several regularization terms based on the aforementioned class prediction probabilities. First, we maximize the minimal spherical angle among different class prototypes to optimize the distribution of inter-class prototypes further. Specifically, we calculate the cosine similarity matrices to measure the similarity between prototypes in Equation (6):
S. [i, = exp
(Pm
Pm
vi [1, [L,
(6)
Ilpm
where Sm j[] quantifies the pairwise similarity between prototype and prototype j within the hyperspherical space. As shown in Equation categorize intra- and inter-class similarity:
we apply binary masks I'm and r to further
S Sm O PoS
Sm
(7)
where T activates entries corresponding to prototype pairs from the same class (excluding self-pairs), and Im identifies prototypes pairs from different classes. The symbol @ represents the Hadamard product. Finally, we arrive at the following regularization term:
S
[
N Σ
LSTR N
log
pos
[2,
Here, E is small smoothing factor used to prevent division by zero. This regularization term serves two key purposes: on the one hand, boosts inter-class separability, thereby sharpening the decision boundaries; the other hand, it ensures intra-class similarity, reinforcing the class semantic features.
To prevent global shifts in the semantic space formed by multiple prototypes, it is also essential to regulate the spherical angular relationship between embeddings and their associated prototypes. To this end, we encourage the minimization of the average spherical angle between node embeddings and their corresponding prototype cluster centers. This reinforces semantic consistency within each class, This regularization term can be modeled as:
1 LSEM
D wh exp(xppl uick Zm.)) 2om1 2k-1 wm exp(xpm km;)
log
(9)
N
o.
3.3 Hyperspherical Consistency Noise Calibration (HS-CNC)
Motivation. Due to the unique neighborhood diffusion mechanism in graph data, noisy labels tend to propagate along the edges. Therefore, it is crucial to design a noise node filtering mechanism that considers both structural and semantic aspects.
Hyperspherical Robust Node Selection. To effectively filter out potential noisy nodes whose mapping trajectories exhibit sigmificant fluctuations, we assess the stability of nodes across different augmented graph views. Specifically, inspired by previous works [7] 27). we introduce data augmentation techniques: edge dropping and feature masking. These techniques randomly drop edges and certain feature values in the graph 9m (Xm, Am):
Am Am P Xim Xm r
(10)
where I m E {0, 1j KNm is the randomly generated edge mask matrix, and r E {0, 1) m is the randomly generated feature mask matrix.
By applying the two mask matrices, the augmented graph 9. (L'.. Am) contains perturbed structural and semantic information. We assess the stability of each node by calculating the consistency of its representation across different views. Nodes that exhibit high consistency across both views considered "clean nodes" because they maintain stable and consistent semantic representations under different perturbation conditions. We train using the subset of stable nodes X which fundamentally ensures that learned representations are grounded in mcaningful:
LCNC
Σ 1i. EVm
(fm vin) fm (Gm, win)) Io (fm of (mn))))))))))
(11)
5

---

where 1 is an indicator function that outputs if the predicted results for node e in both graphs are consistent, and 0 otherwise. The loss function L computes the discrepancy between the node's predicted value and its true label and then calculates the gradient to update the parameters.
This design naturally connects two theoretical perspectives. From the information bottleneck viewpoint, random edge dropping acts as compression: by pruning away connections, the model retain relationships that consistently survive perturbations, which more likely reflect the task- relevant core structure [5] From the generalization under noise viewpoint, these augmentations inject structured noise that preserves global characteristics
prevents overfitting fragile details. Consequently, the model is nudged toward flatter minima in the loss landscape, leading to improved robustness and generalization. Such principles align with findings in graph contrastive learning 62 53
As an example, HYPERION algorithm is shown in Algorithm
Algorithm 1 HYPERION Framework
Communication rounds T, participant scale M, m-th client private model Om. m-th client local data 9m. m-th client prototype cluster Pm and loss weight a, The final global model 0 Ogtobal for i = 1,2,-..
do
Client Side: for m 1 to M in parallel do LoNc + arpersphrriclNoiieeaaaatttttotoi nodes and train with them
Pm Pm) Equation
11 // Select robust
Sm CalculateSimillrity(M.. Jby Equation (6) // Calculate prototypes similarity metrix
LSTA + StructLoss(S,,, by Equation ( // Calculate loss with inter- and intra-class proto-
types
LSEM SemanticLLsssPPm.)) Equation ( // Calculate loss with embedding vector and
prototypes
01+1 + LocalUpdaaing((: LONG + aLSTR + 3CsEM) Backward propagation m
Server Side:
(poos ,0neg +- GMM(P) // Client classification by prototypes
Gmegr NeighbortoodSSpasfficcttooo/// (Oneg, (m.) Vm by Equation 18 // pruning with Maha-
lanobis distance
G gloobul: Polobal < Aggregate (070s ppos), Vm Clean clients hyperspherical aggregation
P' Palobal + ServerDistillation Bi PPgobalt) Vm by Equation [3] // Wassertein distance Server Distillation
6m +- fglobat, Vm Distribute parameters to clients
return 0 global
3.4 Geometric-Awarr Hyperspherical Purification (GA-HSP)
Motivation. is exceptionally challenging to entirely remove noisy nodes relying only on the client-side. Due to the non-IID distribution, each client has a limited perspecitve, meaning that some noisy nodes are detected as normal locally, but are considered anomalous when viewed globally. To address this issue, we propose GA-HSP, which performs knowledge purification on the server side.
Prototype-based Client Classification. Existing research suggests that, practical scenarios, the data noise ratio at each client may vary to some extent (60). To effectively identify noisy clients, we design an unsupervised detection method that fully exploits the global distribution characteristics of local prototypes the client side, distinguishing between benign and malicious clients.
After training each client, the local prototype clusters P uploaded to the server, where the server computes the Gaussian Mixture Model (GMM) of the prototype clusters P from all M clients.
p(P) Σ
N(P Pi: Ec).
(12)

---

where Ti, M4 and Σ, represent the weight, mean, and covariance matrix of the 1-th Gaussian com- ponent, respectively. The client set partitioned into two subsets: clean clients Ppos and ma- licious clients Onegg To ensure the reliability of the global knowledge, we apply GMM to re- aggregate the prototypes from the clean clients Opos. For each class, we aim to aggregate the prototypes into K clusters and take the center of each cluster as the global prototype for that class: Patobal = {pc* & [C], E [K]]. Through this unsupervised approach, we can not only effectively filter out malicious clients, but also integrate the local information from all clean clients, thereby constructing a more precise global prototype cluster Potobat.
Wasserstein- based Server Distillation. To further filter out erroneous information in the prototypes, inspired by recent work on negative distillation [33], we adopt a "negative distillation" strategy: each malicious client prototype cluster Pee serves as the teacher. while the global prototype cluster Paoobal functions as the student. The core of this distillation is to constrain the angular distance between the teacher and the student in the hyperspherical space, thereby suppressing and correcting anomalous representations in the global prototype cluster.
Traditional distillation methods fail to capture cross-dimensional similarities, leading to the underuti- lization of dimensional information. To address this issue, we propose Wasserstein distance-driven prototype negative distillation method, which deploys the discrete Wasserstein distance to compre- hensively measure the distributional differences between the teacher and student models. For the m-th teacher prototype cluster Pargg we define the discrete Wasserstein distance DwAS as follows:
Dw As Potohhtt min Th3
Σ Enccht +
+
Thi log 765
(13)
where "hy represents the mass transferred from the teacher prototype cluster's dimension qi to the student prototype cluster's dimension 9j. subject to the constraints:
Σ The 7 Σ 7a
pt Yis 2 0,
(14)
where À is hyperparameter controlling the entropy regularization term. A key component of this formulation is the cost matrix em which encapsulates the dissimilarity between prototype dimensions:
Pmcy Patobal.s
(15)
|| [ll] slobal.)]
The higher the similarity between prototype dimensions, the lower the transfer cost. Conversely, when there significant difference in the direction of the dimensions, the cost increases considerably. By minimizing Dw AS. the probability mass is effectively reallocated between proximate dimensions in the feature space, thereby naturally reinforcing the correlation of benign features between global prototypes, while effectively diminishing the impact of anomalous features.
Mahalanobis-Baan Based Neighborhood Sparsification. Building on the negative distillation from malicious clients, we attempt to incorporate data pruning to purify them. Our dutt---distribution (OOD) detection method is based Mahalanobis distance 32 42). which is equivalent to the Euclidean distance scaled by the eigenvalues in the feature space. By introducing the inverse of the covariance matrix, the Mahalanobis distance can automatically adjust the importance of each dimension, avoiding certain dimensions dominating the distance calculation due to scale differences. In geometry, Mahalanobis distance transforms the data space into a standardized space, making distance calculations more equitable. We identify and remove noise nodes that deviate from the normal distribution by calculating the Mahalanobis distance between node embeddings and the global prototype distribution (as shown in Equation (18) This method effectively filters out noise that is not detected by clients due to local perspective limitations. Inspired by the work M. we simultaneously measure both inter-class and intra-class distances of prototype clusters to comprehensively assess the outlier risk of nodes. The inter-class prototype distance defined as:
(Zmm;)
1
Σ ea
(16)
Cl
ming (Ezm - piak (T A. (2m, pin +
7

