Compactness and Consistency: A Conjoint Framework for Deep Graph Clustering
Authors: Wei Ju
Siyu Yi
Kangjie Zheng
Yifan Wang
Ziyue Qiao
Li Shen
Yongdao Zhou
Xiaochun Cao
Jiancheng Lv
Venue: ICLR 2026 Oral (2026)
Keywords: Graph Neural Networks
Graph Clustering
Representation Learning
Compactness Learning
Consistency Learning
Primary area: learning on graphs and other geometries & topologies

Abstract: Graph clustering is a fundamental task in data analysis, aiming at grouping nodes with similar characteristics in the graph into clusters. This problem has been widely explored using graph neural networks (GNNs) due to their ability to leverage node attributes and graph topology for effective cluster assignments. However, representations learned through GNNs typically struggle to capture global relationships between nodes via local message-passing mechanisms. Moreover, the redundancy and noise inherently present in graph data may easily result in node representations lacking compactness and robustness. To address these issues, we propose a conjoint framework CoCo, which captures compactness and consistency in the learned node representations for deep graph clustering. Technically, our CoCo leverages graph convolutional filters to learn robust node representations from both local and global views, and then encodes them into low-rank compact embeddings, thus effectively removing the redundancy and noise as well as uncovering the intrinsic underlying structure. To further enrich the node semantics, we develop a consistency learning strategy based on compact embeddings to facilitate knowledge transfer from the two perspectives. Our experimental results indicate that our CoCo outperforms state-of-the-art counterparts on various datasets. The code is available at https://github.com/juweipku/CoCo.

Published as a conference paper at ICLR 2026
COMPACTNESS AND CONSISTENCY: A CONJOINT FRAMEWORK FOR DEEP GRAPH CLUSTERING
Wei Ju', Siyu Yi* Kangjie Zheng', Yifan Wang', Ziyue Qiao², Li Shen", Yongdao Zhou"?
Xiaochun Cao Jiancheng Lv!
'College of Computer Science, Sichuan University
Coolleee of Mathematics, Sichuan University
3 WWelccome Sanger Institute *University of International Business and Economics
"School of Computing and Information Technology. Great Bay University
Schoool of Cyber Science and Technology, Shenzhen Campus of Sun Yat-sen University
"NTFFDD, School of Statistics and Data Science, Nankai University
j jueei (tssuu.ed..co siyuyi eew.ccueuu.c.. ydzhou@mankai..ero
ABSTRACT
Graph clustering a fundamental task in data analysis, aiming at grouping nodes with similar characteristics in the graph into clusters. This problem has been widely explored using graph neural networks (GNNs) due their ability to lever- age node attributes and graph topology for effective cluster assignments. However, representations learned through GNNs typically struggle to capture global rela- tionships between nodes via local redundancy and noise inherently present in graph data may easily result in node representations lacking compactness and robustmess. To address these issues, we propose a conjoint framework CoCo, which captures compactness and consistency in the learned node representations for deep graph clustering. Technically, our CoCo leverages graph convolutional filters to learn robust node representations from both local and global views, and then encodes them into low-rank compact embeddings, thus effectively removing the redundancy and noise as well as uncov- ering the intrinsic underlying structure. To further enrich the node semantics, we develop consistency learning strategy based on compact embeddings to facilitate knowledge transfer from the two perspectives. Our experimental results indicate that our CoCo outperforms state-of-the-an counterparts on various datasets. The code is available at http:///gthhb.comm/wveepk/CCCCCC
mechanisms. Moreover, the
INTRODUCTION
Graph clustering, as a critical task in network analysis and machine learning, plays a significant role in organizing and understanding complex relational data (Liu al., 2022b). It involves partitioning the nodes of graph into clusters, aiming to group nodes that exhibit similar characteristics or share related patterns. Its application spans across various domains, including social network analysis, biological networks, and recommender systems, among others. By identifying cohesive groups of nodes, graph clustering provides valuable insights into the underlying structure and connections within the data, enabling more effective knowledge extraction and -aaking processes.
