Federated Node-Level Clustering Network with Cross-Subgraph Link Mending
Authors: Jingxin Liu
Renda Han
Wenxuan Tu
Haotian Wang
Junlong Wu
Jieren Cheng
Venue: ICML 2025 poster (2025)
Keywords: Federated Graph Learning
Muti-view Learning
Deep Clustering
Primary area: general_machine_learning->clustering

Abstract: Subgraphs of a complete graph are usually distributed across multiple devices and can only be accessed locally because the raw data cannot be directly shared. However, existing node-level federated graph learning suffers from at least one of the following issues: 1) heavily relying on labeled graph samples that are difficult to obtain in real-world applications, and 2) partitioning a complete graph into several subgraphs inevitably causes missing links, leading to sub-optimal sample representations. To solve these issues, we propose a novel $\underline{\text{Fed}}$erated $\underline{\text{N}}$ode-level $\underline{\text{C}}$lustering $\underline{\text{N}}$etwork (FedNCN), which mends the destroyed cross-subgraph links using clustering prior knowledge. Specifically, within each client, we first design an MLP-based projector to implicitly preserve key clustering properties of a subgraph in a denoising learning-like manner, and then upload the resultant clustering signals that are hard to reconstruct for subsequent cross-subgraph links restoration. In the server, we maximize the potential affinity between subgraphs stemming from clustering signals by graph similarity estimation and minimize redundant links via the N-Cut criterion. Moreover, we employ a GNN-based generator to learn consensus prototypes from this mended graph, enabling the MLP-GNN joint-optimized learner to enhance data privacy during data transmission and further promote the local model for better clustering. Extensive experiments demonstrate the superiority of FedNCN.

Federated Node-Level Clustering Network with Cross-Subgraph Link Mending
Jingxin Liu
Renda Han?
Wenxuan Tu
Haotian Wang? Junlong Wu Jieren Cheng?
Abstract
1.0
0.5
Subgraphs of a complete graph are usually dis- tributed across multiple devices and can only be accessed locally because the raw data cannot be directly shared. However, existing node-level fed- erated graph learning suffers from at least one of the following issues: 1) heavily relying on labeled graph samples that are difficult to obtain in real- world applications, and 2) partitioning a complete graph into several subgraphs inevitably causes missing links, leading to sub-optimal sample rep- resentations. To solve these issues, we propose a novel Federated Node-level Clustering Network (FedNCN), which mends the destroyed cross- subgraph links using clustering prior knowledge. Specifically, within each client, we first design an MLP-based projector to implicitly preserve key clustering properties of a subgraph in a denoising learning- like manner, and then upload the resul- tant clustering signals that are hard reconstruct for subsequent cubss-subgraph links restoration. In the server, we maximize the potential affinity between subgraphs stemming from clustering sig- nals by graph similarity estimation and minimize redundant links via the N-Cut criterion. More- over, we employ GNN-based generator to learn consensus prototypes from this mended graph, enabling the MLP-GNN joint-optimized learner to enhance data privacy during data transmission and further promote the local model for better clustering. Extensive experiments demonstrate the superiority of FedNCN.
Original Destroyed Mended structure
Figure I. Research motivation for cross-subgraph link mending. We visualize the encoded raw graph structure using heat maps. The similarities of the destroyed and mended topologies relative the original topology are 44.74% and 78.45%, respectively.
1. Introduction
Graph machine learning (GML) is a well-established tech- nique that harnesses the structure of graph data to facilitate various learning tasks. A large amount of GML. methods have recently been developed and achieved satisfactory per- formance (Liang et al., 2023). Commonly, almost all of them assume that the graph data is centralized and avail- able unconditionally. However, numerous real-world sce- narios, such medical data analysis (Huang al., 2024), smart transportation (Chen et al., 2024), and social networks (Chen et al., 2022), a complete graph is usually partitioned into multiple subgraphs, with each one distributed to an isolated client without directly exposing the raw data. In this background, node-level federated graph learning (FGL) has emerged as a promising solution and has attracted lot of research attention in recent years.
To be brief, node-level FGL is machine learning paradigm composed of multiple isolated clients and an independent server (Back et al., 2023; Zhu et al., 2024; Li et al., 2024) to group the distributed subgraphs into categories while en- suring private data not directly shared. Typically, each client trains local model parameters with its own label sig- nals and sends them to the server, while the server integrates the uploaded signals compute consensus subsequently sent back to all clients to update the local models. Despite their encouraging successes, we find that these studies have at least one non-negligible limitation: 1) heavily relying on labeled graph samples that are difficult to obtain in real-world applications (Zhang et al., 2021; Tan
'School of Cyberspace Security, Hainan University, Haikou, China "School of Computer Science and Technol- ogy, Hainan University, Haikou, China. Correspondence to: Wenxuan Tu <twx @ www.aaau.cccuuc.......... Jieren Cheng <992730@ wwwnaau.cuu.cnn
that is
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

Federated Node-Level Clustering Network with Cross-Subgraph Link Mending
et al., 2024; Wan et al., 2024). Once reliable supervised signals are unavailable, existing methods struggle to effec- tively learn high-quality representations, which adversely affects the quality of multi-source information negotiation at the server; and 2) partitioning a complete graph into sev- eral subgraphs of random sizes inevitably causes substantial amounts of missing links between clients (Zhu et al., 2024; Back et al., 2023). As illustrated in Fig. 1, taking the results on Citeseer an example, more than 55% of the sample con- nections are destroyed during the graph partitioning process. As the number of clients increases, the disruption of links between samples would become more severe. Consequently, it is urgent to develop novel unsupervised node-level FGL framework to handle the aforementioned issues.
Our main contributions can be summarized as follows:
New research task. To the best of our knowledge, we make the first attempt to tackle the issue of link missing caused by graph partition in federated node- level clustering.
Novel FGL framework. A Federated Node-level Clustering Network (FedNCN) is proposed. It not only effectively mends cross-subgraph links, but also pro- motes the great encoding capacity of the local model for better clustering.
Better clustering results. Extensive experiments on five graph benchmark datasets demonstrate the effec- tiveness and superiority of the proposed FedNCN com- pared its competitors.
An intuitive solution involves leveraging clustering signals to facilitate the restoration of missing links across subgraphs on the server, where the mended graph is in turn utilized to boost the performance of each client. To fulfill this, there are two key challenges to be solved: 1) how to collect and preserve key clustering properties from multiple unlabeled subgraphs without directly sharing private information; and 2) how to establish correct topology relationships among subgraphs with the aid of provided pseudo-supervised in- formation. For the first challenge, inspired by the proto- type learning (Liu et al., 2025; Wan et al., 2024), we first select representative nodes from each cluster within the client, and then design a clustcring projector to generate hard-to-reconstruuct counterparts of the selected samples as well as preserve the clustering characteristics from the local model. For the second challenge, we attempt to construct the potential affinity between cross-subgraphs stemming from uploaded clustering signals via graph similarity estimation.
2. Related Work
2.1. Federated Graph Learning
Federated Learning (FL) is widely used distributed ma- chine learning framework that enables multiple clients to collaborate in training global model through central server, while avoiding direct sharing of raw data (Wang et al., 202 a0b;;;b; 2023a; 2024b;c;e; Meng et al., 2024; Wang et al., 2023b; 2024a;d). Due to its ability to effectively lever- age various technologies for application extension, FL has been rapidly developed in recent years, particularly in graph data analysis. For example, FedSage+ (Zhang al., 2021) utilizes node labels to design missing neighbor generator, addressing the issue of missing links in distributed subgraph systems. Moreover, FedPUB (Baek et al., 2023) leverages labeled data to generate a global random graph, which is then fed back to clients to obtain multiple functional embed- dings. These embeddings are used to compute similarities for identifying different communities, thereby mitigating multi-source heterogeneity. Similarly, FedTAD (Zhu et al., 2024) assesses the reliability of node class knowledge in a topology-awar manner, which is then uploaded to the server to guide pseudo-grand -gaph generation for better classification. Previous studies have demonstrated that the FGI. frame- work can be integrated with some advanced techniques and achieve promising performance with the aid of data labels. In contrast, to adapt to the unlabeled scenario, we design a clustering projector that generates hard-to-recom carostttreconttructt coun- terparts while preserving the clustering properties of each cluster for better clustering.
Based on the above observations, we propose a novel fed- erated node-level clustering framework termed FedNCN. The core idea of FedNCN is to mend the destroyed links using clustering prior knowledge. Specifically, within each client, we develop MLP--based projector implicitly pre- serve the key clustering properties (i.e., prototypes) of each subgraph through a denoising- learning approach. Sub- sequently, in the server, we maximize the potential affinity of subgraphs derived from uploaded clustering signals that are hard to reconstruct into the raw data via graph similarity estimation. Meanwhile, we employ the improved N-Cut operation decrease the redundant sample connections within the mended graph much as possible. After that, GNN-based generator is designed to learn consensus pro- totypes based on the mended graph, where the MLP-GNN joint-optimized learner transmits the updated prototypes and model parameters back to each client while protecting data privacy during transmission. As shown in Fig. I, with the proposed cross-subgraph link mending scheme, our method can effectively restore the missing links from 44.74% to 78.4 33.4%% for better clustering performance (see Section 4.2).
2.2. Attributed Graph Clustering
Benefiting from the strong generalization ability of the GNN in handling graph data, significant progress has been made in node-level clustering tasks in recent years (Tu et al., 2021; Li et al., 2022; Gong et al., 2022a;b; Pan & Kang, 2023;
2

---

Federated Node-Level Clustering Network with Cross-Suggraph Link Mending
Guan et al., 2025). CCGC (Yang et al., 2023) constructs two views of the complete graph and utilizes siamese encoder to guide the generation of positive and negative sample pairs, thereby improving clustering performance. Similarly, MAGC (Lin et al., 2021) mines complementary information from multi-view data using weight factors to learn consistency and discriminative relationships. Further- more, AMGC (Tu et al., 2024a) designs a unified framework that alternately optimizes clustering and attribute imputa- tion processes on single graph. Commonly, these methods assume that the graph data is centralized. However, this assumption is overly strict in real-world scenarios, as sub- graphs of a complete graph are usually distributed across multiple devices and are accessible only locally, which in- evitably leads missing links between subgraphs. To this end, we construct cross-subgraph potential affinity relation- ships based prior clustering knowledge to restore the destroyed links.
weight matrix the 1-th layer, d' is the dimension of the latent representations, and a(-) is the activation function. The final layer generates embeddings Ž RNN''', We then employ the K K-means algorithm (Xu & Lange, 2019) to generate multiple prototypes C E ROxd', thereby obtaining representative samples from each cluster. Specifically, the Euclidean distance (Mafakheri et al., 2018) between the final embeddings of each node and the prototype of its corresponding subspace is calculated as:
fa(z,, C2) = 11 - -2
(2)
where 2; E Z is the attribute vector of the i-th node, e2 E C is the j-th prototype. In this way, for each prototype, we select the top n² = kN representative nodes approximated to the prototype, where is hyperparameter that represents the ratio of samples selected. Subsequently, we collect the attribute matrix B E O="Oxd of these selected nodes.
Next, to project directed signals with clustering properties without directly sharing the raw data, we generate a masked matrix R E IR" 'xd directly from Gaussian noise, which is hard to be reconstructed into raw graph data. After that, we utilize an MLP model (Liang et al., 2023; 2024; Liu et al., 2021a) as a projector to extract the meaningful clustering signal in denoising-like manner, which makes R possibly approximate to B:
3. Methodology
Fig. shows the architecture of the FedNCN. The core idea mend the cross-subgraph missing links to enhance the clustering performance of each client in an unlabeled circumstance. This process mainly consists of three stages: local model learning, cross-subgraph link mending, and global knowledge sharing.
R = MLP(R; mmlp):
(3)
3.1. Notations
where R E RnrOxd is the reconstructed masked matrix,
Denote by G E) complete undirected graph with O cluster centers, where V and È are the sets of nodes and edges, respectively. Suppose the G is divided into M subgraphs, which are assigned to M clients. For simplic- ity, we take the local learning over a single subgraph g as an example and consider other graphs similarly. Sup- pose that G contains Ô cluster centers and Ñ nodes. Let X E Nxd and A E N N denote the attribute and origi- nal adjacency matrices, where d is the attribute dimension. The corresponding degree matrix is D E NNN With D, the A is further normalized as A E NNN by calculating D (A + I,D 2 where R.Nx indicates identity matrix. All used notations are summarized in Appendix A.
mmlp represents the parameters of the local MLP. The opti- mization objective as follows:
Smlp = n
B R112.
(4)
Finally, we upload R and the well-trained 6mup to the server in preparing for cross-subgraph link mending.
In summary, the proposed local model learning strategy offers two major advantages: it 1) collects and uploads key clustering signals that are hard to reconstruct into raw data, and 2) preserves more trustworthy clustering properties of each cluster within the client, serving as reliable signals for subsequent information sharing between cross-subgraphs.
3.2. Local Model Learning
This section introduces the local model learning for a single subgraph, and the goal is to preserve the key information of each cluster within each unlabeled subgraph. We first employ a GNN model to capture the node embeddings of the subgraph as follows:
3.3. Cross-Subgraph Link Mending
Since a complete graph is divided into multiple subgraphs, leading missing links across subgraphs. Therefore, after the server receives the clustering signals from each client, we propose the cross-subgraph link mending strategy.
(1) where 7 E N xd' denotes the node embeddings of the th encoder layer, and (0)) equals to X. W()) is the learnable
ツ (AZ ())).
In the first step, to infer the global samples on the server, we adopt R and 6mlp uploaded by each client, which en- sures that these representative nodes are approximately to
3