---

Table 1: Comparison with the state-of-the-art methods on five selected real-world datasets. For each dataset, we report accuracy (%) and F1-macro (%) (with red/green markers indicating regression/mpprveemnn over FedAvg). The noise type is set to 50%-uniform (upper) and 50 % %-piir (lower). The best and second-best results are highlighted with bold and underline, respectively. Additional experimental results on more settings can be found in Appendix D
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where A- represents the inverse of the covariance matrix of all prototypes in class c, and e is used to avoid division by zero errors. The intra-class prototype distance is calculated as:
IPo.
Ciutral (Zmr.) )
(17)
P
(Am; pk Aw, (Zm;
pia +
Finally, we calculate the comprehensive outlier score and rank all nodes based on their outlier risks, pruning the top 17% of the high-risk samples in each training round:
(mn.)((mm;))
Score(zm., (in + dhaara (Zm).)
v++1
{am, E Vt, rankk > Ln * (1tU)) (18)
where VL represents the sample nodes in the f round, and 1y denotes the pruning ratio. Further discussion and limitations can found in Appendix Fand Appendix
4
Experiment
In this section, we comprehensively evaluate HYPERION through four key axes: Q1 (Superiority), Q2 (Resilience), Q3 (Efffectiveness), and Q4 (Sensitivity).
4.1 Experimental Setup
Datasets. To effectively evaluate the performance of our approach, we utilize five benchmark graph datasets of various scales and distributions with different characteristics, including Cora [36), CiteSeer [11], PubMed R. Physics 43, and paazzon_aatins..... These datasets represent a wide range of domains and are commonly used in graph-based machine learning tasks. Detailed descriptions and dataset splits for these datasets can found in Appendix C.T. Furthermore, the implementation
details and parameter settings can be found in Appendix
8

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unfvm
bm
NO0L mIn
Adves.
Karseer
Figure 3: We report the performance of different methods at various noise ratios. with datasets including Cora, Citescer, and Pubmed, and noise types of uniform and pair. The red color represents the performance of HYPERION. (Firsr): 20% mild noise. (Second): 70% severe noise. (Third); We compare HYPERION with several FL and FGL methods on the Cora dataset under uniform noise, as the number elients ranges from 5 to 20,
par
pur
Counterparts. We compare HYPERION against several traditional FL methods: (1) FedAvg [AS-
TAT17] (37,. (2) FedProx [MLSys20] (25). (3) FedNova [NeurIPS20] [54]. (4) MOON [CVPR 21] 24 : four popular FGL approaches: (5) FGSSL [ICCA123] [66]; (6) FedTAD [UCAI24] 70),
(7) FedGTA [VLDB24] [28]; three Robust FL methods: (8) FedNoRo [ICAA13]] 588 (9) Ned
[AAAI24] 33). (10) FedCorr [CVPR22] three Robust GragnLearning methods: (11) CRGNN
[NN24] 27. (12) RTGNN [WWW23] 44), (13) CLNode [WSDM23) [56]. Detailed descriptions
of all the baselines can be found in Appendix
4.2 Superiority
To answer Q1, we conducted systematic evaluations in a variety of noise environments, including two typical noise types (uniform noise and pair noise) and three noise intensity levels (0.2 for low noise ratio, 0.5 for medium noise ratio, and 0.7 for high noise ratio). The perfmance results are presented in Tab. and Figure 3 Several observations from these experiments are summarized (Obs.):
Obs. Existing approaches exhibit suboptimal performance in FGL scenarios with noisy labels. For instance, in the uniform noise mode with 50% noise ratio, most previous methods achieve accuracy rates below 40% on the Cora dataset. Notably, under a 70% noise ratio, the performance of most of these methods deteriorates significantly, worsening with average accuracy rates consistently dropping below 25%. Moreover, the existing robust methods do not demonstrate substantial improvements noisy FGL scenarios, with the performance most approaches being comparable to that of traditional FL and FGL methods.
Obs. HYPERION demonstrates remarkable robustness across various noise scales. Under moderate noise conditions (0.5), HYPERION shows a clear and significant advantage. As shown in (upper,, HYPERION consistently outperforms both FL and FGL baselines across various datasets and noise types. In the Citeseer-uniform setting, it achieves an accuracy of 47.1 1,, surpassing the best baseline, RTGNN (41.93%) by 5.18 percentage points. Additionally, (Second), HYPERION consistently outperforms both FL and FGL baselines across various noise scales. It is evident that, under a noise scale of 0.2, HYPERION achieves varying degrees of performance improvement over all baselines. In niihe environments, HYPERION also demonstrates an average performance gain of 1.1% to 10.1% % compared to the baselines, including RTGNN.
shown in Figure
4.3 Resilience
To address Q2, we evaluate the performance of each method on the Cora dataset under the 0.5 uniform noise setting, across different client scales. Figure (Third)) illustrates that HYPERION consistently achieves robust performance gains across varying client numbers (5-20), outperforming FedAvg by at least 13.69% while maintaining minimum 6% advantage over the top-performing baseline method. This demonstrates that HYPERION effectively identifies noise and maintains stable performance, even under challenging conditions with varying client populations.