For the past decades, many efforts have been devoted to gaining profound understanding of this fundamental problem, traditional methods typically rely on hand-crafted features (Yan et al., 2006) or graph partitioning algorithms (Ng et al., 2001; Vidal, 201 1), which aim to project data samples into a low-dimensional space while incorporating constraints to ensure clear separation between the samples. However, the conventional training paradigm tends to yield unsatisfactory outcomes, as the limited model capacity restricts their potential, failing to fully exploit the abundant structural information contained in the graph. This has led us to study deep graph clustering, offering superior adaptability and expressive power by automatically extracting informative features from graphs (Bo et al., 2020; Ju et al., 2023; Liang et al., 2025b).
"Corresponding Authors.

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Published as a conference paper at ICLR 2026
Recently, graph neural networks (GNNs) have emerged as effective approach, achieving remark- able success in capturing complex dependencies in graph-structured data and serving as a promising tool for deep graph clustering. Based this strength, many researchers have explored the potential of GNNs for deep graph clustering (Liu et al., 2022a; Yang et al., 2023; 2024; Liu et al., 2024a). For example, SDCN (Bo et al., 2020) is the first integrate structural information into deep clus- tering by bridging autoencoder representations with GCN layers through a delivery operator, while MAGI (Liu et al., 2024a) introduces a community-aware graph clustering framework that uses mod- ularity maximization as contrastive pretext task to uncover communities and mitigate semantic drift. In addition. DMGC-GTN (Wang et al., 2025a) explores a novel multi-modal graph clustering method that integrates structural and feature information via graph smoothing and transformer to exploit their complementarity. These GNN-based methods offer a data-driven way to learn node representations inherently capturing node attributes and the relational dependencies among neigh- boring nodes, potentially leading to more meaningful cluster assignments.
Despite the tremendous success of previous methods, there still exist some inherent limitations. First, existing graph clustering methods often struggle to effectively capture global relation- ships among nodes without any intervention (Chen et al., 2020a). Since effective local message passing mechanisms (Gilmer et al., 2017) in GNNs typically propagate information for only a few layers, they often overlook long-range dependencies, limiting their ability capture the underlying node distribution and resulting in suboptimal clustering. For instance, existing works (MAGI and DMGC-GTN) typically only perform local augmentation or random walks on the original graph, which fails to capture longer-range dependencies and consequently prevents clusters from accu- rately representing the underlying community structure. Second, the inherent redundancy and noise present in graph data poses challenges in learning compact and informative node repre- sentations (Kang al., 2019). Existing graph clustering methods often overlook the inherent re- dundancy and noise in data, which inevitably skews the training process and hinders the exploration of the underlying structure among data, thereby obscuring important relationships and patterns, and ultimately leading to less discriminative embeddings.
To address these challenges, we propose a conjoint framework capturing both Compactness and Consistency in the learned node representations, abbreviated as CoCo. Specifically, to explore both neighbor information and long-range relationships between nodes from local and global views re- spectively, our CoCo first leverages the graph convolutional filters encode the node attributes and graph topology based on the original graph and graph diffusion matrix, thus effectively learning complementary node representations. Then, encode node representations into low-rank compact embeddings, which learns the optimal low-dimensional subspace to characterize the intrinsic under- lying structure, thereby fully eliminating redundancy and noise. To well facilitate the knowledge sharing of the compact embeddings from the two perspectives, we introduce consistency learning strategy to encourage the model to produce consistent similarity distributions for each node, thus further enabling the learned node representations with richer semantics from both local and global information. Comprehensive experimental results across various graph datasets demonstrate the superior performance and effectiveness of our method over previous approaches.
2 METHODOLOGY
Notations. Let G = EV,E,,, X, A) denote an attribute graph of N nodes, where V = (u1,-... uN} represents the set of nodes and E V x V is the set of edges. Denote by X E NxF the attribute matrix of all nodes, where F is the dimension of attributes. A E (0, 1) NXN is the adjacency matrix, where A.j- = 1 (v;, 21y) E E. The normalized adjacency matrix is denoted as À = D 1/2 AD 1/2 where Á equals to A + IN with added self-connections, and D is the diagonal degree matrix with DH Σm= Ay. Then the symmetric normalized graph Laplacian matrix is defined as L L A.