---

Federated Node-Level Clustering Network with Cross-uugrraph Link Mending
Global knowledge sharing
Cross-subgraph link mending
GNN
N-Cut
Prototypes
Model parameters
Server
Minimize redundant links
Local model learning
C
Rebuilt structure
High similarity
Maximize correlated links
Low similarity
ht M
Alignment
Graph kernel
Graph kernel
Graph kernel
Graph kernel
Original samples
True nodes
Predicted nodes
GNN
Node sampling
Optimization
Node affinity construction
Server
Node affinity construction
O
K-meens
Clusters
Clustering signals
Uploaded signals
di
dtn
Uploaded signals M
din
chtn
Figure 2. The framework of FedNCN is composed of three major components, i.c., the local model learning that collects and preserves more trustworthy clustering signals for destroyed sample connection restoration, the cross-subgraph link mending that establishes correct connections among subgraphs with the aid of prior learned clustering knowledge, and the global knowledge sharing that learns high-quality consensus features based on the mended graph and ensures reliable feedhack to each client. Three parts are seamlessly integrated into a unified optimization framework. Notably, the cross-subgraph link mending is the core component that includes four steps, i.e., @ global sample inferring; 2 intra-cluster link mending; @ inter-cluster link mending; and @ global structure refining.
B. Specifically, the server first optimizes the MLP projec- tor based on obtaining multiple masked matrices and MLP model parameters. Then, we inference the global sample matrix R& E pnd within the client by:
In the third step, to restore the cross-subgraph missing links as much as possible, we identify cross- cross-subgraph relation- ships by the graph kernel method and then mend the cross- subgraph link in an approximately optimal manner (Borg- wardt et al., 2020). Specifically, we first calculate the graph- level affinity matrix S E ER x O, to measure relationships between subgraphs generated from clusters:
R = fp R: mlp)),
(5)
where fp(-) denotes the projector function.
Súj GK(GG,
(8)
In the second step, after inferring the R, we cannot di- rectly obtain the intra-cluster links due to privacy constraints. Therefore, attempt to calculate the potential affinity be- tween nodes within cluster in low-dimensional proximity manner (Yang et al., 2024). Specifically, we first divide the nodes belonging to the same cluster within each client into a single space. Then, we calculate the node similarity:
where GK(.,)) denotes graph kernel similarity function (e.g., COS (Choromanski, 2023), WL (Ju et al., 2023) and SP (Borgwardt et al., 2020)). See Appendix B for de- tailed descriptions of graph kernels. Sij S represents the affinity between G; and 93. Then, we construct edges between any two subgraph pairs generate global graph G (Vo. E... Here, the total number of the G is No EM KN,O,, and the corresponding nodes set is D (ep, ug u2..... 28. ). the edges set is E represented by adjacency matrix A N
F; (T,)
Sim(f., Fy)
(6)
Fi ||y|||
where (im(..) is similarity function, i; E R and it e R
are attribute vector of i-th node and j-th node, respectively. Subsequently, we construct the refined subgraph Poub (Vont ) with n nodes from each cluster by Eq. (6). Here, Eab is the edge set represented by the adjacency matrix A E xx*
In the last step, we design an improved N-Cut method to ensure that the 9 focuses on the links that preserve the key structure, while eliminating redundant links. Specifically, we calculate the contribution of all cross-subgraph edges in the global graph as:
sub
(1, if (02, (g) E Vo and, 0g E TT otherwise
d
= 0,
Con(2 = fa( Vo)
(9)
fa Doj'
where T the target node set of the i-th node the Po
where Con(-) is the contribution function, fa Vo)

---

Federated Node-Level Clustering Network with Cross-Subgraph Link Mending
Algorithm 1 Training Procedure of FedNCN
consensus prototypes C E BOxd and well-trained GNN parameters i @gnn: achieved by minimizing the reconstruction loss LISE and the clustering loss LKL. as:
Input: Complete graph G. For each subgraph: attribute
matrix X; adjacency matrix A; server epoch E client epoch Fellien.
L = LMSE + LAI.
(10)
Output: Clustering results Y at clients.
Initialize ( 0 and o of all clients.
Here, LMSE and LKI. are calculated by:
for i 1 to Eserver do / Local model learning
1 No
MMSE
X 1
(11)
for 1 to Eelient do
CKL = KL(PIIQ) = ΣΣ Piy log
Pij
Generate 2 for each client by Eq. (1).
(12)
qu
end for
Obtain C from Ž by K K-means for each client.
where X E N is reconstructed node attributes matrix of the 9 by GNN decoder, the soft assignment distribution Q E R N x of latent representations is calculated by the
Obtain B with C and by Eq. (2).
Generate R with a shape similar to B using noise. Train the MLP with R. and B by Eqs. (3) and (4). Upload R and 0 mlp to the server.
Student's s t-distribution (Gong al., 2022a), while the target distribution P E R x0 is obtained by sharpening Q.
// Cross-subgrape
link mending
Infer R with R and 0 mip by Eq. (5).
strat- egy provides significant benefits: 1) with the aid of the mended topology information, the global model is enabled to learn consensus sample representations, which ensures reliable feedback to each client to learn features; and 2) the distinet nature of the sub-model pa- rameters, where the locally uploaded mip and the globally returned Ögnn are not identical, which further guarantees privacy security during data transmission.
In summary, the proposed global knowledge-
Generate the subgraph 9 by Eqs. (6) and (7).
Generate global graph 9 by Eq. (8).
Refine the mended graph G by Eq. (9).
// Gl obal knowl edge sharing Train global GNN model by Eqs. (10)-(12).
frienddly
Backpropagate the C and Ognn to each client.
end for
Obtain clustering results Y at clients by K-means.
return Y
3.5. Client-Server Collaborative Learning
After the global model learns the G on the server, each client receives the updated C and ggnn Subsequently, C is employed as the cluster center for the next round of local model training, and gmn is utilized to initialize the parame- ters of the GNN. In this way. the discriminative ability of the local representations could be further enhanced. Finally, client-server collaborative learning is achieved by minimiz- ing the total loss function the local GNN (Similar to Eqs. (10)-(12)) Algorithm provides a detailed description of the FedNCN learning process. The Detailed algorithms for the client and the server provided in Appendix C.
Eiery of and (2 yo) = Eievyg Jev of repre- sent the association of the two subgraphs connected by edge og with the Go, respectively. V and Vy correspond to the node sets of two subgraphs within the GO. In this way, we re- tain the key edges based on their contribution, obtaining the mended graph G and its adjacency matrix A t R XN=
The merits of the proposed cross-subgraph link mending strategy can be summarized as: 1) it effectively leverages the prior learned clustering knowledge to enable the model to accurately conduct cross-subgraph link mending, and 2) a topological refinement approach that maximizes trustworthy connections while minimizing redundant ones improves the topology between subgraphs for better clustering.
4. Experiments
4.1. Experiment Setup
Benchmark Datasets Following the experimental setup from FedTAD (Zhu et al., 2024), we construct distributed subgraphs by dividing the dataset into 5 clients, 10 clients, and 20 clients, respectively, where each client has sub- graph that is part of complete graph. Specifically, CiteSeer (Liu et al., 2023a), PubMed (Jiang et al., 2024), Amazon- Amazon-Computte Computer, Amazon- -hoto (Lin et al., 2021), and Questions (Platonov et al., 2024) as our experimental bench- mark datasets. For a detailed description of the datasets, please see Appendix D.1.
3.4. Global Knowledge Sharing
After achieving the cuoss-subgraph links mending, we aim to learn the missing structure information from the g (V>, Ed with O cluster centers, where 8dh is the edges set, X E No xd denotes the attribute matrix, A E x*"" x is the adjacency matrix. Inspired by previous work (Liu et al., 2025), the prototypes often retain the clustering prop- erties. Consequently, we construct a GNN generator that consistent with the local model at the server to learn
5