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4.4 Effectiveness
To address Q3, we conduct an ablation study on the key components of our method, both at the client-side and server-side, under a noise scale of 0.5. Tab. rereorts the performance of HYPERION and its variants by removing specific components from the TP-HSL and HS-CNC modules-namely, structural loss, robust node selection, and semantic loss. Tab. presents the results for HYPERION and its variants derived from the GA-HSP module, where we ablate server-side distillation (SD), node pruning (NP), and client classification with server distillation (CC+SD). Individually, both TP-HSL and HS-CNC contribute significantly to improving model accuracy. Moreover, GA-HSP demonstrates substantial effectiveness in integrating global reliable information, thereby reinforcing the robustness of our design in mitigating label noise in FGL settings.
Table 2: Ablation study of TP-HSL and HS-CNC Table Ablation study of GA-HSP on the server- the client-side of HYPERION. All results ported under noise ratio and 10-client scale.
side of HYPERION. w/o CC+SD means SD depends on CC, so without CC, SD is also removed.
Cora
Citeseer
Cora
Citeseer
Client
Server
uniform
pair
uniform
pair
uniform
pair
uniform
pair
LSTR
50.27
36.93
42.37
42.07
w/o SD
49.18
37.11
39.85
38.67
w/o Lexc
50.82
35.28
43.56
38.37
w/o
37.29
34.83
30.07
31.26
w/o Lsear
50.09
36,20
40.29
41.04
w/o CC+SD
44.66
36.40
36.07
35.70
HYPERION
56.31
41.50
47.11
43.11
HYPERION
56.31
41.50
47.11
43.11
4.5 Sensitivity
To address Q4, we perform sensitivity analyses on hyperparameters of HYPERION. Specifically, we examine the model's performance under varying values of X, 17. a, and B, as illustrated in Figure where these hyperparameters are fixed at different scales and values. We systematically vary the hyperparameters and 7) within the ranges [0.01, 0.05] and [0.90, 0.98], respectively, the stability of GA-HSP under different settings. For TP-HSL, we vary a and / with in the ranges [0.4, 0.6] and [0.6, 0.8), using step size of 0.05. The results indicate that the choice of and v) has minimal impact on the performance of HYPERION. However, when a is within the range of |0.55, 0.6] and / is within the range of [0.7, 0.75), HYPERION achieves the best performance across all datasets.
evaluate
(a) GA-HISP Parameter À (b) GA-HSP Parameter y) (c) TP-HSS Parameter
(d) TP-HSL Parameter 3
Figure 4: Analysis on hyper-parameter in HYPERION. Node classification results under varying values a, and B. The colors green, blue, and yellow correspond to performance on the Cora, Amazon Amazon_ratinge ratings, and Citeseer datasets, respectively. All experiments conducted using 50%-uniform noise.
We investigate the impact of the hyperparam- eter K on the performance and efficiency of HYPERION. Specifically,, we vary K 21,2,3, 4} in the Citeseer-pal setting and ob- serve the corresponding changes in performance. As shown in Figure 3 setting K = results in under-learning of hypersphere, leading to consis- tently lower performance. In contrast, increas- ing K to 3 or yields marginal performance im- provements. In all experiments, set K 3.
Figure 5: Performance across varying values under different noise ratios.
20%
50%
70%
K
ACC
FI-M
ACC
F1-M
ACC
FI-M
59.41
52.13
38.81
36.78
20.48
20.53
2
59.56
52.24
39.70
37.92
21.33
21.39
61.93
56.58
43.11
41.11
24.59
24.48
4
61.04
55.96
38.07
36.02
24.65
25.02
10

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Conclusion
In this work, propose innovative exploration of robust FGL in noisy environment. To achieve this goal, we project nodes onto hyperspherical embedding space, thereby introducing a novel framework, HYPERION. For robust representation, TP-HSL employed to project nodes onto hyper- spherical space, effectively addressing the coupling problem associated with complex topologies and malicious noise. Leveraging hyperspherical representations, we also introduce HS-CNC, which filters out potential noisy nodes by considering both structural and semantic factors. Specifically, for effective global collaboration, we further design GA-HSP to facilitate knowledge purification. By integrating these three strategies, HYPERION outperforms various fate-of-the-an methods in node classification tasks across different noisy scenarios.
Acknowledgement
This work is supported by National Natural Science Foundation of China under Grant (62225113 13, 623B2080), and the Innovative Research Group Project of Hubei Province under Grants 2024AFA017. The supercomputing system the Supercomputing Center of Wuhan University supported the numerical calculations in this paper.

---

A Notations
We present a comprehensive review of the commonly used notations and their definitions in Tab.
Notation
Definition
Gm
Graph data for the m-th client.
N
The number of nodes for the m-th client.
Vm
The node set of Gm.
on
The feature matrix of 9m.
Am
The adjacency matrix of 9m.
Vm
The one-hot label matrix of 9m.
Xma
The feature of node in Gm.
Ama
The edges of node
9m-
M
The number of clients.
d
The dimension of the node feature.
C
The number of node classes.
The number of prototypes in each class per client.
VL
The labeled node of Gm.
DU
The unlabeled node set of 9m.
N
The number of labeled nodes for the mn-th client.
m
N
The number unlabeled nodes for the m-th client.
m
fem
The classifier of the m--th client.
Dn
The prediction matrix of Gm-
The noisy label matrix of Gm.
C
The loss function.
The fixed probability that the true label is flipped to specific paired class. The hyperspherical embedding representation of node in 9m
2m;
cck
The k-th prototype of the e-th class of the m-th client.
wer
The k-th prototype weight of the e-th class of the m-th client.
m
k
The concentration parameter.
S S
The similarity metrix between prototypes in the m-th client.
S Spos
The similarity matrix between intra-class prototypes in the m-th client
S
Sheg
The similarity matrix between inter-class prototypes in the mitth client The covariance matrix of all prototypes in class
n
A
n
The pruning ratio detected nodes.
poos
The clean clients set.
gneg
The malicious clients set.
The struct loss weight.
3
The semantic loss weight.
X
The hyperparameter controlling the entropy regularization term.
Table Notation and Definitions.
B Related work.
Federated Graph Learning(FGI). FGL. enhances Federated Learning (FL) by extending it graph-structured data, facilitating decentralized training while safeguarding raw graph data, thereby bolstering privacy protection| 19 30 4 53 52.. Current research primarily focuses on addressing the non-lID data problem in FGL. For instance, FedGCN [61] employs an attention mechanism to dynamically reweight local model parameters, mitigating the impact of data distribution heterogeneity, FGSSL [16] further decomposes the Non-IID issue into node-level semantic divergence and graph- level structural discrepancy, calibrating them separately. However, these approaches overlook critical practical challenge: label noise caused by annotator negligence or bias may extensively exist local data, significantly degrading the global model's generalization performance. To address this limitation, we propose robust FGL. framework grounded in hypersphere representation learning, which enhances the model's capacity to capture subtle structural differences graph data, thereby maintaining stable classification performance under label noisc.
12

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Robust Federated Learning and Graph Learning. The existing solutions to the noisy label problem FL can be broadly classified into two categories: label correction and self-supervised learning. The first category involves label correction mechanisms, which reassign noisy labels based on representations extracted from the training data. These include methods like nearest neighbors in the embedding space [48] and predictions from the global model [60]. The second category leverages self-supervised learning to obtain more robust representations, as seen in methods such as RoFL [10] and FedNed [33]. For instance, FedNed (33) reduces the risk of propagating incorrect
information by using noisy client negative distillation, while FedDPCont [ promotes robust learning by randomly selecting contrastive labels and sharing them with the server. Analogous challenges
also prevalent in the domain of graphs, where noisy labels and structural complexities similarly hinder model performance. In recent years, growing body of research has focused developing GNN methods tailored for robust graph learning under lahel noise. Some methods have achieved significant success by incorporating techniques such loss modulation
29 9 robust training strategies [56). graph structure augmentation 6 40 67 and contrastive learning 64 27] However, all these methods are not conducive to more fine-grained structural learning, leading to the mixing of valid
[39 66
signals and noise components in the feature space, and our method is the first attempt at leveraging hypersphere learning for robust federated graph learning.
C Experimental Details.
C.1 Dataset Details
To assess the effectiveness of HYPERION, we conduct experiments on eight real-world graph datasets: Cora, CiteSeer, PubMed, Physics, and Amazon-ratings. Each dataset is split into training, validation, and test sets in a fixed 20%/40%%/40% ratio. The key statistics of these datasets are summarized in Tab. A detailed description is provided below:
Cora, CiteSeer, and PubMed. These three citation network datasets are standard benchmarks in graph-based machine learning, especially for tasks like node classification and link prediction. In these datasets, nodes correspond to academic papers, while edges represent citation links. Each node is assigned class label, and its feature vector is constructed from textual information such as words in the title or abstract. These datasets exhibit sparsity and high dimensionality, making them well-suited for evaluating the effectiveness and scalability of graph neural networks (GNNs).
Amazon-ratings This dataset is derived from the Amazon product co-purchasing network metadata in the SNAP repository. Nodes represent products (books, music CDs, DVDs, VHS tapes), and edges connect products that are frequently purchased together. The task is to predict the average rating given to product by reviewers. The authors categorize the possible rating values into five classes. For node features, they use the average of fastText embeddings the words in the product descriptions. To reduce the size of the graph, only the largest connected component of the 5-core subgraph is considered.
Seauthor-Physice is academic network containing co-authorship relation- ships based on the Microsoft Academic Graph. Nodes in the graph represent authors and edges represent co-authorship relationships. In the dataset, authors are categorized into five classes based on their research areas, and the nodes are characterized as bag-of-words representations of keywords of papers.
Seauthor-Physicee
Dataset
#Nodes
#Edges
#Classes
#Features
Cora
2,708
5,278
7
1,433
Citeseer
3,327
4,552
6
3,703
Pubmed
19.717
44,324
3
500
Amazon-rating
24.492
97,050
5
300
Coauthor-Physics
34,493
530,417
5
8,415
Table 5: Statistics of datasets used in experiments.
C.2 Counterpart Details
This section provides comprehensive overview of the bascline approaches employed in study.
13