Deep Graph Clustering, The task of deep graph clustering is to partition an unlabeled graph with N nodes into C disjoint clusters, denoted as (C,..... Cc) based on a well-trained node representation matrix Z E RRN In general, a self-supervised loss is developed to guide the training process to learn informative node representations. Then, a clustering algorithm such as K-means, spectral clustering, or a eeurall-etwoof clustering layer, is performed output the clustering results. In this section, we present a novel framework CoCo for deep graph clustering. The complete framework is depicted in Figure 1.
the trained node representations to
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Local
Embeddings Z
Compactness Learning
Consistency Learning
Low-rank Subspace Training
Upiate
Shep
Step
|10
Ancher
Smoothing
Filter
Uaskared
Z
7
Consistency Loss L
Fecture Reconstruction
Auchoz
Weight 9(Y
Optimal Subepace A
1.1.1-22
Emboddings 2
Global
Figure 1: Illustration of the proposed framework CoCo.
2.1 LOCAL- AND GLOBAL-VIEY FEATURE EXTRACTION
To learn effective node representations, most existing methods employ graph convolution on the adjacency matrix, which propagates messages between one-hop neighbors. For capturing long- range neighbor information, the convolution layers are deepened, which inevitably leads to the over- smoothing issue, i.e., indistinguishable node representations of different clusters, due the over- mixing of features and noises (Chen et al., 2020a). To alleviate this issue and fully explore graph topology, we leverage graph diffusion in this paper to smooth out the neighborhood over the graph. Formally, the graph diffusion matrix personalized PageRank (Page et al., 1999) with teleport probability E 1). The elements in S measures the influence/corrclation between all pairs nodes. Compared with the adjacency matrix A. S characterizes the soft relationships among the nodes, thus achieving the ability to globally exploit the long-range neighbor information. To reduce the computational complexity, there are fast approximations also sparsify (e.g., set values below a certain threshold to zero) to obtain S', and modify it as a (S' + 3/2 to maintain symmetry. In the following, we treat the tuples [X, A) and (X as the local and global views to retrieve different knowledge over the given graph.
defined as: alIx - (1 - -))A)-1 which adopts the
achieves linear runtime (Andersen al., 2006; Wei et al., 2018) and
Inspired by Wu et al. (2019), the entanglement of graph convolutional filter and weight matrix in GNN will harm both the performance and robustness when learning node representations. Instead, we adopt the disentangled architecture to encode the attribute and structure information from local and global views. First, we utilize the generalized Laplacian smoothing filters to denoise the high- frequency components and integrate node attributes and structure information:
(1) where k., k> 0) are real values and & is the number of the filter layers. i- L. and É. are the local- and global-view normalized graph Laplacian matrix, in which L can be acquired like the expression of L except replacing A with S. Theoretically, we can derive the following conclusion and the proof shown in Appendix B.l.
X = (IN - L'/ x= (IN (IN--= L*
Theorem kd' X/mar and Sinax are the optimal choice to pursue low-pass filters, where Aeur and Xmar are the maximal eigenvalues if and LF, respectively.
Then, to make node representations trainable, we learn weight parameters by feeding the filtered features into two unshared multi-layer perceptrons (MLP; and MLP2) for two views
z = MMPP (X'), z = (X.).
2.2 COMPACTNESS LEARNING FOR REDUNDANCY ELIMINATION
Low-rank representations can effectively exploit the inherent underlying correlation structure among data and suppress the impact of noise under the assumption that high-dimensional data points often intrinsically lie on a low-dimensional subspace (Ren et al., 2019: Tang et al., 2019; Jin et al., 2022a). For self-supervised learning, achieve better fusion of the semantics from the local and global
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views, expect to utilize the same set of low-dimensional subspace to abstract and reconstruct the low-rank node representations of the two perspeetives, which closes the gap between their semantic spaces and eliminates the redundancies in features, thereby uncovering the underlying data structure and yielding compact node representations from the two views.
Low-rank Subspace Training. Technically, we leverage the Gaussian mixture model (GMM, in Appendix A) (Richardson Green, 1997) to learn the optimal low-dimensional subspace that can represent the local- and global-view node embeddings in Equation 2. Let N = 2N,, denote the em- bedding space as Z (Z') Z )T E D the learnable subspace A E R (K < D), the j-th column of Z as . = (21j...... "N₂)² and the k-th column of A as X. Ak =
We introduce the latent variable matrix Y E DDxK where the element Yık in Y indicates whether z.j is related A. f. Under GMM, the goal is to maximize log p( (Z|A) and the likelihood functions for the observed data Z and the complete data {Z, Y) are proportionally formulated as:
J=11 (Ex k== ((.,,AA,A.