---

Federated Node-Level Clustering Network with Cross-Suggraph Link Mending
Table 1. Performance comparison across different federated node-level clustering methods. Notably, all compared methods are evaluated
under unsupervised settings on five benchmark datasets to ensure fair comparison.
CiteSeer (5 Clients)
CiteSeer (10 Clients)
CileSeer (20 Clients)
Methods
ACC
NMI
ARI
F1
ACC
NMI
ARI
F1
ACC
NMI
ARI
F1
FedSago-"
16.13411.74
2.69==0:40
0.18=0.59
11:41+0.755
17.18=2..6
5.43=0.32
-0.13=0.55
01.27=00.2
13.56+2..03
9,54+0.21
0.26+0.52
10.06=0.84
FedPUB
16:40:+3.40
0.00
0.00:20.00
4.11=:0.56
9.05::5.17
0.00:00.00
0.00-10.00
3.05=1662
11.17:6.48
0.00:00000
0.0000.00
3.75::2.044
FedTADD
17.59.1.333
1,21:11443
0.6011.00
7.13:12.55
19.12.1.57
4.011.6633
1.571127
9.8912.69
17.77.1154
3.2233600
1.81
6.77.12.40
FedGTA*
24.05:10.80
1.68:::1.0
3,54
19.95441.05
26.34400.1
8.36+11.60
1.74:::995
19.67-00.71
28.05-4:.62
12.35=2.088
4.3242.088
17.13-2.21
FedUllI"
14.21:12.1 0.00:10:00
0.00::0000
3.73:10.62
67.5
0.69:10.79
0.42:10.99
4.8221100
18.5417.52
0.02::0.05
0.07:0.133
5.2811193
FedNCN
1.78
13.27.12.45
19.6713.71
2.47
2.99
17.53:1.311
15.74.14.01 2..12:1
5..47:1
19.69-1.92
1.72 32.82
Methods
PubMed Clients)
PubMed (10 Clients)
PubMed Clients)
FedSage-
49.53-+6.33
8.67::3777
4.8344.72
33.75++4449
40.71
2.22:44.47
0
26.16::6.01
43.77#8.32
1,96+1,74
-0.0243.557
30.81=3.20
FedPUB'
31.97:19444
0.00:00000
0.00:00.00
14.07:13.11
32.03:99.04
0.001:0.00
0.00:10:00
34.99:3.119
28.30119.25
0.00:00.00
0.00
12.43-3.30
FedTAD"
34.06.14.71 1.16.10:99
0.84.10.4
26.88 1..47
37.63
0.39.10.51
0.54.10.89
2..74.1..87
36.5111.223
0.44
0.72.11.19
18.18:1.664
FedGTA
46.22:42.50
6.15:
1,3
39.60:42.77
46.87
1.66
2.08::0.83
1.79:1119
1.92
1..754
2.52+0..33
2.63
32.24=0.97
FedlIII"
32.30.19.71 0.04.10.09
0,131:0.27
34.55
7.77
0.02.10.03
0.02.1..03
1.84
28.30
0.001.0.00
0.00100.0
32.41
FedNCN
1.542:1.59
10.53-13.88
13.85 44.03:4:2.30
1..95
1.75 13.59 44.37:3.45
66.3341..72
7.46:10.87
11.02::.221
1..28=1..31
Methods
Amazon-Computer (5 Clients)
Amazon-Computer (10 Clients)
Amizon-Cmmputer (20 Clients)
FedSage="
21.42:40:44
5.08::2.78
3.98
14.65+22.06
23.04:00.44
8.19::0.88
6,33-11.04
17.39:10.18
20.65:+1.28
1.36
4.73422000
21.16:00.93
FedPUB
1279_ 0.00.1..00
0.00..0.00
2.24
12.39:12.22
0.00110:00
0.00100:00
4951..11
14.25:12.14
0.00.10.00
0.00.10.00
3.04
FedTADD
9.82=4.38
3.19::1.73
1.34
3.26:11.47
12.84
12.41
3.68:2.03
4.71
1.93
17.29
1.08
3.99-10.38
1.824:.44
FodGTA
11.40=#1..6 12.07:43.95
8.38+33.27
8.96+1001
20.96=#1446
6.46+1.22
3.71+1033
6.92=0.52
22.08+11225
7,25+0.96
3.46=0.76
6,30+:0.28
FedlTH"
13.52:17.09 0.00:10:00
0.00:100.00
2.62:11.54
7.811:4.25
0.09:10.11
0.06:1:0:9
1.87
5.58114558
0.02
0.05:10.10 10
1.8
FedNCN
1..17=1111 23.47::1.01
21262 24.52++2.13
71.21
21.33:2.111
23.76:2..54 24.00-10.81
70.20:22.61
21.64=1.45
22.86::3.20 02.54=0.77
Methods
AmazoonPhottn (5 Clients)
Amazon-Photo (10 Clients)
Amazon-Photo (20 Clients)
FedSago="
12.45=1199
10.95+2..26
11,43
18.73+1.11
14,0
11.10+1.23
9
18.47:00.2
36,75+00.97
11.28=0.24
9,80=00334
26.52=0.54
FedPUTB
74.89.1..33
0.00:1::000
0.00
3.44
8.42.13.17
0.00110.00
0.00:10:00
1.91 00.49
12.71116881
15:20=00.00
15.20:10.00 6.02.13.58
FedTAD*
13.50:14.99 4.11=:6880
4.03=6680
3.6742.3
13.07=3.559
5.4242222
5.48+2.50
1.02
11,78:+4.6
1.34
8.63 1.21
1.25
FodGTA"
35.54=#1.96 14.30::4.04
11.88=4.09
11.99:1115
32.68=1.122
11.84:10.76
1.59
10.83:00.99
29.40=
10.82=0.75
0.84=0.87
9,60+:0.34
FedITH
24.33:14420 0.00:10.00
0.00::0.00
1.09
11.6647.77
0.00+00000
0.00=+0.00
1.74
12.11477778
15:25-00.00 10 15.22:4:0003
4.144:3.48
FedNCN
74.05=22.55
37.22+:5.34 43.05=7.02 29.11+
1.,4
29.16++2.13 31.11+3443 17.60=
66.09+2.06
19,85=162 16.72+22.01 32.46=0.92
Methods
Questions (5 Clients)
Questions (10 Clients)
Questions (20 Clients)
FedSage="
78.08.12.10
1.31 1.56
39.24110.80
7..31
0.82.12.18 0.84
38.29.11244
76.00.10.63
0.6410.72
0.57.10.96 36.25 0.76
FedPUTR*
59.40=46.04
0.00:10.00
0.00=00.00
20.70:222.09
59.411146.11
12
0.00-10.00
0.00-10.00
20.68
40.59-145.99
0.01
-0.044:0.07
22.45:222.6
FodTAD*
56.05=27.81 0.18+0013
-0.83:0.78
12.82
66.30+331.50
0.10=#0.6
-0.28=00.2
34.57=14.78
69
34.36
0.06+0.07
-0,3240.1
38.74+12.99
FedGTA
88.47:14.45
0.62:10.33
3.17
50.50-11..7
88.0743.04
0.3810.006
1.29-1:044
0..56:10.52
86.69-12.39
0.724:0.9
0.6722.12
1.48
FedlIH"
77.40:26.22
0.00:::000
0.00:20.00
30.70=22..69
38.2643.44
0.00=20.00
0.00=:0:00
30.68=22.74
80.06::2.77
0.38
0.63=1..3
41.63=2.85
FedNCN
94.64:12.04
1.88::0.86
6.44::2.93 52.89-:1.73 91.27:4..9
164::0.82
4.62:t-2.34 51.4==2.15 93.03+5.14 1.08::.811 3.23
50.80=1.299
Note that these supervised node-level FGL methods are adapted to the unsupervised scenario.
Clients
10 Clients
20 Clients
FedSage+
90
FedSage+
FedSage+
90
FedPUB
FedPUB
FedPUB
80
70
FedNCN
FedTAD
70
FedNCN
FedTAD
70
80
FedTAD
FedNCN
60
ACC
50
50
50
40
40
30
30
30
20
20
10
10
10
0
CiteSeer
PubMed Computer
Photo
Questions
GiteSeer
PubMed Computer
Questions
CiteSeer
PubMed Computer
Photo
Questions
Figure 3. Performance comparison of FedNCN with three advanced federated node-level classification methods on five graph datasets.
Baseline Methods To demonstrate the superiority of Fed- NCN, we conduct comprehensive comparisons with two groups of baseline methods. The first group consists of five advanced FGL methods, namely FedSage+ (Zhang et al., 2021), FedPUB (Back et al., 2023), FedTAD (Zhu et al., 2024), FedGTA (Li et al., 2024), and FedlIH (Yu et al.,
2025). The second group comprises three classical FL ag- gregation strategies, including: FedAvg (McMahan et al., 2017), FedProx (Li et al., 2020), and FedPer (Arivazhagan et al., 2019), where the clients utilize our proposed local model to ensure fair and consistent comparison. We provide detailed descriptions of baseline methods in Appendix D.
6

---

Federated Node-Level Clustering Network with Cross-Subgraph Link Mending
0.8
0.8
Table The performance of different Fl. aggregation strategics in
B5-GXS
federated node-level clustering.
03-0OSS-A CLM
23-0KS-OO CUM
Datasets
Methods
ACC
Fl
avg.
Local
44.111.77
8.6
4.12211
23.5.0..5
0.2
0.2
FedAvg
55.942..8
4.240.5
0,4088
16.5400.7
-0.8
CiteSeer
ACC
N941
ABI
ACC
ARI
(10 Clients)
FedProx
45.043.00
8.120.9
3.3413
23.710.7
-0.1
0.5
Cladder (18 Client)
0.8
PuoMeed (28 Client)
FedPer
48.7+10
7.842233
4.7430
22841.
+0.9
FedNCN
59.0::3.0
17.5:+1.3
15.7444.0
27.1:2299
+9.7
0.6
0.0
Local
54.4+11.5
8.0
7.1
37.041.0
PubMed
FedAvg
63.74:299
5.24:1.7
4.820.7
23.9:66.9
-2.
02
0.2
(10 Clients)
FedProx
FedPer
55.51:19 7.51:100
56,5:009
2.1:1.66
3.2:166
6.711.6
35.34.0.7
36.8:112
-2.1
0.0
0.0
ACC
ANI
ACC
NAI
ASI
FedNCN
65.0:t:.5
10.6:+1.88
13.624.6
44.4:3.5
+6.8
Local
45.9.10.3
4.811..1
1.911
15.7.10.3
FedAvg
67.11:0.0
0.8.10.0
.02.000
13.51:0.0
+3.2
Amazom Computer
Figure 4. Ablation studies for CLM strategy and GKS strategy in federated node-level clustering scenarios.
(10 Clients)
FedProx
43.1113.0
2.810.33
0.7.00.7
15.9100.4
-1.5
FedPor
50.2+1111
2.5
1.1+10
15.7+0.5
+0.3
FedNCN
71.2+:2.0
21.3+3111
23,8:45
24.0=1..8
+18.0
Local
48.2.10.8
5.7.0..5
2.41099
15.8.0..5
80
Computer
Photo
FodAvg
63.3+113
3.120.8
1.620.3
15.0::0.6
+2.8
Amazon Photo
(I0 Clients)
FodProx
49.7:118
5.3:0..1
4,2112
16.9::0.3
+1.0
60
FedNCN
FedPer
57,0+1144
73.51:8
29.2:1..1
5.3+0088
3.1334
4.7+10
1.1
17.7#0.7
+22.4
+3.2
Ade
40
Clients
Clients 10
Clients
Local
78.14:1.8
0.5:0..1
-0.7::0.7
45.8::0.2
20
Clients
20
Clients 20
Questions
FedAvg
57.6:00.8
0.0:0..0
0.0:000
46.2::0.0
+2.5
4
6 8 10 12
Iterationn
14
16
18
20
2
4
10 12 Iterations
14 16 18 20
(10 Clients)
FedProx
84.41133
0,340
199:02
49.2::03
+3,0
FedPer
82:242.1
0.320.1
1.620.7
48.410.9
+2.1
FedNCN
91.3+447 1.62088
4.6+2.
51.44:222
+6.2
Figure 5. Algorithm convergence on Computer and Photo datasets.
Implementation Details To ensure experimental fairness, all methods are evaluated under identical hardware and set- tings. We utilize a four-layer GNN on both the client and the server to obtain node embeddings, with hidden layer dimensions are 500-500-2000-10 10. Moreover, we use one- layer MLP to obtain the local clustering signals, which are then uploaded to the server. During model optimization, we adopt the Adam optimizer (Xiao et al., 2024) with a learning rate of 1e-3. The client-server interaction conducted 20 times, with the local model training 10 epochs during each interaction. All methods are implemented using Py Torch 2.4.0 and single NVIDIA GeForce RTX 4090 GPU.
node-level classification methods into the corresponding graph clustering versions to ensure fair comparison. To reduce randomness, each experiment is repeated five times, and the mean and standard deviation of the four clustering metrics are presented in Table We can observe that, under different client partitions, FedNCN consistently outperforms the compared methods on five datasets. The reasons behind this are two-fold: 1) these approaches such FedSage+, FedTAD, FedPUB, FedGTA, and FedlIH are specifically designed to enhance the quality of local-globe -global interactions for better classification. As a result, their effectiveness is limited when dealing with unlabeled graph data; and 2) on the one hand, FedNCN tailored for node-level clustering scenarios. On the other hand, we design a cross-subgraph link mending strategy that restores missing links the server while not directly sharing the raw data, enabling the global model to generate clustering frienndlyrenyyyy prototypes for better clustering.
Clustering Metrics To comprehensively evaluate the clus- tering performance of all methods, we adopt four widely used evaluation metrics, including Accuracy (ACC) (Liu et al., 2022b; Wang et al., 2022b;atc: Cai et al., 2022), Normalized Mutual Information (NMI) (Liu al., 2021b; 2022a; Wang et al., 2021c; Liang et al., 2023), Adjusted Rand Index (ARI) (Yang et al., 2023; Tu et al., 2022), and Fl Score (F1) (Liu et al., 2023b; Tu et al., 2024c;b; 2025a;b) to test clustering results from multiple perspectives.
Comparison with Supervised FGL Methods To further verify the effectiveness of FedNCN, we compare it with five representative supervised FGL methods: FedSage+, Fed- TAD, FedPUB, FedGTA, and FedlIH. Each method of these methods is trained with 5% of the labeled data. As shown in Fig. 3, the experimental results indicate that, under differ- ent client partitions, FedNCN consistently maintains strong competitiveness in clustering tasks, even when compared to federated node-level classification tasks (i.e., FedSage+,
4.2. Experimental Results
Comparison in Unsupervised Learning Settings To vali- date the effectiveness of the proposed FedNCN, which is the first federated node-level clustering framework with cross- subgraph link mending. We extend five advanced federated
7