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FedAvg [ASTATI7], A foundational algorithm in Federated Learning, FedAvg operates by allowing clients independently train models on their local datasets and subsequently transmit their model updates to a central server. The server performs a weighted aggregation of these updates to refine the global model, which is then redistributed to the clients for further local training. By transmitting only model parameters instead of raw data, FedAvg reduces communication costs and enhances privacy. However, it struggles with performance degradation in scenarios where client data distributions are highly non-IID 126 38).
FedProx [MLSys20]. As an enhancement of FedAvg, FedProx is specifically designed to address the challenges posed by statistical heterogeneity in federated learning. It introduces an additional regularization term that constrains local updates, preventing excessive divergence from the global model. This proximal term mitigates the impact of local data distribution shifts, leading to more stable convergence. By ensuring consistency in updates across clients, FedProx demonstrates improved robustness in non-III settings.
FedNova [NeurIPS20]. FedNova refines the FedAvg framework by introducing normalization to local updates before aggregation. Unlike standard averaging methods, FedNova ensures that each client's 's contribution to the global model is proportional to the amount of data it possesses. This approach addresses the issue of unequal client influence, leading to more balanced and efficient convergence. FedNova is particularly beneficial in federated environments where data distributions are skewed across clients.
FGSSL [IICAI23]. FGSSL addresses local client distortion caused by both node-level semantics and graph-level structures. It improves discrimination by contrasting nodes from different classes, aligning local nodes with their global counterparts of the same class while pushing them away from different classes. To handle structural information, transforms adjacency relationships into similarity distributions and distills relational knowledge from the global model into local models. This approach preserves both structural integrity and discriminability. achieving superior performance on multiple graph datasets.
FedTAD [1CA]24]. FedTAD addresses subgraph heterogeneity in FL by decomposing local graph variations into label and structural differences, preventing inconsistent model aggregation. It enhances knowledge transfer via hopology-award distillation, boosting FL reliability and efficiency. FedGTA [VLDB24]. FedGTA is tailored for large-scale graph federated learning, tackling issues of slow convergence and suboptimal scalability. Unlike prior methods that focus on either optimization strategies complex local models, FedGTA integrates topology-awarc local smoothing with mixed neighbor feature aggregation to improve learning efficiency [69. By leveraging graph structures in aggregation, it enhances scalability and performance in federated graph learning.
MOON [CVPR21]. MOON adopts a model-contrastive approach to address data heterogeneity in federated learning. The framework utilizes similarities between model representations to correct local training through model-level contrastive learning, providing an effective solution for collaborative training with deep learning models on image datasets while preserving data privacy. FedNoRo [IICAI23]. FedNoRo udopts a two-stage framework to address class- dass-immalanceed global data with heterogeneous label noise in federated learning. The method first identifies noisy clients through per-class loss indicators and Gaussian Mixture Modeling, then performs noise-robust federated updates via joint knowledge distillation and distance-aware aggregation, specifically designed for realistic medical scenarios with data imbalance and complex noise patterns.
FedNed [AAAI24]. FedNed adopts a negative distillation framework to effectively leverage extremely noisy clients in federated learning. The method first identifies noisy clients, then inno- vatively utilizes them as "bad teachers' through a dual-training approach: one model trained on original noisy labels for reverse knowledge distillation, and another on global model-generatee pseudo-labels for conditional participation in aggregation. This approach transforms noisy clients from detrimental elements into valuable contributors while progressively enhancing their trustwor- thiness through pscudo-label refinement
FedCorr [CVPR22]. FedCorr adopts a multi-stage framework to address heterogeneous label noise in federated learning while preserving data privacy. The method first dynamically identifies noisy clients through model prediction subspace analysis and per-sample loss evaluation, then employs an adaptive local proximal regularization to handle data heterogeneity. After fine-tuning on clean clients and correcting labels for noisy ones, FedCorr performs final training across all clients to fully utilize available data, effectively handling varying noise levels without requiring prior assumptions about client noise models.
CRGNN [NN24]. CRGNN addresses label noise in GNNs by combining neighborhood-bat label correction and contrastive learning. It utilizes message passing neural networks update
14

---

Calpey
Netab
275 Nut
70% Nate
Lelkee
Par
Ledomr
Pdr
Underm
Par
Ladone
Parr
Lederrs
Parr
Uniiere
Palr
18
80
91.31
90.5huuax
(1
18 ts
64.38
16.12.00
22.05
24224
Podhis (41
52 20
38,64
45
76443
To 70.0
22.06
20.20 19226.o
24,04
25 26
24.33
16.5
MOOK (CV9221
68.01
76-35
98.48
21.64
POATTA (VL2034)
10.00
45,00
57.32
76.09
16.00
21.73
22.18
23.20
25.40
F01.
07 07
16.00
65.30
76:57
50 50
21.14
51.10
00.30
00,30
1647
Podteko (XCA12
56.70
80:22
10.34
21.30
22.77 oz
23.3
25.0
16.77 is
Pobled (AAN244
75:68
21.29
24.46
25.71
25.0819030
CRONN 30611
28.18
61.8
14.30
03.02
25.82
23.00
26.11
22,00
GL
IIGIN (W9922
6408
10:30
E1701100
63:15
56.47 60.35 35
40 60.47
29.75
2000
27.38
20.00
300
31
Table 6: Comparison with the state-of-the-an methods on three selected real-world datasets. The noise is set 20% and 70,, and the number of clients M is set 10 throughout all experiments. The best and second -ooot results are highlighted with bold and underline. respectively.
node representations, integrating graph contrastive learning for consistent representations across augmented graph views. Finally, CGNN employs an MLP for prediction distributions and iteratively corrects noisy labels by comparing them with their neighbors and choosing the most labels.
RTGNN [WWW23]. RTGNN proposes noise governance framework that combines self- reinforcement supervision for noisy label correction and consistency regularization to prevent overfitting. The method categorizes labels into clean and noisy types, then applies adaptive supervi- sion by rectifying inaccurate labels and generating pseudo-label, for unlabeled nodes, enabling effective learning from clean labels while mitigating noise impact.
CLNode [WSDM24). CL.Node adopt curriculum learning strategy to mitigate the impact of label noise. To be specific, first utilize a pulti-perspecttiv difficulty measurer to accurately measure the quality of training nodes. Then employ a training scheduler that selects appropriate training nodes to train GNN in each epoch based on the measured qualities. The authors demonstrated this method enhances the robustness of backbone GNN to label noise.
C,3 Implementation Details.
The experiments are conducted using NVIDIA GeForce RTX 4090 GPUs as the hardware platform, coupled with Intel(R) Xeon(R) Platinum 8336C CPU @ 2.30GHzz. The deep learning framework employed was Pytorch, version 2.5.1, alongside CUDA version 12.2. Our network features a four- layer GCN backbone with uniform 384-dimensional hidden layers throughout the first three layers, each employing symmetric normalization (normalize=True) and ReLU activation, followed by 0.2 dropout for regularization. The final GCN layer produces compact 32-dimensional graph embeddings without activation. These embeddings are processed through a two-layer MLP head with ReLU activation in the hidden layer. The architecture optionally incorporates prototype learning with configurable parameters: each class maintains multiple 32- 32-dimensional prototype vectors, and the prototype contrastive loss operates with temperature parameter r = 0.07 to control the similarity distribution sharpness. All GCN layers implement symmetric normalization (normalize=Truue, and consistent dropout (p- = 0.2) is applied after each intermediate layer to prevent overfitting. TP-HSL parameter is set in the range (0.40, 0.50, 0.60), B Is set in the range {0.65, 0.70, 0.75).As for the range {0.03, 0.04], 7 in the range {0.92, 0.94, 0.96). The number of communication rounds is 100 for all methods. The number of clients M is set to 10 throughout all experiments, except for Figure (Tirdd).
GA-SHP parameter À and 7,, we set
D Additional Experimental Results.
We place additional F1-macro score results under 0.2 and 0.7 noisy label ratios in Tab.
E Broader Impact.
Our work is an important step in overcoming the widespread and imperceptible labeling noise in FGL. This approach can effectively enhance the topological attention of the model to discriminate the
15