II, J(.,,AA.AA y
p(Z, YIA)
where a is a hyper-parameter to adjust the normal distribution. By implementing the expectation- maximization (EM) algorithm (Dempster al., 1977), introduced in Appendix A we can obtain in the E step that the posterior probability of the latent variable (p( (Y IZ, A)) is calculated by:
N(z A oIn)
TWj+) p(ys* = zz A
Σ N (Z... Aold aIg)
and the posterior expectation Q(A, Aold is: Q(A, Andd)
E (lo YJA)). In the M step,
maximize 2(A,Aob)) and the subspace is updated by:
Σ D
now
(3)
2j=1 (Mgnk)
Here, the qualities of the Gaussian means A. and the posteriors ((ysk) are critical. To simplify the model, we fix the mixture weights (priors) be equal and the covariance matrices be isotropic. does not compromise the model's generality, since the equal prior does not affect the posterior trends, which are primarily data-driven. also has advantages: (1) equal weights mitigate cluster collapse and promotes uniform coverage of the embedding space; and (2) fixing the covariance focuses the model's fitting capacity on the "mean-defined subspace", since based on the negative ELBO bound, maximizing the log-likel in the M-step is equivalent to minimizing the weighted squared- distance objective Σ, (8)x)) llz., - x= Theoretically, the iterative algorithm is guaranteed to converge (Dempster et al., 1977), as stated in Remark 1 and proven in Appendix B.2. In the experiment, verify that the subspace searching algorithm can achieve good performance within 10 iterations across different datasets, incurring negligible additional computational cost.
Remark 1. In each iteration, we have log (ZZ)A new) log p( A
By repeatedly iterating the E step and M step until converging. the algorithm enforces the trained subspace to effectively represent the core characteristies the embedded representation Z, as only the principal and cluster-level directions captured by the GMM means are retained while cor- related or weakly informative variations are removed. As such, the intrinsic data relationship is preserved in A while removing the redundancy. Discussion on the comparison with other rank reduction ways can be found in Appendix
Feature Reconstruction. Further, to hold the main energy of the "clean" data (Ren et al., 2019), we perform data reconstruction produce low-rank and compact representations stripped of redun- dancy. Concretely, we use the well-trained posterior probability of the latent variable (1yk) and the optimal subspace Å (Asz) to linearly reconstruct the original features, i.e., each entry in the reconstructed embedding Ż = (2ir) N is formulated as:
Eig k=1
Ast (Ujz))
(4)
Since the noise or unstable fluctuations the original embeddings cannot be expressed within the constrained subspace and thus vanish when reconstructing onto the original dimensions (mathemat- ically, we have rZn rank(ArT)) & min(rank(A) rank(())) K, which indicates that Ž
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maintains the low-rank property). Further, argue in Theorem 2 that the proposed approach constructs embedding Z in way that optimally preserves individual information and total variation of the original embedding Z. The proof is shown Appendix B.3.
Theorem 2. Under the low-rank feature reconstruction defined in Equation 4, the following two conservation properties hold:
(1) Individual Mass Conservation: the aggregated information for each individual is pre- served:
D Σ,= -.
for all i (1.....N)
(2) Maximal Variation Preservation: the reconstruction Z maximally preserves the total vari- ation of the original embedding Z among all low-rank factorizations. Specifically, is the solution the optimization problem:
Z
= arg min
Σ Ñ Z'EROxD
Ei= ZiglZig 2gg) subject to Z' ATT,
where A E R N and r E R with K E.
Theorem 2 (1) guarantees the invariance of each individual's total signal mass. This is erucial for fairness and interpretability, as it prevents the model from systematically biasing the reconstructed profiles of any individual; while Theorem (2) ensures that our reconstruction prioritizes the reten- tion of the significant variations (with large =gy) due to the cross-term En, Zeg2iy. This enables our low-rank reconstruction to align strongly with these dominant patterns. By leveraging the optimal low-dimensional subspace to reconstruct both the local- and global-view node embeddings, the se- mantic gap between them is also well alleviated. The obtained compact representation from both views can express the underlying data structure better, which is promising and beneficial to enhance the graph clustering. However, the above implementation is performed outside the gradient flow, hence inject the tuned representations back into the gradient path using a residual connection:
(5) On the one hand, it ensures the model remains trainable by allowing gradients flow through the residual path; on the other hand, it combines the global trends captured by the low-rank component with the local details preserved in the original Z, mitigating the over-smoothing that may result from relying solely on low-rank constraint and preventing model collapse (He et al., 2016).