---

Federated Node-Level Clustering Network with Cross-Subgraph Link Mending
FedTAD, and FedPUB) with limited labeled data, further highlighting the effectiveness of FedNCN. The comparative results of other methods can be found in Appendix D.2.
0.6
0.6
Score
Comparison of FL Aggregation Strategies To verify the effectiveness of the global aggregation strategy, we compare FedNCN with the classical FL aggregation strategies. In our setup, local models are utilized to generate clustering signals, which are subsequently uploaded for global sharing via FL aggregation strategies including FedAvg, FedProx, and FedPer. We report the average of four clustering metrics and their average gains compared to local model testing in Table 2. As seen, compared to these FL. aggregation strategies, FedNCN achieves noticeable improvements in clustering performance. For example, it achieves average improvements of 9.7%, 6.8%, 18.0%, 22.4%, and 6.2% on the CiteSeer, PubMed, Amazon- Computer,, Amazon- Photo, and Questions datasets, respectively. These findings indicate that the integration of our global aggregation strategy signif- icantly enhances its generalization ability across different datasets. In addition, for the federated node-level clustering tasks with clients and 20 clients, the detailed experimental results can be found in Appendix D.2.
0.2
0.2
0.0
0.0
ACC
ARI
5
F1
NMI
Metrics
5
10 15
NMj
Metrics
k(%)
(1%)
CiteSeer (10 Clients)
PubMed (10 Clients)
Score
0.4
0.4
0.2
0.2
0.0
0.0
ACC ARI NMI
ACC
ARI
R(%)
5 10
15
20
Metrics
(%)
10
15 20
NMI
Metrics
Computer (10 Clients)
Photo (5 Clients)
Figure 6. The sensitivity of FedNCN with the variation of a hyper- parameter & on CiteSeer, PubMed, Computer, and Photo datasets.
Analysis of CLM Strategy and GKS Strategy In this part, we evaluate the effectiveness of the proposed Cross- subgraph Link Mending (CLM) strategy and GKS (Global Knowledge Sharing) strategy. In our setup, "BS" denotes the local model of FedNCN. "BS+GKS" denotes the Fed- NCN variants with the CLM strategy being removed, and "BS+GKS+CLM" denotes the FedNCN. As summarized in Fig. 4, observe that 1) the four clustering metrics of "BS+GKS+CLM"" are consistently better than both the "BS+GKS" and "BS"; 2) compared with the "BS", the per- formance of "BS+GK5" has a slight improvement. These findings can be attributed to the following points. Firstly, the cross-subgraph link mending strategy effectively restores the destroyed links on the server, thus facilitating better in- formation sharing across clients; Secondly, the updated pro- totypes and model parameters obtained through the global knowledge-sharing strategy guide each local model to gen- erate a robust target distribution.
We conduct experiments on four datasets to analyze the im- pact of this & by varying its value from 1% 20%. From the results in Fig. 6, we can observe that 1) the hyperparam- eter k indeed plays crucial role in FedNCN, suggesting that searching & from a reasonable hyper-parameter region could benefit clustering performance; 2) uploading a few samples may result in sub-optimal performance, as it fails to reconstruct a meaningful global graph structure, which reduces the quality of the model collaborative learning; and 3) uploading too many samples may hinder clustering per- formance due to excessive noise, leading to unreliable links mending on the server.
5. Conclusion
In this paper, we explore a new research task, federated node- level clustering with crbss-subgraph link mending, which is the first attempt restore the destroyed links between clients due to graph partition under unsupervised settings. To tackle this task, we propose novel unsupervised node- level FGL framework, called FedNCN. In our method, the proposed local model learning method, the cross-subgraph link mending strategy, and the global strategy collectively succeed in restoring missing links with- out reliance on annotations, thereby facilitating the informa- tion negotiation among multiple clients for better clustering. Extensive experimental results across five graph datasets demonstrate that FedNCN can effectively mend missing links caused by sample partition in an unsupervised setting. In future work, we plan to enhance the generalization capa- bilities of FedNCN, extending its application to a broader
Convergence Analysis In Fig. 5, we plot the ACC metric during the training iterations to monitor the model conver- gence on five random experiments on the Amazon-Phout and MrazzonCComputer datasets to evaluate the robustness of the model. Under different client settings, the ACC met- ric consistently converged to stable value within the 20 iterations, with only minor fluctuations. This further high- lights the advantages of FedNCN in handling node-level clustering tasks.
wwwledge-sharing
Analysis of Hyper-Parameter k FedNCN introduces prper-parameter// k. leveraged to determinee the ratio of key clustering samples uploaded to the server from each cluster.
8

---

Federated Node-Level Clustering Network with Cross-Suggraph Link Mending
range of FGL tasks, such as incomplete federated graph
Gong, L.,, Tu, W., Zhou, S., Zhao, L.,, Liu, Z., and Liu, X. Deep fusion clustering network with reliable structure preservation. IEEE Transactions on Neural Networks and Learning Systems, 2022a.
clustering and federated multi-view clustering.
Acknowledgement
Gong, L., Zhou, S., Tu, W., and Liu, Attributed graph clustering with dual redundancy reduction. In IJCAI, pp. 3015-3021, 2022b.
This work is supported by the Key Research and De- velopment Program of Hainan Province (Grant No. ZDYF2024GXJS014 ZDYF2023GXJS163 the National Natural Science Foundation of China (Grant No. 62162022,
Guan, R., Tu, W., Wang, S., Liu, J., Hu, D., Tang, C., Feng, Y., Li, Xiao, B., and Liu, X. pruuture-adaptte multi-view graph clustering for remote sensing data. In Proceedings of the AAAI Conference Artificial Intelli- gence, pp. 16933-16941, 2025.
62162024), the Natural Science Foundation of Hainan Uni- versity (Grant No. XJ2400009401), the Youth Foundation Project of Hainan Natural Science Foundation (Grant No. 624QN279), and the Innovative research project for Gradu- ate students Hainan Province (Grant No. Qhyb2023-28).
Huang, W., Ye, M., Shi, Z., Wan, G., Li, H., Du, B., and Yang, Q Federated learning for generalization, robust- ness, fairess: A survey and benchmark. IEEE Transac- tions on Pattern Analysis and Machine Intelligence, 46 (12):9387 99066 2024.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Jiang, B., Zhang, Z., Ge, S., Wang, B., Wang, X., and Tang, J. Learning graph attentions via replicator dynam- ics. IEEE Transactions on Putlern Analysis and Machine Intelligence, 46(12):7720-772) 2024.
References
Arivazhagan, M. G., Aggarwal, V., Singh, A. K., and Choud- hary, S. Federated learning with personalization layers. arXiv preprint 1912.00110000/ 2019.
Johansson, F., Jethava, V. Dubhashi, D., and Bhattacharyya, C. Global graph kernels using geometric embeddings. In Proceedings of the 31st International Conference on Machine Learning, volume 32, pp. 694-702, 2014.
Baek, J., Jeong, W., Jin, J., Yoon, J., and Hwang, S. J. Per- sonalized subgraph federated learning. International conference on machine learning, pp. 1396-1415, 2023.
Ju, W., Gu, Y., Chen, B., Sun, G., Qin, Y., Liu, X., Luo, X., and Zhang, M. Glee: A general framework for graph- level clustering. Proceedings of the AAAI Conference on Artificial Intelligence, 37(4):4391-4399 2023.
Borgwardt, K., Ghisu, E., Llinares-López. F., O'Bray, L., and Rieck, B. Graph kernels: State-of-the-art and fu- ture challenges. Foundations and Trends@ in Machine Learning, 13(5-)):311-122, 2020.
Li, T., Sahu, A. K., Zaheer, M., Sanjabi, M., Talwalkar, A., and Smith, V. Federated optimization in heterogeneous networks. Proceedings of Machine learning and systems, 2:429-4501 2020.
Cai, J., Fan, J., Guo, W., Wang, S., Zhang, Y., and Zhang, Z. Efficient deep embedded subspace clustering. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1-10, June 2022.
Li, X., Wu, Z., Zhang, W. Zhu, Y., Li, R.-H., and Wang, G. Fedgta: Topology- aware averaging for federated graph learning. arXiv preprint arXiv:240. 2024.
Li, Z., Tang, C., Zheng, X., Liu, X., Zhang, W., and Zhu, High-order correlation preserved incomplete multi- view subspace clustering. IEEE Transactions on Image Processing, 31:2067-2080. 2022.
Chen, F., Li, P., Miyazaki, T., and Wu, C. Fedgraph: Fed- crated graph learning with intelligent sampling. IEEE Transactions on Parallel and Distributed Systems, 33(8): 1775-1786, 2022.
Liang, K., Liu, Y.. Zhou, S., Tu, W., Wen, Y., Yang, X., Dong, X., and Liu, X. Knowledge graph contrastive learning based on gelatio--symmetrice structure. IEEE Transactions on Knowledge and Data Engineering, 36 (I):226-236 2023.
Chen, H., Wang, H., Long, Q., Jin, D., and Li, Y. Ad- vancements in federated learning: Models, methods, and privacy, ACM Computing Surveys, 57(2):1-39, 2024.
Choromanski, K. M. Taming graph kernels with random features. In Proceedings of the 40th International Con- ference on Machine Learning, pp. 5964 5977, 2023.
Liang, K., Meng, L., Liu, M., Liu, Y., Tu, W., Wang, S., Zhou, S., Liu, X., Sun, E, and He, K. survey of
9

---

Federated Node-Level Clustering Network with Cross-Subgraph Link Mending
knowledge graph reasoning on graph types: Static, dy- namic, and multi-modal. IEEE Transactions on Pat- tern Analysis and Machine Intelligence, 2024. doi:
Meng, L.,, Liang, K., Yu, H., Liu, Y., Zhou, S., Liu, M., and Liu, X. Fedean: Entity-aware adversarial negative sampling for federated knowledge graph reasoning. IEEE Transactions on Knowledge and Data Engineering, 2024.
DOI:10.1 00l:10.1109/TPAN 109/TT0M..222.33411111
Lin, Z., Kang, Z., Zhang, L., and Tian, L. Multi-view attributed graph clustering. IEEE Transactions knowl- edge and data engineering, 35(2):1872-188)) 2021.
Pan, F. and Kang. 7. Beyond homophily: Reconstructing structure for graph-agnostic clustering. In Proceedings of the 40th International Conference on Machine Learning, volume 202, pp. 26868-22822 26867, 2023.
Liu, J., Liu, X., Yang, Y., Guo, X., Kloft, M., and He, L., Multiview subspace clustering via co-training robust data representation. IEEE Transactions Neural Networks and Learning Systems, 33(10):5177-518, 2021a. doi: DOI:10.1 10l:10.1109/11 1011/1NN9NNN S 2022 .30699244
Platonov, O., Kuznedelev, D., Diskin, M., Babenko, A., and Prokhorenkova, L. A critical look at the evaluation of gnns under heterophily: Are we really making progress?, 2024.
Liu, J., Cheng, J., Han, R., Tu, W., Wang, J., and Peng, X. Federated graph-level clustering network. In Proceed- ings of the AAAI Conference on Artificial Intelligence, volume 39, pp. 18870-18878, 2025.
Tan, Z., Wan, G., Huang, W., and Ye, M. Fedssp: Federated graph learning with spectral knowledge and personalized preference. arXiv preprint arXiv:2410.20100 2024.
Liu, S., Wang, S., Zhang, P., Xu, K., Liu, X.. Zhang, C., and Gao, F. Efficient one-pass multi-view subspace clus- tering with consensus anchors. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pp. 7576-7584, 2022a.
Tu, W., Zhou, S., Liu, X., Guo, X., Cai, Z.,, Zhu, E., and Cheng, J. Deep fusion clustering network. In Proceed- ings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 9978-999899 2021.
Tu, W., Zhou, S., Liu, X., Liu, Y., Cai, Z., Zhu, E., Zhang, C., and Cheng, J. Initializing then refining: A simple graph attribute imputation network. In Proceedings of the International Joint Conference on Artificial Intelligence, pp. 3494-3500, 2022.
Liu, X., Liu, L., Liao, Q. Wang, S., Zhang, Y., Tu, W., Tang, C., Liu, J., and Zhu, E. One pass late fusion multi-view clustering. In Meila, M. and Zhang, T. (eds.), Proceed- ings of the 38th International Conference on Machine Learning, volume 139, pp. 2850-6859, b021b.
Tu, W., Guan, R., Zhou, S., Ma, C., Peng. X., Cai, Z., Liu, Z.,, Cheng, J., and Liu, X. Attribute-missing graph clustering network. Proceedings of the AAAI Conference on Artificial Intelligence, 38(14): 1534)):):222113:5 15401, 2024a.
Liu, Y., Tu, W., Zhou, S., Liu, X., Song, L., Yang, X., and Zhu, E. Deep graph clustering via dual correlation reduction. In Proceedings of the AAAI conference artificial intelligence, volume 36, pp. 7603-7611, 2022b.
Tu, W., Liao, Q., Zhou, S., Peng. X.,, Ma, C., Liu, Z., Liu, X., Cai, Z., and He, K. Rare: Robust masked graph autoencoder. IEEE Transactions on Knowledge and Data Engineering, 36(10):5340-534 53500055333300 2024b.
Liu, Y., Yang, X., Zhou, S., Liu, X., Wang. S., Liang. K., Tu, W., and Li, L. Simple contrastive graph clustering. IEEE Transactions Neural Networks and Learning Systems, 2023a.
Tu, W., Zhou, S., Liu, X., Ge, C., Cai, Z., and Liu, Y. Hier- archically contrastive hard sample mining for graph self- supervised pre-training. IEEE Trunsactions on Neural Networks and Learning Systems, 35(1 15:7777:661111111.11 2024c.
Liu, Y., Yang, X., Zhou, S., Liu, X., Wang, Z., Liang, K., Tu, W., Li, L., Duan, J., and Chen, C. Hard sample aware network for contrastive deep graph clustering. Proceed- ings of the AAAI Conference on Artificial Intelligence, 37 (7):8914-8922, 2023b.
Mafakheri, B., Goratti, L., Riggio, R., Buratti, C., and Reisenfeld, S. Lte transmission in unlicensed bands: Evaluating the impact over clear channel assessment. In 2018 27th International Conference on Computer Com- munication and Networks (ICCCN), pp. 1-8, 2018
Tu, W., Xiao, B., Liu, X., Zhou, S., Cai, Z.,, and Cheng, J. Revisiting initializing then refining: An incomplete and missing graph imputation network. IEEE Transactions on Neural Networks and Learning Systems, 36(3):3244- 3257, 2025a
McMahan, B., Moore, E., Ramage, D., Hampson, S., and Arcas, B. A. Communication-efficin learning of deep networks from decentralized data. In Artificial intelli- gence and statistics, pp. 1273-1282 2017.
Tu, W., Zhou, S., Liu, X., Cai, Z., Zhao, Y., Liu, Y., and He, K. Wage: Weight-sharing attribute misutte graph autoencoder. IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 1-18, 2025b.
10