---

noise. This could lead to more robust and trustworthy graph learning systems in real-world federated environments, where data quality and consistency are often difficult guarantee.
F
Discussion on Limitations.
Although HYPERION has demonstrated significant success in efficiently capturing subtle topological differences between nodes of the same class and mitigating malicious noise through a hyperspherical representation, it still faces some limitations. Specifically, our current formulation primarily addresses class label noise, while other noise types (e.g., feature noise or adversarial edge perturbations) may require additional mechanisms beyond the proposed purification framework.
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26


Reviews:
[{"review_id": "hR0w0M7I5u", "reviewer": "Reviewer_4hFk", "summary": "This work presents HYPERION which is a collection of 3 complementary techniques to deal with graph learning in the presence of noisy/incorrect node labels. The basic foundation of their approach relies on projecting the node features into a hypersphere representation  where nodes with similar features end up on the same region on the hypersphere. Based on that assumption the second step randomly drops edges in the graph and re-assesses the node's hypersphere neighborhood. For nodes that are stable the neighborhood stays the same in presence of small perturbations in the graph. When such stable nodes are identified they are used as the clean nodes whose labels can be more trusted. Finally, the third technique uses the global knowledge of information gained from each client in a federated setting to provide a mechanism to identify noisy nodes and propagate that knowledge to each client so that the global model stays robust. These 3 techniques are combined into a single framework called HYPERION which is then evaluated extensively over multiple datasets to quantify the performance.", "questions": "I have two questions I posed regarding the assumptions and theoretical foundation for the assumptions made in the work. I would appreciate some insight or additional commentary on the two above mentioned weaknesses.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "+ The work presents a novel approach to projecting the node features into a hyperdimensional space to identify node clusters. \n+ The 3 components are tied nicely together to generate a unified framework. \n+ Results are sufficiently detailed. \n\n- The main criticism is that the assumption that node features projected into hyperdimensional space can automatically form clusters that can be used to identify node label noise. It is unclear whether graph nodes with nearly identical features can potentially have different labels. For instance, consider two nodes that share many common features except their gender value. Their labels may infact differ vastly given that some features contribute  more to the overall label than other features. But that is treated as noise in the presented setting which may not be a good assumption. \n- On the same vein, identifying stable nodes by randomly deleting edges or other nodes in the graph is also an empirical observation without much sound theoretical basis. In fact a few key edges may change the learning behavior. \n\nAddressing the above two assumptions is important to make the paper more interesting to the audience.", "was_revised": true, "final_justification": "Overall the work has some promising results in how GNNs can be trained more efficiently using hyperspherical embeddings. I must add the caveat that my understanding of the formal proofs provided is not strong. Hence, I don't have a strong opinion on whether the paper's formal contributions are worthy of publication or otherwise. I will leave that judgement to other reviewers. From my view I thought the paper presented an interesting solution to robust GNNs using a technique (hyperspherical embeddings) that I did not know about before. Hence, I would like to see the paper be accepted, but as I noted in my review my confidence level is low.", "rebuttal": "Thank you for your thorough review and encouraging feedback. We are grateful for your positive assessment about novelty and importance of our work. We address your concerns below and hope our responses will help update your score:\n\n> `Weaknesses 1`: The assumption that projecting node features into a hyperdimensional space forms clusters to identify label noise is flawed, as nodes with similar features may have different labels, which could be misinterpreted as noise.\n\nThank you for this question. We sincerely appreciate the opportunity to clarify the inner workings of the HYPERION method.\n\nDuring training, if two nodes have highly similar features but different labels, we do not necessarily consider this as noise. In fact, **the noise detection mechanism in HYPERION primarily comes from HS-CNC, as introduced in Section 3.3.** This mechanism determines whether a node has label noise based on the stability of node predictions after actively adding noise. **TP-HSL, described in Section 3.2, is mainly designed to achieve clustering of nodes from different classes and does not specifically identify label noise.** After projecting the node features into the hyperdimensional space, TP-HSL uses a corresponding loss function to encourage significant differences in the hyperdimensional representations of nodes from different classes, thus achieving clustering. In the case you mentioned, where two nodes have highly similar features but different labels within a client, HYPERION can effectively address this issue. These two nodes will ultimately be projected to different locations in the hyperdimensional space due to TP-HSL, as the projection function has learned higher weights for the features that contribute to the difference in labels for these similar nodes. **If label noise indeed exists for either of these nodes, HS-CNC will remove the noise based on the stability of the node after perturbations.**\n\nWe sincerely appreciate this insightful observation. In response to the valuable feedback,we will enhance our methodological presentation in the revised manuscript to provide a more comprehensive explanation of the node projection mechanism and its theoretical underpinnings.\n\n> `Weaknesses 2`: The approach of identifying stable nodes by randomly deleting edges or other nodes lacks a strong theoretical foundation, as removing a few key edges could significantly alter the learning behavior.\n\nAlthough the identification of stable nodes through perturbation (such as random deletion of edges or nodes) is empirical, it serves as a heuristic method that addresses the practical challenge of noise in graph data within federated learning environments. The empirical approach we adopt is designed to mitigate the impact of noise on the model. While this method is not grounded in traditional graph theory, **its effectiveness has been demonstrated in our experiments.** As shown in *Table 1* and *Figure 3* of the paper, HYPERION consistently outperforms existing methods across various noise levels, supporting the validity of this empirical strategy. Additionally, **previous works have also employed similar methods such as random edge deletion or feature modification [1, 2],** and these approaches have been widely validated as effective.\n\nFurthermore, this method does not involve the actual removal of edges in the graph; instead, it randomly deletes certain edges or modifies node features to assess the stability of the nodes. **The graph input to the model remains the original graph, i.e., the undisturbed graph.** The loss function is computed solely based on those nodes that exhibit stability under perturbations. We sincerely thank you for this question. We will include more detailed analyses in the revised version.\n\nWe hope this clarifies your questions.\n\n[1]: Deep graph contrastive representation learning. In ICML, 2020. \n\n[2]: Contrastive learning of graphs under label noise. In Neural Networks, 2024."}, {"review_id": "sVryaOlCSr", "reviewer": "Reviewer_kQxL", "summary": "This work introduces HYPERION, a robust federated graph learning framework designed to handle complex topologies and resist malicious noise by leveraging hyperspherical embeddings.\nClients embed local node samples onto a hypersphere with class prototypes and generate perturbed views to identify noise-sensitive regions. During global training, a Gaussian Mixture Model filters clean clients. Prototypes from these trusted clients are aggregated and distilled into a global hypersphere, while biased nodes are pruned to ensure model integrity. Experiments confirm that HYPERION outperforms existing methods and shows remarkable resilience to severe label noise.", "questions": "Insufficient methodological justification. The authors employ Mahalanobis distance distillation in GA-HSP but do not provide a thorough rationale for this choice. A more detailed discussion comparing it with alternative distance metrics (e.g., standard L2 distance) would strengthen the methodological foundation.", "limitations": "I have put all my doubts in Weaknesses and Questions.