2= )T - Z.
2.3 CONSISTENCY LEARNING FOR SEMANTIC ENHANCING
On the basis of compact node representations Z = (...... T and 2 = -Z)... ,,)) ) in Equation 5, develop the consistency learning to facilitate the exchange of knowledge between the two complementary perspectives. Meanwhile, we expect that the final representations can finely reflect the inherent relationships among nodes, thereby enhancing the label-free graph clustering. Toward this end, we share the compact semantics by comparing the similarities of each node to other samples in the embedding spaces of the two views.
We first randomly select a set of nodes over the given graph with indices (ai...., am) } as the anchor samples. Then, we calculate the cosine similarities between each node and these anchor samples and formulate the similarity distribution by the softmax operation. Mathematically, for the i-th node representations from the local and global views, the similarity scores of the m-th anchor are:
cxp(cos(Z)) (2), Zo )/T)
exp(cos E Z 2.. )/T)
I'm m exp(cos(2 Z2,, Z. )/T)*
>M exp(
oo )/T)
where cos(a, b) a aTa/////lla b/([|a] [[bl] is the cosine similarity, denotes the temperature parameter. For a comprehensive similarity measure, we need a large number of anchor samples so that they have large variations to cover the neighborhood of any node. However, it requires high computational costs to process too many samples in a single iteration. To address this problem, we maintain a memory bank with size M as a queue defined the fly by random nodes and calculate the similarity scores for each node and the samples in the queue. By dynamically updating the queue, we improve the diversity of the anchors with low complexity.
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Table 1: Clustering performance on five benchmark datasets (mean standard deviation). The top two results for each method are marked in bold and underline, respectively.
Dataset Metrle
SDCN
DFCN
AutoSSL
AFGRE
GDCL
ProCCi
COGC
Graphl.carner
MAGI
CoCo (Ours)
ACC
35.60=2 83
36.13-00.99
638144057
26,25e1 24
7083=047
57. 13.13.1.
7338+1.22 .02
74.1+1175 78
26.21+1.5
79.3640.69
Cora
NMI
04
14.28=191
04.6722110
19,36=0887
47.62x0.45
56.30=0.36 12.36+1554
14.32+1.87
48.05.0.72
0.36
30.71a2.70
3..021.33
52.51
46,5540044
53.8242
58. 16.164083
59,8420.45
57.62=0.81
58.7611.4F
60.71
F1
24.3711.01 26.16_0.50
56.4210.21
30.2011.15
52.88.0.97
15.68.1.29
70.9812.79
73.3311.86
24.0740.45
77.9510.72
ACC
53.44-0 81
7682-00.3
54 5540.9
75.51+0.77
43.75=0.78
51 53-03-00. 38
77 47.25+00.1
77.24+037
75.42+3.22
79.27+0.77
AMAP
NMI
31,214122 23
44,85 83
66.23=1.21
58.28=0.74
26.8720.33
54,4540.4 48
64 (4.05+0.10 15
21.57=0.51
97.32=0228
39 50.0..00.3
18=0.89
07 45444 48
57 67.9999.66
67.12+092
84.04
55.68:2.88
64.98=1 .2
68A5+11.5
60.94:1.11
50.66a1449
71.25.0.11
54.4740.83
69.9910.34
38.37
31.9720.44
73.0343.30
22.3641.15
ACC
53.05
55.73-0.06
42.43a0.47
50,9240.44
45.42=0554
55. 73.70.0
75.0141.78
75. 50
59,5423.90
78.8519.91
BAT
NMI
ARI
25.7115.71
91.04::997
18.7720.51
37.76.0.223
17.8110.98
13 81
21 1918910 74
27. 67
19.33:0.57
12
28 6920.92
21 8.41.34
50.2312.13
46.9513.09
47.45 53
50.50000
23.91
29.8325.13
53.52.1.15
55.0010.87
46.45+5.90
50.90=01.12
14.