---

Federated Node-Level Clustering Network with Cross-Suggraph Link Mending
Wan, G., Huang, W., and Ye, M. Federated graph learn- ing under domain shift with generalizable prototypes. In Proceedings of the AAAI Conference on Artificial Intelli- gence, volume 38, pp. 15429- 15429-15437, 2024.
Wang, S., Liu, X., Liu, L.., Zhou, S., and Zhu, E. Late fusion multiple kernel clustering with proxy graph refinement. IEEE Transactions on Neural Networks and Learning Systems, (4(8))4359-43))) 2021c
Wang. H., Guo, S., Tang, B., Li, R., Yang. Y., Qu, Z., and Wang, Y. Heterogeneity-awann gradient coding for tolerating and leveraging stragglers. IEEE Transactions on Computers, 71(4):779-794, 2021a.
Wang. S., Liu, X., Liu, L., Tu, W., Zhu, X., Liu, J., Zhou, S., and Zhu, E. Highly-efficient incomplete large-scale multi-view clustering with consensus bipartite graph. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 9776-9785, 2022a.
Wang, H., Qu, Z., Guo. S., Wang, N., Li, R., and Zhuang, W. Losp: Overlap synchronization parallel with local compensation for fast distributed training. IEEE Journal on Selected Areas in Communications, 39(8):241122577, 2021b.
Wang, S., Liu, X., Liu, S., Jin, J., Tu, W., Zhu, X., and Zhu, E. Align then fusion: Generalized large-scale multi- view clustering with anchor matching correspondences. Advances in Neural Information Processing Systems, 35: 5882-5895, 2022b.
Wang. H., Y., Xu, W., Li, R., Zhan, Y., and Zeng, Z. Dafkd: Domain aware federated knowledge distillation. In Proceedings of the IEEF/CVF conference on computer vision and pattern recognition, pp. 20412-20421. 2023a.
Wang. S., Liu, X., and Zhu, E. Late fusion multi-view clustering via global and local alignment maximization. arXiv preprint arXiv:2208. 01 198, 2022c
Xiao, N., Hu, X., Liu, X., and Toh, K.-C Adam-family methods for nonsmooth optimization with convergence guarantees. Journal of Machine Learning Research, 25 (48):1-53, 2024.
Wang, H., Xu, W., Fan, Y., Li, R., and Zhou, Aoce-fi: Fed- erated learning with aligned overlapping via calibrated compensation. In IEEE INFOCOM 2023 IEEE Confer- ence on Computer Communications, pp. 1-10, 2023b.
Xu, J. and Lange, K. Power k-means clustering. In Proceed- ings of the 36th International Conference Machine Learning, volume 97, pp. 6921-6931 2019.
Wang, H., Jia, Y., Zhang. M., Hu, Q., Ren, H., Sun, P., sub-
Wen, Y., and Zhang. T. Feddse: Distribution- model extraction for federated learning over resource- constrained devices. In Proceedings of the ACM Web Conference 2024, pp. 2902-2913. Association for Com- puting Machinery, 2024a.
Yang, L.,, Huang, Z., Zhang, Z., Liu, Z.,, Hong, S., Zhang, W., Yang. W., Cui, B., and Zhang, L. Graphusion: Latent diffusion for graph generation. IEEE Transactions on Knowledge and Data Engineering, 36(11):63586636, 2024.
Wang, H., Jia, Y., Zhang, M., Hu, Q.. Ren, H., Sun, P., Wen, Y., and Zhang. T. Feddse: Distribution-awe aware sub- model extraction for federated learning over resource- constrained devices. In Proceedings of the ACM on Web Conference 2024, pp. 2902-2913, 2024b.
Yang, X., Liu, Y., Zhou, S., Wang, S., Tu, W., Zheng, Q.. Liu, X., Fang, I.,, and Zhu, E. Cluster-guided contrastive graph clustering network. Proceedings of the AAAI Con- ference on Artificial Intelligence, 37(9):10834-100 10842, 2023.
Wang, H., Xu, H., Li, Y., Xu, Y., Li, R., and Zhang. T. Fed- cda: Federated learning with cross-rounds divergence- aware aggregation. In The Twelfth International Confer- ence on Learning Representations, 2024c.
Yu, W., Chen, S., Tong, Y., Gu, T., and Gong. C. Modeling inter-intra heterogeneity for graph federated learning. In Proceedings of the AAAI Conference on Artificial Intelli- gence, volume 39, pp. 22236-22244, 2025.
Wang, H., Zheng, P., Han, X., Xu, W., Li, R., and Zhang, T. Fednir: Federated learning with neuron-wise learn- ing rates. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 3069-3080. Association for Computing Machinery, 2024d.
Zhang, K., Yang, C., Li, X., Sun, L., and Yiu, S. M. Sub- graph federated learning with missing neighbor genera- tion. Advances in Neural Information Processing Systems, 64:6671-6683 2021.
Zhu, Y.. Li. X., Wu, Z., Wu, D., Hu, M., and Li, R.-H. Fedtad: Topology-ave data-fres knowledge distilla- tion for subgraph federated learning. arXiv preprint arXiv:2404.1406. 2024.
Wang, H., Zheng, P., Han, X., Xu, W., Li, R., and Zhang, T. Fednir: Federated learning with neuron-wise learning rates. In Proceedings of the 30th ACM SIGKDD Con- ference on Knowledge Discovery and Data Mining, pp. 3069-3080, 2024e.

---

Federated Node-Level Clustering Network with Cross-Subgraph Link Mending
A. Notations
In this section, we provide all notations of the proposed FedNCN are summarized Table 3.
Table 3. Basic notations for the proposed FedNCN.
Notations
Meaning
Notations
Meaning
X E xd
Attribute matrix of each subgraph
A &
Original adjacency matrix of each subgraph
DE RNN
Degree matrix of cach subgraph
A E Nxx
Normalized adjacency matrix of each subgraph
I R
Identity matrix of each subgraph Final latent representations of each subgraph
E R Wxa'
Latent representations of each subgraph Prototypes of each subgraph
Z IR Mxdd
C R Oxd
B E RaaOxd
Truth samples attribute matrix of each subgraph
R E R Ra*
Oxd
Masked matrix of each subgraph
R e Oxd
Reconstnucted masked matrix of each subgraph
R R Oxd
Global samples attribute matrix of each subgraph
S RILL O 6.
Graph-level affinity matrix
Q RV=*
Soft assignment distribution of the mended graph Target distribution the mended graph Consensus prototypes
X E xd
Attrihute matrix of the mended graph
No
A N*XN"
Adjacency matrix of the mended graph
C R
B. Different Graph Kernel Methods
In this section, we provide detailed description of different graph kernel methods. Common graph kernel methods include the cosine kernel, the weisfeiler-lehman kernel, the shortest path kernel, and the lovász-ther kernel. The specific descriptions are as follows:
The Cosine (COS) Kernel (Choromanski, 2023) calculates the similarity by compressing each graph as a feature vector and then using cosine measure the angle between these feature vectors. This method combines the graph structure and the node attributes.
The Weisteiler-Lhmman (WL) Kernel (Ju et al., 2023) calculates the similarity between the two graphs by re-marking the nodes of the graph through iteration, thus generating the hierarchical structure representations.
The Shortest Path (SP) Kernel (Borgwardt et al., 2020) calculates the similarity between the two graphs by evaluating the distribution of shortest path lengths between corresponding nodes, which captures the global structure information of the graphs.
The Lovász-Thett, (LT) Kernel (Johansson et al., 2014) calculates the similarity between the two graphs by solving Lovász theta function, which reflects the size of the largest independent set in the graph.
C. Algorithms
In this section, we provide the algorithms for the local model learning method executed on the client, the cross-subgraph link mending strategy, and the global snowledge-hharing strategy on the server in our proposed FedNCN. Specifically, the process executed on the client is shown in Algorithm 2, while the process executed on the server is presented in Algorithm 3.
D. Experiments
In this section, we first provide a detailed description of the five benchmark datasets used in Subsection D.1. Then, we elaborately introduce the baseline methods and our proposed FedNCN method. Finally, we further report detailed information on the comparative experiments and ablation studies of the FedNCN method, as in Subsection D.2.
D.I. Experiment Setup
Datasets We report the detailed information of five benchmark datasets, as shown in Table 4.
CiteSeer and PubMed (Jiang et al., 2024) are both used for citation graphs. CiteSeer is a dataset containing academic papers and their citation relationships, primarily used for analyzing citation networks and is especially suitable for node-level graph
12