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:  \nS1: Novel and well-motivated. The authors identify the limitations of coarse-grained representations and the neglect of intra-class topological differences in existing robust FGL methods. It innovatively proposes using hyperspherical embeddings to capture fine-grained structural patterns, providing an approach to mitigate noise propagation.  \nS2: Effective and thought-provoking. The introduction of multiple topological prototypes per class allows the model to capture subtle structural variations within the same class, a crucial aspect of graph data that is often overlooked. This design significantly heightens the ability to differentiate noisy signals from valid ones. Good job!  \nS3: Comprehensive framework. The interplay between local noise calibration and global geometric-aware purification creates a holistic defense mechanism against label noise . Furthermore, there has been little work done in this area previously, and the author has indeed solved the problem.  \nS4: Strong empirical performance. The experimental validation is extensive and compelling. HYPERION consistently and significantly outperforms a wide array of state-of-the-art methods across multiple datasets and under various noise conditions.  \n\nWeaknesses:  \nW1: The authors use Mahalanobis distance when pruning and Wasserstein distance when distilling after clustering, but did not explain in detail the characteristics of these two distances and the differences between them.  \nW2: The authors compare the proposed approach with several traditional FL/FGL and robust FL and GL methods, but additional comparisons with more FGL and robust FL and GL methods could strengthen this study.", "was_revised": true, "final_justification": "After rebuttal, some of my concerns have been addressed. overall, this paper is novel, and I will keep the rating.", "rebuttal": "Thank you for the feedback. We address your concerns below and hope our responses will help update your score:\n\n> `Weaknesses 1`: Clarify Mahalanobis vs. Wasserstein distance characteristics.\n\nWe appreciate the opportunity to elaborate on these distance metrics and their respective roles in our approach.\n\n1. The **Mahalanobis distance** is used to measure the distance between a point and a distribution, taking into account the covariance structure of the data. By introducing the inverse of the covariance matrix, Mahalanobis distance can automatically adjust the importance of each dimension, avoiding certain dimensions dominating the distance calculation due to scale differences.In geometry, Mahalanobis distance transforms the data space into a standardized space, making distance calculations more equitable.We identify and remove noise nodes that deviate from the normal distribution by calculating the Mahalanobis distance between node embeddings and the global prototype distribution (as shown in Formula 18). This method effectively filters out noise that is not detected by clients due to local perspective limitations.\n2. The **Wasserstein distance**, measures the minimum cost of transforming one probability distribution into another. It is particularly effective when comparing distributions that are not necessarily aligned or of the same type, making it well-suited for our distillation phase after clustering. By minimizing Wasserstein distance, the probability mass is effectively reallocated between proximate dimensions in the feature space, thereby naturally reinforcing the correlation of benign features between global prototypes, while effectively diminishing the impact of anomalous features.Unlike Mahalanobis distance, which focuses on individual point-to-point distances, Wasserstein distance captures the \"flow\" of information between distributions, making it ideal for distilling information from the clustered nodes while preserving their relational structure.\n\nWe will include more detailed analyses in the revised version.\n\n> `Weaknesses 2`: Add comparisons to more FGL and robust FL/GL methods.\n\nWe agree that expanding the set of methods for comparison would provide a more comprehensive evaluation of our proposed approach. We have conducted additional experiments and included comparisons with a broader set of FGL and robust FL/GL methods. These additional comparisons further strengthen our study and provide a more detailed analysis of the performance of our proposed approach in different settings. The newly added experiments include comparisons with methods FedDC, FedDyn, FedProto, Scaffold and Co-teaching, which we believe are highly relevant to the challenges addressed in our work. The noise type is set to 50%-uniform and we report accuracy (%) and F1-macro (%).\n\nNew Experimental Results:\n\n| Acc         | Cora      | Citeseer  | Pubmed    | Physics   | Amazon-ratings |\n| ----------- | --------- | --------- | --------- | --------- | -------------- |\n| FedDC       | 45.25     | 37.48     | 52.73     | 50.73     | 36.43          |\n| FedDyn      | 46.53     | 32.15     | 74.34     | 72.52     | 36.28          |\n| FedProto    | 40.22     | 32.89     | 45.68     | 56.12     | 37.41          |\n| Scaffold    | 45.8      | 35.26     | 70.37     | 50.26     | 36.51          |\n| Co-teaching | 36.01     | 32        | 65.99     | 52.71     | 37.12          |\n| Ours        | **53.56** | **47.11** | **74.85** | **75.21** | **38.96**      |\n\n| F1-macro    | Cora      | Citeseer  | Pubmed    | Physics   | Amazon-ratings |\n| ----------- | --------- | --------- | --------- | --------- | -------------- |\n| FedDC       | 42.25     | 34.37     | 37.17     | 13.46     | 11.04          |\n| FedDyn      | 39.82     | 30.5      | 73.65     | 44.92     | 18.04          |\n| FedProto    | 36.43     | 29.67     | 30.16     | 40.24     | 21.16          |\n| Scaffold    | 43.53     | 33.28     | 70.6      | 21.23     | 11.13          |\n| Co-teaching | 33.56     | 30.5      | 64.69     | 14.34     | 20.23          |\n| Ours        | **51.15** | **40.25** | **73.74** | **45.71** | **22.77**      |\n\nWe acknowledge the value of testing on diverse systems and will include more comprehensive results in the revised version.\n\n> `Questions`: Justify Mahalanobis over L2 distance in GA-HSP.\n\nWe are happy to provide additional details on why we opted for Mahalanobis distance and how it compares with alternative metrics, such as the standard L2 distance.\n\nMahalanobis distance is particularly suitable for our method due to its ability to account for the covariance structure of the data. Unlike the standard L2 distance, which assumes that the data points are independent and identically distributed (i.i.d.), the Mahalanobis distance adjusts for correlations between features. This is important in graph-based learning settings, where node features often exhibit dependencies that cannot be captured by simple Euclidean measures.\nIn our case, using Mahalanobis distance allows the GA-HSP method to better capture the relationships between nodes in the graph, considering their high-dimensional feature space. By incorporating the covariance matrix of the data, Mahalanobis distance normalizes the feature space, making it more robust to differences in feature scaling and correlations between features. This makes it more appropriate for distilling node representations in the graph, where such dependencies are common.\n\nThe L2 distance is a simpler metric that calculates the straight-line distance between two points in the feature space. While it is computationally efficient and commonly used, it assumes that all features are equally important and independent, which may not be the case in graph learning tasks. In settings where feature dependencies exist (e.g., in graph data with highly correlated features), L2 distance might not capture the true relationships between the nodes effectively, potentially leading to less accurate results.\nIn contrast, Mahalanobis distance is sensitive to the correlation between features and adjusts for the underlying structure of the data. It takes into account the covariance matrix, allowing it to measure distances in a more meaningful way when dealing with correlated or non-i.i.d. data. In our experiments, we observed that using Mahalanobis distance for distillation provided more accurate and stable results, particularly in scenarios where feature correlations played a significant role in the graph structure.\n\nWe hope this clarifies your questions."}, {"review_id": "xkSwqyPPiX", "reviewer": "Reviewer_MNA6", "summary": "HYPERION proposes a federated graph learning framework that handles noisy labels and complex topologies using hyperspherical embeddings. Its multi-prototype design captures fine-grained structures, while noise calibration and purification (via Wasserstein distillation and Mahalanobis-based methods) enhance robustness. Experiments show strong performance across datasets and noise levels, outperforming existing methods.", "questions": "See Strengths And Weaknesses.", "limitations": "In the experiment, the authors compared several robust graph learning methods, but they did not explain how to apply these methods in a federated learning scenario.I hope the authors will introduce how they applied these methods to the federal framework.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\n1. This paper points out several main reasons why noise problems in FGL are difficult to solve. Subtle differences in graph structures and the classic differences in client data distribution in federated learning indeed make it difficult to effectively identify noise. The problem illustration clearly shows this phenomenon. The author's writing and illustrations made it easy for me to understand his intentions.\n\n2. The proposed method of robust FGL appears interesting and innovative. The paper provides a detailed explanation of the methodology. The paper considers the difference between normal topological variations and subtle noise, introducing Topological Prototypes Hyperspherical Learning to fully capture the rich topological differences between nodes of the same class. I find this approach inspiring.\n\n3. This paper conducted comprehensive experiments, compared multiple methods, and provided a persuasive explanation of the experimental results. The proposed method makes sense for the problem. The node classification task is widely used as a downstream task in FGL methods and is practically applicable. Overall, the approach appears to be both methodologically sound and relevant to real-world scenarios.\n\n\nWeaknesses:\n\n1. On page 9, the author discusses the impact of different numbers of prototypes in prototype clusters on performance under different noise conditions. However, the author does not introduce the experimental results under the F1-macro metric, nor does he provide information on other hyperparameters besides K for this experiment.\n\n2. The TP-HSL component of your approach introduces multiple prototypes per class. Why does this method enable fine-grained structural learning?", "was_revised": true, "final_justification": "I have carefully read the author's response and the other reviewers' comments. The rebuttal has addressed the major concerns raised during the initial review phase. Therefore, I maintain my original score of accept.", "rebuttal": "Thank you for the feedback. We address your concerns below and hope our responses will help update your score:\n\n> `Weaknesses 1`: Add F1-macro results and full hyperparameter details for prototype clusters (p9).\n\nThank you for your constructive feedback regarding the presentation of experimental results. We appreciate your suggestion to include the F1-macro metric and provide more details about the hyperparameters used in the experiment.We have now included the F1-macro results for the experiment involving different numbers of prototypes in prototype clusters under various noise conditions.The experimental results are shown in the table below.\n\n| F1-macro | 20%       | 50%       | 70%       |\n| -------- | --------- | --------- | --------- |\n| K=1      | 52.13     | 36.78     | 20.53     |\n| K=2      | 52.24     | 37.92     | 21.39     |\n| K=3      | **56.58** | **41.11** | 24.48     |\n| K=4      | 55.96     | 36.02     | **25.02** |\n\nFrom the perspective of F1-macro, it has been confirmed once again that setting *K* = 1 results in under-learning of hypersphere, leading to consistently lower performance.In contrast, increasing *K* to 3 or 4 yields marginal performance improvements.\n\nWe also understand the importance of clarifying the impact of other hyperparameters on the experiment. Specifically, we split the dataset into training, validation, and test sets in a 20%, 40%, and 40% ratio. We used the Louvain partitioner to induce data skew. The experiments were run for 100 rounds, involving 10 clients. Each client was trained for 3 local epochs with a batch size of 24. The learning rate was set to 0.005, and we introduced pairwise label noise to simulate real-world imperfections. Our network features a four-layer GCN backbone with uniform 384-dimensional hidden layers throughout the first three layers,each employing symmetric normalization (normalize=True) and ReLU activation, followed by 0.2 dropout for regularization. The final GCN layer produces compact 32-dimensional graph embeddings without activation. These embeddings are processed through a two-layer MLP head with ReLU activation in the hidden layer. The architecture optionally incorporates prototype learning with configurable parameters: each class maintains multiple 32-dimensional prototype vectors, and the prototype contrastive loss operates with a temperature parameter *τ* = 0*.*07 to control the similarity distribution sharpness. All GCN layers implement symmetric normalization (normalize=True), and consistent dropout (*p* = 0*.*2) is applied after each intermediate layer to prevent overfitting.\n\nWe acknowledge the value of testing on diverse systems and will include more comprehensive results including F1-macro in the revised version.\n\n> `Weaknesses 2`: Clarify how TP-HSL's multi-prototype design enables fine-grained structural learning (e.g., through prototype diversity or cluster separation).\n\nIn our TP-HSL method, we utilize the von Mises-Fisher (vmF) distribution to model the distribution of node embeddings. The vmF distribution is well-suited for accurately measuring the angular differences between embeddings, especially in high-dimensional spaces. In graph data, node embeddings are typically high-dimensional, and the vmF distribution, through its concentration parameter, effectively quantifies the similarity and dissimilarity between node embeddings. This angular difference measure not only helps capture distributional differences between classes but also provides a deeper understanding of each node's local structural characteristics in the high-dimensional space. In particular, in graphs with complex topological structures, the vmF distribution offers a precise and efficient way to assess subtle differences between nodes.\n\nAdditionally, traditional graph learning methods often assign a single prototype to each class, which overlooks the structural variations within the nodes of the same class. In our TP-HSL method, we design multiple prototypes for each class, which help capture the diversity among nodes within the same class in the high-dimensional space. Each prototype represents a distinct topological feature within the class, allowing the class to be partitioned into multiple subspaces, each represented by one prototype. This approach ensures that the node embeddings within each class are better aligned with the corresponding prototypes, thereby improving both intra-class cohesion and inter-class distinction. By doing so, we can finely capture the structural differences within each class.\n\nWe sincerely thank you for this suggestion. We will enhance the description of TP-HSL in the revised version.\n\n> `Limitations`: Clarify how the compared robust graph learning methods were adapted for federated learning in the framework.\n\nIn our experiments, each robust GL baseline (e.g., CRGNN, RTGNN, CLNode) was applied independently on each client. Specifically:\n\n**Client-Side:** On each client m, the robust GL method was employed to train a local model using the client’s private graph data G_{m}. The original noise-resilience mechanisms proposed in these methods—such as label correction, contrastive learning, or topology-aware regularization—were preserved and applied locally without modification.\n\n**Server-Side:** After local training, model updates from clients were aggregated on the server using the standard FedAvg algorithm. No changes were made to the server-side aggregation logic for these baselines, ensuring consistent treatment across all compared methods.We selected FedAvg as the aggregation strategy due to its simplicity, stability, and widespread use in FL research. More importantly, using a uniform aggregation protocol eliminates the influence of advanced server-side techniques, allowing us to isolate and fairly compare the effectiveness of each robust GL metho’s local training design under federated conditions.\n\nWe sincerely thank you for this suggestion. We will enhance the description of implementation details in the revised version.\n\nWe hope this clarifies your questions."}, {"review_id": "ewuNxO5fVx", "reviewer": "Reviewer_XQTy", "summary": "This study tackles challenges such as non-IID graph data distribution and adversarial noise perturbation in federated graph learning. It introduces a robust strategy where clients collaboratively learn hyperspherical embeddings to enforce geometric consistency across heterogeneous local graphs, mitigating feature skew. To address the classic differences in client data distribution in federated learning, the approach incorporates Geometric-Aware Hyperspherical Purification. Lastly, adequate experiments are conducted to verify the model effectiveness.", "questions": "1. The author did not introduce how to create enhanced views in HS-CNC.\n2. Why did the author project the nodes onto the hypersphere? Can the author explain the benefits of doing so?", "limitations": "My complete set of queries and concerns is captured in the 'Weaknesses' and 'Questions' part.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. It is novel and meaningful to propose a robust FGL approach. This study precisely identifies the noise challenges of FGL and tackles them effectively. The overall approach is well-motivated, and the motivation is also explained with clarity.\n\n2. The figures of problem and framework illustration are clear and detailed, and the equations and explanations are reasonable. I believe this is a comprehensive framework for the graph data.