---

Federated Node-Level Clustering Network with Cross-Subgraph Link Mending
Algorithm 2 FedNCN for Client Algorithm
Input: Complete graph G; attribute matrix X; adjacency matrix A; GNN epoch Egnni MLP epoch Emir- Initialize model parameters C @mip mip and Ogna from server.
Output: Masked matrix R. and MI.P parameters mmlp.
for j = 1 to Egnn do Generate Z by Eq. (1).
end for
Obtain C from Z by -mmaas.
Obtain B with C and Z by Eq. (2).
Generate with a shape similar to B using gaussian noise.
for j = 1 to Empp do
Obtain R. from R by Eq. (3).
Optimize the local MLP model by minimizing Eq. (4).
end for
Upload R. and mlp to the server.
Algorithm 3 FedNCN for Server Algorithm
Input: Masked matrix R: MLP parameters mmip: GNN epoch Egan-
Output: Consensus prototype C and well-trained GNN parameters 0 ggnn. Infer R with R and mllp by Eq. (5).
Generate the refined subgraph Ggub by Eqs. (6) and (7).
Calculate the graph-level similarity S by Eq. (8).
Generate the global graph g with S.
Generate the mended graph G by Eq. (9). for i Egon do
Calculate the global reconstruction loss LSEE by Eq. (11). Calculate the global clustering loss LKI. by Eq. (12). Optimize the GNN model by minimizing Eq. (10).
end for
Obtain consensus prototype Č and well-trained GNN parameters 0 0gmn.
Share C and gg each client.
learning tasks. Each node represents each paper, and the edges represent the citation relationships between papers. Similarly, the PubMed dataset an academic citation dataset sourced from the PubMed database, containing literature from the fields of medicine. PubMed dataset is mainly suitable for training and evaluating large-scale node-level graph learning models.
Amazon- Computrr and Amazon-Photo (Lin et al., 2021) datasets are part the Amazon product graph. Amazon-Computer dataset focuses on products related to computers and is part of the Amazon product review dataset. Each node represents product (related to computers), and the edges represent the correlation between products. Amazon-Photo dataset is another subset of the Amazon product review dataset, focusing on products related to photos and cameras. Similar to the Computer dataset, each node represents a product (related to photos and cameras), and the edges represent the correlation between products. Both of these datasets are commonly used for node-level graph learning tasks.
Questions dataset (Platonov et al., 2024) is commonly used in the field of graph learning, especially in applications such as question recommendation systems or question graph analysis. In these applications, questions abstracted as nodes in the graph, and the correlation between questions is represented by the edges in the graph.
Baselines and Our Model
FedSage+:
This method (Zhang et al., 2021) designs a missing neighbor generator based on FedSage to compensate for the missing graph structure information. In this process, each client first receives the node representations from the other clients and then computes the gradient of the distance between these representations and local node features. Finally, the gradient used update the graph generator.
13

---

Federated Node-Level Clustering Network with Cross-Subgraph Link Mending
Table 4. Dataset summary.
Dataset
Nodes
Edges
Dimensions
Clusters
Type
Domian
CiteSeer
3327
9104
3703
6
Graph
Citation
PubMed
19717
88648
500
3
Graph
Citation
Amazon-Computer
13752
491722
767
10
Graph
Computer Product
Amazon-Photo
7650
238162
754
Graph
Photo Product
Questions
48921
153540
301
Graph
Q&A System
5 Clients
10 Clients
20 Clients
FedGTA
90
FedGTA
FedGTA
90
90
FedlIH
80
FediIH
FediIH
FedNCN
FedNCN
FedNCN
70
ACC (%)
60
60
60
40
40
30
30
20
20
10
0
CiteSeer PubMed Computer
Photo
Questions
CiteSear
PubMed Computer
Photo
Questions
CiteSeer PubMed Computer
Photo
Questions
Figure 7. Performance comparison of FedNCN with advanced federated node-level classification methods five graph datasets.
This method (Baek et al., 2023) proposes novel personalized subgraph FL framework. In this approach, the central server constructs a random graph under the guidance of real labels and distributes it to each client. Subsequently, each client learns this random graph to obtain the functional embeddings, which are then uploaded to the server for identifying the community consisting of a group of densely connected subgraphs. Moreover, FedPUB learns a personalized sparse mask on each client to select only the relevant subset of the subgraph for parameter updates, enhancing the aggregation performance.
FedPUB:
This method (Zhu et al., 2024) is a topology-awan subgraph federated learning framework that addresses subgraph heterogeneity by enhancing the knowledge transfer from local models to the global model. Each client first decouples node and structure variation, revealing their difference in structure homophily and label distribution. Then, a knowledge distillation process applied to transfer reliable local knowledge the global model, which optimizes the performance of the local model.
FedTAD:
FedGTA;
This method (Li et al., 2024) is a novel optimization strategy for FGL, which is designed to address scalability and performance issues in existing FGL methods when dealing with large-scale graph data. This method introduces a topology-awel aware mechanism that uses a mixed aggregation of neighbor features enhance the capacity of the local model to capture graph structure information.
FedlIH: This method (Yu et al., 2025) is a federated learning framework that addresses both inter- and intra-subgraph hetcrogeneity. It uses a hicrarchical variational model to capture inter-subgraph similarities and then disentangles subgraphs into multiple latent factors to handle intra-heterogeneity which improves the overall model robustness.
FedNCN: This is our Federated Node-level Clustering Network with inter-client subgraph mending. On the client, an ML.P-asedd projector is designed to implicitly obtain clustering signals, which are then uploaded for subsequent inter- client link mending. On the server, the cross-subgraph link mending strategy is proposed to maximize correlated links through graph similarity estimation and minimize redundant links by using a method similar to N-Cut. Subsequently, a GNN-based generator is developed to learn the consensus prototypes from this mended graph, which enhances the clustering performance of each client.
Local: This method is a non-FL approach, which is the proposed FedNCN local model.
14

---

Federated Node-Level Clustering Network with Cross-Subgraph Link Mending
Table The performance of different federated learning strategies in federated node-level clustering. Here, "Local"" denotes the use of only our local model, while in "FedAvg", "FedProx", and "FedPer", different aggregation strategies are applied by the server. with the client all using our local model.
Datasets
Methods
ACC
NMI
ART
F1
avg. A
Datasets
Methods
ACC
NMI
ART
FI
avg. A
Local
43.3:22.8
6,5422
5.144.2 21.1118
Local
45.5:20.6
10.8:0.8
4.0
28.041.0
FedAvg
53.3:1222
39112
1.11.7
16.611
0.1
FedAvg
53.91111
8.0.0.88
1.41088
22.3115
0.7
CiteSeer
CiteSeer Clients)
Clients)
FedProx
41.2+119
49+10
3.2+15
21.0=0077
-1.5
FedProx
48.343.6
10.1+0..5
2.8+0.3
27.1+111
FedPet
53.14:1.8
7.3+2222
8.6
21.6::12
+3.6
FedPer
47,8+13
11.00:88
4.3+009
28.0+1.
+0.7
FedNCN
55.0=+1.8
13.342,
19.7+3.7
2.
+9.2
FedNCN
57,3+1.6
19.7=119
18.0=1.7
32.
+9,9
Local
55.5.12.1
9.5.1
9.5.1.5
38.11.0.7
Local
54.3110.9
6.3.1.0.8
1.9
35.4.1.0
FedAvg
61.7:40.5
9.6:004
9.010.9
1..8.1.88
-0.4
FedAvg
64.2:1.322
4.6:22.3
1.1
20.0
-1.8
PuhMed
PubMed
(5 Clients)
FedProx
60.5:::177
10.3::4.2
11.043.6
39.8:12.00
+2.3
(20 Clients)
FedProx
6088::14
6.1::1.3
7.
36.74:0.8
+2.5
FedPer
56.1:11.7
8.71007
10.6120
38.711..3
+0.4
FedPer
53.3:10.4
6.2.1..8
4.51088
35.2.00.5
-0.4
FedNCN
63.4:1:.6
10.5::355
13.9.3.6
44.0.12.3
+4.8
FedNCN
66.3:11.7
7.5:100,9
11.01:12
1.3
+6.3
Local
37.7
2.9:1-22
1. 1.3
13.24.0.7
Local
48.1::0.5
3.3:4..11
0.0+0.5
18.240.3
FodAvg
59,62003
0.94002
0.1
11.44:.4
4,3
FedAvg
64.2#2.0
1,7402
4..420.1
17920.2
+3,5
Amazon-Compunn
Amazon-Compont
(5 Clients)
FedProx
39.2.13.9
2.2.10.3
1.5.1.4
13.11.0.7
.0.1
(20 Clients)
FedProx
50.2:1:1. 3.1.1..2
1.7.0044
19.910.44
+1.3
FedPer
4..7:1..8 2.11:02
0.7.008
1..
+1.8
FedPer
52.9:11.5 3.6:10.1
2.4088
20.2:...6
+2.4
FedNCN
63.4:22.3
17.0=99.0
15.348.7
21.41..99
+15.6
FedNCN
70.2:22.6
21.6:1.5
22.9:322
32.540.8
+19.4
Local
47.0=0.6
4,7403
38+099
16.3+00.5
Local
55.1=0.9
9,74002
8.6+0.3
25.5+05
FodAvg
63.42:0.5
0.9:00.3
0.0:0.3
12.3::0.6
+1.2
FedAvg
63.5:22.2 6.4:22.
5.0:2.55
23.24-99
+0.3
Amazon-Photo Clients)
FedProx
49.1::1.2 5:2::-10
7.74114
17.1
+1.8
Amazea-Photo Clients)
FedProx
58.0+22.1 10.2+:04
8.4
26.74:0.6
+1.1
FedPer
55.2::1.6
5.5+1111
1.161.
16.4+00.9
+3.1
FodPer
63.4+0.8 12.2+0.9
11.7=11
27.9+088
+4.1
FedNCN
74.1::3.0
37.2+5.33
43.1=7.0
29.1
+27.9
FedNCN
66.1+2211
19.9::1.6
16.7=2200
32.5+0.9
+9.1
Local
73.1:22.4
0.4-20.3
-0.7+:0.5
44.72:10
Local
80,7#166
0,7::002
-0.4-20.5
46.3+:0.4
FedAvg
87.4:10.8
0.0::0.0
0.0:0.0
46.2::0.0
+3.5
FedAvg
88.2:1155 0.0::0.0
0.1100.0
46.240.0
+1.7
(5 Clients)
Questions
FedProx
85.3.14.1 0.3110.1
1.6111
48.8.1..
+4.1
(20 Clients)
Questions
FedProx
81.71008
0.6.1.22
...11.2
47.0.0.5
+0.4
FodPer
84.1#20
0.3:5..1
11+09
48.4+08.
+3.6
FodPer
81.3+055
0.5:t:02
40.2::.88
46.94:066
+0.3
FedNCN
94.6:22000 1.9:1-99
6,4:229
52.9+117
+9.1
FedNCN
93.0:15.1
111:0.88
3,33333 50.8+2.33
+5.1
D.2. Experiment Results
Comparison with Supervised FGL Methods In the main text, we compare the proposed FedNCN with FedSage+, FedPUB, and FedTAD, demonstrating the superiority of our method. Here, we compare FedNCN with FedGTA and FedllH to further validate the competitiveness of FedNCN in clustering tasks. The experimental setup remains consistent with that in the main text, where each supervised federated node-level classification method is trained using 5% of labeled graph data from the dataset. As shown in Fig. 7, the experimental results indicate that, under different client partitions, FedNCN consistently outperforms federated node-level classification tasks (i.e., FedGTA and FedIIH) with limited labeled data.
Comparison of FL Aggregation Strategies In the main text, demonstrate that the proposed FedNCN method significantly outperforms three classical FL aggregation strategies (including FedAvg, FedProx, and FedPer) in 10 clients. Here, as shown in Table 5, we report the performance of FedNCN in 5 clients and 20 clients across multiple clustering metrics on five unlabeled benchmark graph datasets. As seen, we can observe these findings 1) in the 5 clients, FedNCN achieves average improvements of 9.2%, 4.8%, 15.6%, 27.9%, and 9. % on the CiteSeer, PubMed, Amazon-Computer Amazon-Photo, and Question datasets, respectively; 2) in the 20 clients, the average improvements are 9.9%, 6.3%, 19.4%, 9.1%, and 5. 1% on the same datasets, respectively. These findings indicate that under different client partitions, our method significantly enhances the clustering performance on all datasets compared other aggregation strategies with our local model. These improvements could be attributed to the following points. Firstly, the cross-subgraph link mending strategy restores links as much as possible, facilitating better information communication. This allows the model to learn consistent prototypes and model parameters from the mended graph, thereby eliminating the constraints caused by the destroyed structural information and enabling each local model to generate a caastering--rienddi target distribution.
Graph Kernel Ablation Analysis In this part, we evaluate the effectiveness of different graph kernels in FedNCN. In our setup. "COS" denotes the integration of the COS graph kernel with FedNCN, while "WL," " "SP." and "LT" denote the combinations of the WL, SP, and LT graph kernels with FedNCN, respectively. As shown in Fig. 8, we observe that 1) the ACC remains consistently above 50% across all experiments different datasets; 2) under different datasets and client partitions, the choice of graph kernels has little impact on clustering performance. These findings may be attributed to the fact that different graph kernels focus on different paradigms and the inherent patterns of graph structures. However, in our method, which benefits from the effective recovery of intra-cluster links, these graph kernels can always identify representative features of refined subgraph structures to accurately measure the affinity between these subgraphs. In summary, each graph kernel method has its own strengths, but the overall performance differences are minimal. Therefore,
15