\n\n3. The approach proposed in the study is consistent and natural. The key modules and their components are innovative and logically coherent. I believe this approach is inspiring.\n\nWeaknesses:\n\n1. The work lacks analysis of model performance with different local training epochs. Can the author supplement the training effects of his method under different training rounds?\n\n2. It would be better if the reasons for the selection of comparison algorithms and datasets were explained.", "was_revised": true, "final_justification": "My initial review was generally correct, and the authors addressed my concerns well, so I am leaving my original score unchanged.", "rebuttal": "Thank you for the feedback. Regarding your questions, we would like to clarify the following points:\n\n> `Weaknesses 1`: Add analysis of model performance across different local training epochs/rounds.\n\nWe greatly appreciate these insightful suggestions.We have conducted additional experiments to analyze the performance of our method under different local training epochs. Specifically, we varied the number of local training epochs and observed how this impacted the overall model performance:\n\n| Acc      | 25        | 50        | 75        | 100       |\n| -------- | --------- | --------- | --------- | --------- |\n| FedAvg   | 38.48     | 34.1      | 33.27     | 32.91     |\n| MOON     | 38.48     | 35.47     | 33.76     | 33.64     |\n| FedNed   | 34.64     | 34.10     | 33.82     | 32.82     |\n| CRGNN    | 49.27     | 46.98     | 45.78     | 44.61     |\n| HYPERION | **55.24** | **54.34** | **53.75** | **53.56** |\n\nWe report Acc values on Cora dataset with 50%-uniform noise. The results clearly show that HYPERION consistently outperforms the other methods across all local training epochs. As the number of epochs increases, we observe a slight decline in performance for all methods, which is typical due to the overfitting risk and the noise in the dataset. However, HYPERION maintains a significant margin over the competing methods even at higher epoch counts, indicating its robustness to noisy labels and its ability to generalize better across different training configurations. We will include more detailed analyses in the revised version.\n\n> `Weaknesses 2`: Justify the selection of comparison algorithms and datasets.\n\nWe compare our method, HYPERION, with several state-of-the-art federated learning (FL) and federated graph learning (FGL) methods to comprehensively evaluate its robustness and effectiveness under noisy conditions. These comparison methods include FedAvg, FedProx, FedNova, MOON, and others, which are widely recognized in the FL and FGL communities. We included them because they represent the current standards in federated learning, especially in settings with label noise.Additionally, Robust FL methods like FedNoRo, FedNed, and FedCorr are also included to evaluate how HYPERION compares to methods specifically designed to address noise in federated settings. In addition, we also considered several robust graph learning algorithms, which have demonstrated excellent robustness in the presence of noisy labels in graph data, but have not addressed issues such as the client-side limitations in federated learning.The comparison with these algorithms helps us demonstrate HYPERION’s superior robustness in scenarios where noise is a major challenge.\n\nAs for the choice of datasets, we used five benchmark datasets: Cora, CiteSeer, PubMed, Physics, and Amazon_ratings. These datasets were selected because they represent a range of real-world graph data types with varying complexities and noise profiles.\nCora, CiteSeer, and PubMed are well-established citation networks commonly used in graph learning tasks, and they exhibit challenges such as node sparsity and high feature dimensionality, making them suitable for testing the robustness of our method under label noise.The Amazon_ratings dataset introduces a real-world, user-item co-purchase graph, which has more complex structural properties and is frequently used to benchmark collaborative filtering algorithms. This dataset helps us evaluate how HYPERION performs in scenarios involving collaborative data.The Coauthor-Physics dataset provides a more specialized network of co-author relationships, testing HYPERION’s ability to handle graphs from specific academic domains, demonstrating its applicability to different types of graph-structured data.The details of the datasets are shown in the table below.\n\n| Dataset          | Nodes  | Edges   | Classes | Features |\n| ---------------- | ------ | ------- | ------- | -------- |\n| Cora             | 2,708  | 5,278   | 7       | 1433     |\n| Citeseer         | 3,327  | 4,552   | 6       | 3703     |\n| Pubmed           | 19,717 | 44,324  | 3       | 500      |\n| Amazon-ratings   | 24,492 | 97,050  | 5       | 300      |\n| Coauthor-Physics | 34,493 | 530,417 | 5       | 8,415    |\n\nWe sincerely thank you for this suggestion. We will enhance the description of algorithms and datasets in the revised version.\n\n> `Questions 1`: Clarify the enhanced view generation process in HS-CNC.\n\nIn the HS-CNC module, enhanced views are created through a dual perturbation strategy that combines structural and feature-level augmentations to identify noisy nodes. Following prior work [1, 2], our method first applies random edge dropping to the adjacency matrix with a predefined probability, disrupting local graph connectivity while preserving global structure. Simultaneously, it randomly masks a portion of node features to simulate semantic variations, generating perturbed versions of the original graph data. These augmented views are then processed through the same model to obtain hyperspherical embeddings, where nodes demonstrating consistent cluster assignments across different perturbed views are retained as reliable samples while inconsistent nodes are filtered as potential noise. This approach effectively amplifies the difference between stable clean nodes and unstable noisy nodes by evaluating their robustness to controlled perturbations, enabling more accurate noise identification without requiring explicit label correction. The view generation process is designed to maintain the intrinsic graph properties while introducing sufficient variation to test node stability, striking a balance between preserving meaningful patterns and exposing noise-sensitive regions of the data.\n\n> `Questions 2`: Explain the advantages of hyperspherical node projection in the method.\n\nBy constraining embeddings to a unit sphere, the model leverages angular distance (cosine similarity) rather than Euclidean distance, which naturally amplifies inter-class separation while compacting intra-class clusters. This geometric property makes the model more sensitive to meaningful semantic relationships while being less susceptible to noise and feature magnitude variations. The hyperspherical space enables more stable prototype learning through von Mises-Fisher distributions, facilitates reliable federated aggregation via directional consistency, and provides a principled framework for noise purification through angular outlier detection. Compared to traditional Euclidean spaces, the hyperspherical projection offers superior noise robustness by encoding semantic similarity in angular relationships rather than absolute positions, which is particularly crucial for handling label noise and structural variations in decentralized graph data. This approach allows the model to better capture fine-grained topological differences while maintaining strong generalization across clients.\n\nWe sincerely thank you for this suggestion. We will enhance the description of hyperspherical node projection in the revised version.\n\nWe hope this clarifies your questions.\n\n[1]: Deep graph contrastive representation learning. In ICML, 2020. \n\n[2]: Contrastive learning of graphs under label noise. In Neural Networks, 2024."}]

Author rebuttal:

Meta-review: This paper proposes a new framework with three complementary techniques for graph learning with noisy labels. It first projects node features onto a hypersphere to group similar nodes. Then, by randomly dropping edges and re-evaluating neighborhoods, it identifies stable nodes as reliable "clean" labels. Finally, in a federated setting, it uses global information gain to detect noisy nodes and shares this knowledge to enhance global model robustness. 

Strengths:

1. The method is novel and well-motivated. The paper firstly proposes using hyperspherical embeddings to capture fine-grained structural patterns, providing an approach to mitigate noise propagation.

2. The method is effective. The experimental validation is extensive and compelling across multiple datasets and under various noise conditions.

Weaknesses:
Two distances including Mahalanobis distance and Wasserstein distance are directly used for pruning and distilling separately, without sufficient explanation of the motivation and the theoretical validation. 

All reviewers keep positive to the paper. The authors also provide a reasonable explanations to the concerns raised by reviewers. Therefore, I recommend the acceptance of the paper.
Decision: Accept (spotlight)