---

Federated Node-Level Clustering Network with Cross-Subgraph Link Mending
Clients)
Clients)
Computer Clients)
Photo Clients)
Questions Clients)
CO5
ACC
CO5
ACC
CO5
CO5
CO5
ACC
WL
WL
SP
17
LT
Scorel's)
Score(%)
(core(%)
Score(%)
Clients]
Clients)
Computer Clients]
[10 Clients)
Questions (10 Clients)
CO5
ACC
C05
ACC
C05
ACC
CO5
CO5
ACC
WL
SP
LT
Score[%)
Score(%)
Score(%)
Score(%)
Score(%)
Computer Clients]
Clients)
Questions (20
CO5
ACC
C05
ACC
CO5
ACC
C05
ACC
COS
ACC
NMI
SP
SP
SP
SP
Score(%)
Scoru(S)
Score(S)
Score(%)
Score(S))
Figure The performance different graph kernel ablation studies on five benchmark datasets (CiteSeer, PubMed, Computer, Photo. Questions). Here, each row represents different client settings (5 clients, 10 clients, 20 clients).
0.8
0.8
0.8
0.8
Base
Base
Base
Base
FedNCH
PedNCN
PedNCN v1 FedNCN
PedNCN
FedNCN
FedNCM
FedNCN
Score
0.4
FedNCN
FedNCN
0.4
PedNCN
0.44
FedNCN
FedNCN v3
Score
0.4
PedNCN
FedNCN
0.0
ACC
NMI
ARI
F1
0.0
ACC
NM4 Publed
ARI Clienta)
F1
0.0
ACC
NMI
ARI
F1
0.0
ACC
HMI
ARI
F1
CitsSeer (3 Clients)
Computer
Clients)
Clients)
FedNCN
FedNCN.
FedNCN
0.6
FedNCN v2
0.6
FedNCN. v2
0.6
FedNCN .2
0.6
FedNCN v2
5c006
FedNCN.vv v3
FedNCN. va
FedNCN, v3
Score
FedNCN, v3
FedNCN
FedNCN
FedNCN
FedNCN
0.2
0.2
0.2
0.2
CiteSeer (10 Clients)
PubMed (10 Clients)
Computerr (10 Clients)
Photo (10 clients)
0.5
0.8
0.8
0.8
Base
Base
Base
Base FedNiNN v1
FedNCN.vi v1
FedNCN.Vv v1
FedNCN v1
FedNCN_y2
FedNCN
FedNCN v2
FedNCN v2
500r0
Score
0.4
FedNCN
0.4
FedNCN
0.44
TedNCN
0.44
redNCN
0.0
ACC
HMI
ARI
F1
0.0
ACC
NMI
ARU
0.0
ACC
NMI
ARI
F1
0.0
ACC
NMI
ARI
F1
Clients)
Computer
Figure 9. The performance of module ablation in FedNCN under the federated node-level clustering scenario.
16

---

Federated Node-Level Clustering Network with Cross-Sugrraph Link Mending
in our approach, we choose the computationally simpler "COS" graph kernel for ease of extension and application.
Module Ablation Study In this section, evaluate the effectiveness of the proposed modules in FedNCN, In setup, "Base" denotes the local model of FedNCN. "FedNCN_vL _v""," "FedNCN.v2" _v2", and "FedNCN_vs .v3 denote three FedNCN variants with the intra-cluster link mending strategy, the inter-cluster link mending strategy, and the improved N-Cut method being removed, respectively. As summarized in Fig. 9, we can observe that 1) the clustering metrics of the FedNCN method are consistently better than the other methods; 2) Compared to the "Base", "FedNCN_vI", "FedNCN.v2", and "FedNCN v3" exhibit better clustering performance in most cases. These findings can be attributed the following points. Firstly, FedNCN can effectively restore the missing links between clients, facilitating better knowledge sharing to enhance each local model for better clustering. Secondly, each module of FedNCN plays effective role in restoring the missing links between clients on the server.
17


Reviews:
[{"review_id": "2BXtLR5I8i", "reviewer": "Reviewer_uoNs", "summary": "This paper introduces federated node-level clustering that achieves cross-subgraph link mending under unsupervised circumstances. The proposed approach is mainly composed of three components, i.e., the local model learning scheme that collects and preserves trustworthy clustering signals for destroyed sample link restoration, the cross-subgraph links mending scheme that establishes correct links among subgraphs with the aid of prior learned clustering knowledge, and the global knowledge sharing scheme that learns high-quality consensus features based on the mended graph and ensures reliable feedback to each client. Abundant experiments on five benchmark datasets have been done.", "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": "# Response to Reviewer:\n\n**i. Some concerns:**\n\n**a) Clustering ground truth:** Thanks for your question. The clustering ground truth is the true cluster category of samples. In our paper, it corresponds to the node label. The ground truth is used only in the evaluation stage of this task.\n\n**b) Parameter initialization:** Thanks. We do the initialization by following the parameter initialization principles in federated learning referring to previous work [1, 2]. The learnable weight matrix **W** is randomly initialized at the global server and subsequently delivered to the local models for learning.\n\n[1] Baek, J.; Jeong, W.; Jin, J.; Yoon, J.; and Hwang, S. 2023. Personalized subgraph federated learning. In *ICML*, 1396-1415.\n\n[2] Zhu, Y.; Li, X.; Wu, Z.; Wu, D.; M, Hu.; and Li, R. 2024. FedTAD: Topology-aware Data-free Knowledge Distillation for Subgraph Federated Learning. In *IJCAI*, 5716–5724.\n\n**c) Federated graph-level clustering:** Thanks. It is indeed an interesting and meaningful task. Federated node-level clustering clusters nodes in only one graph, while federated graph-level clustering clusters multiple graphs. Both clustering methods work in the way of federated learning. The proposed network encounters several difficulties when handling the federated graph-level clustering task. For instance, compared to node-level tasks, graph-level tasks involve data that may originate from different domains and have more complex graph structures. This makes it difficult to capture common patterns across multiple clients. In future work, we aim to overcome these challenges and extend our model to federated graph-level clustering tasks.\n\n**d) Advantages:** Thanks. Due to the non-uniform distribution of instances in the sample space, the traditional KNN graph construction method has inherent flaws [1-3]. Using different values of $K$ for different classes outperforms using a fixed $K$ across all classes, Li et al. have drawn this conclusion through mathematical derivation. Based on this theoretical claim, we further demonstrate the advantages of dynamic graph construction through experiments, compared to the traditional KNN-based approach. The experimental results have been provided at the anonymous link https://anonymous.4open.science/r/FedNCN-rebu-418F/data.png\n\n[1] S, Zhang. 2020. Challenges in KNN Classification. *IEEE Transactions on Knowledge and Data Engineering*, 4663-4675.\n\n[2] S, Kazemi.; R, Goel.; K, Jain.; I, Kobyzev.; A, Sethi.; P, Forsyth.; and P, Poupart. 2020. Representation learning for dynamic graphs: A survey. *Journal of Machine Learning Research*, 1-73.\n\n[3] M, Munir.; W, Avery.; M, Rahman.; and R, Marculescu.; 2024. Greedyvig: Dynamic axial graph construction for efficient vision gnns. In *CVPR*, 6118-6127.\n\n[4] B. Li.; Y. Chen.; and Y. Chen. 2008. The nearest neighbor algorithm of local probability centers\", In *IEEE Transactions on Systems, Man, and Cybernetics: Systems*. 141-154.\n\n**ii. Minor writing issues:**\n\n**Typos:** Thanks for your careful review again! These typos have been revised. a) We have made the dataset names consistent in Table 1, Table 2, and Figures 3-5. b) We have removed unnecessary symbols (e.g., the extra period at line 104 in the left column). Moreover, we have tried our best to correct similar typos and double-checked throughout the paper.\n\n**Essential References Not Discussed:**\n\n**Adding the reference:** Thanks for your careful review again! The FedGCN [1] method that you mentioned has been discussed in the final version.\n\n[1] Yao, Yuhang, et al. \"FedGCN: convergence-communication trade-offs in federated training of graph convolutional networks.\" In NeurIPS 2023.\n\nThanks for your constructive comments, we hope that our responses will make you satisfied.", "claims_and_evidence": "The authors claim that they use graph kernel similarity and N-cut to dynamically construct the mended graph \nHowever, there seems to be a lack of discussion about the intuition behind using such a method. For example, why use dynamic graph construction instead of KNN graph construction?", "methods_and_evaluation_criteria": "The authors employ five benchmark datasets (i.e., CiteSeer, PubMed, Amazon-Computer, Amazon-Photo, and Questions) and four widely used evaluation metrics (i.e., ACC, NMI, ARI, and F1) to evaluate the proposed method.", "theoretical_claims": "Yes. I have checked. It would be better that the authors further prove the advantages of dynamic graph construction compared to the traditional KNN-based approach.", "experimental_designs_or_analyses": "Yes, the experimental results are sufficient, and the corresponding analyses are self-consistent.", "relation_to_broader_scientific_literature": "The paper builds on recent advances in federated graph learning models, extending ideas from prior literature such as FedPUB and FedTAD. It attempts to introduce a novel method of dynamic graph construction in unsupervised settings, using graph kernel similarity and N-cut to enhance the clustering performance of each local model.", "essential_references_not_discussed": "This paper utilizes federated learning to train GCN models with reduced communication overhead, and the work on distributed graph learning with cross-client edges using homomorphic encryption [1] should be discussed.\n\n[1] Yao, Yuhang, et al. \"FedGCN: convergence-communication trade-offs in federated training of graph convolutional networks.\" In NeurIPS 2023.", "other_strengths_and_weaknesses": "Advantages:\n\ni. Novelty and Innovation: This paper proposes a new federated node-level clustering network for cross-subgraph link restoration, which is a fresh perspective in federated graph learning. \n\nii. Technical Contribution: The authors design three components (the local model learning, the cross-subgraph links mending, and the global knowledge sharing) that are seamlessly integrated into a unified optimization framework, offering a more  reasonable way to mend cross-subgraph links. \n\nDisadvantages: \n\ni. Some concerns: a) Can you describe what is the clustering ground truth in this task? Is it simply the node label? b) How are the parameters of each local model initialized in the federated learning framework? How is the initialization of the learnable weight matrix W achieved? More explanations should be given. c) Federated graph-level clustering also represents a variant of unsupervised federated graph learning. Could the proposed network be extended to handle this task? d) It would be better that the authors further prove the advantages of dynamic graph construction compared to the traditional KNN-based approach.\n\nii. Minor writing issues: a) The dataset names are not consistent. For example, Table 1 and Table 2 use 'Amazon-Computer' and 'Amazon-Photo,' while Figures 3, 4, 5, and 6 use 'Computer' and 'Photo.' While these do not significantly affect the content, they should be corrected to improve professionalism and readability. b) The presence of unnecessary punctuation marks (e.g., extra period at line 104 in the left column).", "other_comments_or_suggestions": "N/A.", "questions_for_authors": "a) Can you describe what is the clustering ground truth in this task? Is it simply the node label?\n\nb) How are the parameters of each local model initialized in the federated learning framework? How is the initialization of the learnable weight matrix W achieved? More explanations should be given. \n\nc) Federated graph-level clustering also represents a variant of unsupervised federated graph learning. Could the proposed network be extended to handle this task?\n\nd) It would be better that the authors further prove the advantages of dynamic graph construction compared to the traditional KNN-based approach.", "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": "OLET0QYV3Q", "reviewer": "Reviewer_c9LU", "summary": "The authors propose a Federated Node-Level Clustering Network (FedNCN), which is the first attempt to tackle the issue of link missing caused by graph partition in an unsupervised learning scenario. The core idea of FedNCN is to mend the destroyed links using prior clustering knowledge. Extensive experiments have been conducted to evaluate the performance of FedNCN.", "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": "# Response to Reviewer:\n\n**1. Relationships between three components:** Thanks. In our approach, the local model learning strategy collects and preserves more reliable clustering signals to prepare for the recovery of damaged sample connections. The cross-subgraph link mending strategy utilizes the prior learned clustering knowledge to establish correct links between subgraphs, which provides high-quality data for global consensus learning. The global knowledge sharing strategy learns the clustering-friendly consensus prototypes based on the mended graph and ensures reliable feedback for each client, enhancing the discriminative ability of each local model in graph-level embeddings. These three components are seamlessly integrated into a unified optimization framework. \n\n**2. Fig. 2 revision:** Thanks for your careful review. In Figure 2, 'clustering signals' and 'uploaded signals' refer to the same entities, and we have revised the notation for consistency according to your advice. We will update Fig. 2 in the final version.\n\n**3. Definition of “overlapping”:** The term \"overlapping\" has been widely present in previous studies on federated graph learning [1, 2], referring to the scenario where different subgraphs are distributed across multiple clients and share some nodes. In contrast, since the problem definition in our paper involves the structure loss caused by partitioning a complete graph into several subgraphs, we do not consider node overlapping and instead focus on the scenario where there are no overlapping nodes between subgraphs.\n\n[1]Baek, J.; Jeong, W.; Jin, J.; Yoon, J.; and Hwang, S. 2023. Personalized subgraph federated learning. In *ICML*, 1396-1415.\n\n[2] Zhu, Y.; Li, X.; Wu, Z.; Wu, D.; M, Hu.; and Li, R. 2024. FedTAD: Topology-aware Data-free Knowledge Distillation for Subgraph Federated Learning. In *IJCAI*, 5716–5724.\n\nThanks for your constructive comments, we hope that our responses will make you satisfied.", "claims_and_evidence": "Yes, the claims made in the submitted paper are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "Yes, this paper uses benchmark datasets and widely accepted evaluation metrics in the field.", "theoretical_claims": "The author suppose that connected nodes typically have high feature similarity, which is evident and has been used in previous work.", "experimental_designs_or_analyses": "The experimental setup is clearly presented, and the comparative experiments, ablation studies, and hyperparameter analysis are comprehensively covered.", "relation_to_broader_scientific_literature": "The paper proposes a federated graph learning framework that is capable of mending the destroyed subgraph links across clients.", "essential_references_not_discussed": "All the references that are crucial to the key contributions of this paper have been cited and discussed.", "other_strengths_and_weaknesses": "Strengths\n1. Overall, I think this is a good work that takes a step forward in unsupervised federated node-level graph learning for missing links. This work effectively leverages the prior learned clustering knowledge to enable the model to accurately conduct cross-subgraph link mending, which promotes the great clustering encoding capacity of the local model.\n2. The problem addressed in this paper is evident, and the innovations are novel, demonstrating its research value. Moreover, the overall structure of the paper is well-organized. Necessary illustrative figures are provided to help readers understand the contents. \n3. Experiments on five graph benchmark datasets demonstrated the effectiveness and superiority of the proposed FedNCN against its competitors. Furthermore, ablation studies and convergence analysis further confirm its strong potential for practical applications.\n\n\nWeaknesses\n1. The authors should further explain the relationship among the three proposed components, i.e., the local model learning strategy, the cross-subgraph link mending strategy, and global knowledge sharing strategy.\n2. A few errors need to be checked and corrected. For example, in Figure 2, do 'clustering signals' and 'uploaded signals' refer to the same entities? If so, they should be consistently noted in the figure. \n3. I have some questions about certain contents in this paper. In the left column on page 6, lines 320-322: “Here, we consider the scenario where there are no overlapping nodes between subgraphs. ” What does the scenario with overlapping nodes refer to, and why is this scenario not considered in this paper? Are there any other possible scenarios?", "other_comments_or_suggestions": "All concerns are presented as weaknesses.", "questions_for_authors": "Please check the weaknesses in “Other Strengths And Weaknesses” section.", "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": "9ljXnC4Fcx", "reviewer": "Reviewer_S8JV", "summary": "This paper designs a federated graph learning framework called Federated Node-Level Clustering Network (FedNCN) that mends the cross-subgraph missing links to enhance the clustering performance of each client in an unlabeled circumstance while not sharing private data. The work also conducts experiments on several benchmark datasets using the proposed method.", "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": "# Response to Reviewer:\n\n**1. Clustering centers:** Thanks. In FedNCN, both the client and the server have models that are used to learn the graph embeddings. The method for calculating the cluster centers is the same for both, initialized by K-means and updated iteratively by the model. However, the former (local model) learns the local graph data assigned to each client to obtain key clustering signals for subsequent cross-subgraph links restoration, while the latter (global model) learns the mended graph to obtain consensus prototypes that guide each local model to better cluster.\n\n**2. Explanation:** Thanks. In our manuscript, the methods marked with \\* indicate that supervised methods are adapted to an unsupervised scenario.\n\n**3. Source code:** Thanks. The complete source code and datasets will be available for reproducibility if the paper is accepted.\n\n**4. Limitations and future work:** Thanks. Our method is designed for federated node-level tasks with cross-subgraph link mending in the unsupervised scenario. However, many scenarios involve federated graph-level clustering in practical applications, such as social network analysis and disease prediction. In federated graph-level clustering, each client learns from independent graph-level datasets, which exhibit significant structural heterogeneity. The divergence in multi-source data sharing is further exacerbated due to the lack of label guidance, making it difficult for our current model to effectively address these issues. Future work can explore a cross-domain federated learning framework to be applied to broader scenarios.\n\n**5. Detailed issues:** Thanks for your careful review. We have revised the sentence “the hyperparameter $k$ indeed plays a crucial role in FedGCN, ...” to “the hyperparameter $k$ indeed plays a crucial role in FedNCN, ...”. We will try our best to double-check the manuscript carefully and correct similar typos in our final version.\n\nThanks for your constructive comments, we hope that our responses will make you satisfied.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes, I believe that the methods and evaluation criteria presented in this paper are meaningful for the proposed problem and have been clearly described.", "theoretical_claims": "The stability guarantee provided by FedNCN is both interesting and well-supported.", "experimental_designs_or_analyses": "Yes, the author has provided a detailed analysis.", "relation_to_broader_scientific_literature": "Different from existing supervised federated graph learning methods, the proposed federated learning framework is the first to explore the issue of missing links in an unsupervised setting.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Pros:\n- In contrast to existing unsupervised FGL methods, FedNCN is proposed to address the issue of missing links caused by graph partition, which reveals a certain novelty.\n- The motivations are presented clearly, and each innovation has an intuitive explanation. In particular, the proposed cross-subgraph link mending strategy is a significant technical contribution to the field of node-level FGL.\n- The results show that FedNCN consistently outperforms existing methods, even when fewer real labels are used. This robust experimental evidence supports the paper's claims.\n\nCons: \n- Both the local model and the global model in FedNCN compute cluster centers. What is the difference between these two types of clustering centers? Moreover, are these two cluster centers updated iteratively, or are they calculated only once? The authors need to give an instruction to make the reader understand them.\n- In Table 1, what does the \"*\" next to the methods signify? The paper does not provide an explanation for this. Could the author provide a clear explanation for the readers?\n- To enhance the reproducibility of the experiments, the authors should share the source code of the proposed method.\n- The limitations are not discussed in the paper. The authors need to provide a brief discussion on the limitations of the study to offer a more balanced and comprehensive evaluation of FedNCN. This would provide valuable guidance for future research in unsupervised FGL.\n- Some types need to be carefully checked and corrected, such as, “... crucial role in FedGCN...” --> “... crucial role in FedNCN...”.", "other_comments_or_suggestions": "See the weaknesses.", "questions_for_authors": "See the weaknesses.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "z8b7zsaoDs", "reviewer": "Reviewer_ZDxr", "summary": "In this study, the authors investigate two unexplored issues in federated graph learning (FGL), namely: 1) the heavy reliance on labeled graph samples that are difficult to obtain in real-world applications; and 2) the inevitable missing links caused by partitioning a complete graph into several subgraphs. To address these issues, the authors propose an easy-to-understand federated learning algorithm, named FedNCN, which introduces a dynamic graph construction scheme to mend the missing links among subgraphs. The main results on benchmark datasets show improvements in clustering performance compared to existing 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": "# Response to Reviewer:\n\n**(W1) :**\n\n**Reasons for the low NMI and ARI:** Thanks. The Questions dataset exhibits a significant class imbalance, with 47461 samples for class 0 and 1460 for class 1, which could result in lower NMI and ARI performance. Although class imbalance presents significant difficulties, our approach demonstrates competitive results relative to existing advanced methods. In future work, we will further improve FedNCN to enhance its performance in scenarios with class imbalance.\n\n**Future optimization directions:** Thank you for your valuable suggestion. We agree that our method could indeed be further improved by incorporating adversarial learning [1, 2] in future work. Specifically, there are two steps in our paper about maximizing edge construction and then minimizing edge retention. Transforming these current two steps into a dynamic optimization framework using adversarial learning could enhance the robustness of FedNCN in server mending missing links. We will consider this direction in our future work to further refine the proposed FedNCN. \n\n**(W2) :**\n\n**Expression:** Thanks. We have tried our best to avoid redundant expressions according to your advice. In addition, we will further refine the writing in the final version of the paper to make it more concise.\n\nThanks for your constructive comments, we hope that our responses will make you satisfied.", "claims_and_evidence": "The evidence provided in support of the claims in this paper is convincing.", "methods_and_evaluation_criteria": "Yes, the evaluation criteria make sense.", "theoretical_claims": "The paper primarily focuses on experimental validation, and no major theoretical inconsistencies were identified.", "experimental_designs_or_analyses": "Through a series of experiments, the authors verify the effectiveness of the method. Compared to existing advanced FGL methods, FedNCN achieves better results.", "relation_to_broader_scientific_literature": "This topic and the obtained findings are interesting to the federated graph learning community", "essential_references_not_discussed": "No, the related works that are essential to understanding the key contributions of the paper have been included in the manuscript.", "other_strengths_and_weaknesses": "Strengths: \nThe article is well-motivated. It is meaningful and reasonable to provide some insights about the newly designed choices, which are valuable for researchers in related areas. In addition, some experimental results seem good.\n\nWeaknesses:\n(W1) There are still some issues that should be further discussed. For example, in Table 1 on page 6, although the proposed method achieves the best performance on the Questions dataset compared to the advanced FGL method, its NMI and ARI scores are relatively low compared to other datasets. What are the potential reasons for this phenomenon? FedNCN first maximizes edge construction and then minimizes edge retention. In future work, adversarial learning [1, 2] could be used to further refine the entire process of FedNCN. \n[1] Gong L, Zhou S, Tu W, et al. Attributed Graph Clustering with Dual Redundancy Reduction[C]//IJCAI. 2022: 3015-3021.\n[2] Suresh S, Li P, Hao C, et al. Adversarial graph augmentation to improve graph contrastive learning[J]. Advances in Neural Information Processing Systems, 2021, 34: 15920-15933.\n\n(W2) The text contains some redundant expressions that should be optimized. For example: 1) Left column, lines 331-334: \"Here, 'Local' denotes the use of only our local model, while in 'FedAvg', 'FedProx', and 'FedPer', different aggregation methods are applied by the server, with the client all using our local model.\" This content has already been described in the main text and is reiterated in the table caption, leading to redundancy. It is recommended to express it only once to avoid repetition. 2) Right column, lines 344-346: \"'BS' denotes the local model of FedNCN. 'BS+GKS' denotes the FedNCN without mending the missing links, and 'BS+GKS+CLM' denotes the FedNCN.\" This description is repetitive and can be condensed to improve readability and logical clarity.", "other_comments_or_suggestions": "No further comments or suggestions.", "questions_for_authors": "No", "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: This paper received four effective reviews, and all of them are positive. Overall, the paper is of good quality and should be accepted.
Decision: Accept (poster)