46.53+0.57
39.94-0557
75
78.56
59
19.37=0.19
31.3340:52
33.4620.18
43.36=0.87
57.2240
19.10=1.55
58,8720.499
EAT
NMI
ARI
08.8312 54
06.31
32.90.0.11
23.25_0.18
07 07.6310.85
02 1310.67
06.5711.73
11.4411441
04.31.0.29
13:22
23.9020.45
15.0320.99
27.7110.41
83.85.0.87
26.2110081
33,471034
27.00
21.52.1.00 1.01
27.91=1.52
34.1011.26
33,42e3 10
42.45:00.04
21.82
30.53+1147
25.02=0231
45,5400.45
ST 09.0909994
57,5360.67
40.38=2.20
28.0612.64
92.25=1191
3361:0009
42.5244
41,50m0
06
3..3860.55
56,
55.31
30.35e0.14
59.6810.36
UNT
ARI
21,63=1 49
21.61+1.00
11.87=0.23
41.0141411
13.13+0.71
17.86+0.22
13.6240.57
17.33+0.54
21.76=0011
10-001
22.02-2
14.74e1.99
15+199
22.14+1 667
24.40+1
17.79=0.25
21,45=0.28
30.1+00.11
29 29.46600.4
F1
45.59g3.54 54
25.
79.099
34.9420.87
36.52+0.89 89
45. 15.69:0008
39, 30s
55 (5.2441.69
52.7762.66 61
47.5220.13
58.0350.34
With the local- and global-view similarity distributions p' = (pi.....,ph Phr)) and q (qi.... LM
we encourage the consistency of them facilitate the knowledge transfer and mutually enhance the representation semantics. Formally, we define the consistency learning loss as:
Σ 1=1 (KL Il'') + (I.(''lp'))
L
(6)
2N
where KL(-II-) is the Kullback-Leibe (KL) divergence. In the training, minimize L to optimize our proposed CoCo and enhance self-supervised learning. After converging, we fuse the local- and global-view representations by:
zF = ( + //2
Then, we perform K-means on the fused node representation z" to obtain the clustering results. An outline of the training procedure is provided in Appendix D. A detailed analysis of time and space complexities can be found in Section 3.8 and Appendix E.
EXPERIMENT
3.1 EXPERIMENTAL SETUP
We evaluate our CoCo with five widely used benchmark datasets for deep graph clustering, i.e., Cora (Sen et al., 2008), AMAP (Shchur et al., 2018), BAT (Liu et al., 2023c), EAT (Liu et al., 2023c), and UAT (Liu et al., 2023c). To comprchensively assess the effectiveness of our CoCo, we benchmark it against leading state-of-the-art methods, including untoencoder-basee methods, i.., DEC (Xie et al., 2016), IDEC (Guo et al., 2017), DAEGC (Wang et al., 2019), ARGA (Pan et al., 2019), SDCN (Bo et al., 2020), DFCN (Tu et al., 2021), and contrastive learning-based baased methods, i.e., AGE (Cui et al., 2020), MVGRL (Hassani & Khasahmadi, 2020), GDCL (Zhao al., 2021), AutoSSL (Jin et al., 2022b), AGC-DRR (Gong et al., 2022), AFGRL (Lee et al., 2022), GDCL (Zhao al., 2021), ProGCL (Xia al., 2022), RGC (Liu et al., 2023a), Dink-Net (Liu al., 2023b), CCGC (Yang et al., 2023), GraphLearner (Yang et al., 2024), and MAGI (Liu et al., 2024a). Details on evaluation metries and implementation are presented in Appendix F.
3.2 EXPERIMENTAL RESULTS
In Table 1 and Appendix G, we present the quantitative results of our proposed CoCo, compared with various competitive deep graph clustering baselines. From the tables, we draw the following key observations. On the one hand, compared to autoencoder-based methods, contrastive learning- based approaches show better performance. The reason lies in the ability of contrastive learning to more effectively exploit the intrinsic semantic information of the graph- graph-structured data. By learning discriminative representations in a principled manner, contrastive learning better serves the cluster- ing task. On the other hand, our approach achieves almost the best results all five datasets, and

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significantly outperforms the rumner-ups many datasets. For instance, Cora and BAT in Ta- ble I, our proposed CoCo surpass the runner-ups by [4.13%. 1.45%, 2.00%. 5.23% and (4.16% 8.74%. 12.79%, 4.19% under four evaluation metrics, providing substantial evidence for the su- periority of our approach. These results substantiate the success of the compactness learning and consistency learning embodied in CoCo for graph clustering, while also implicitly suggesting the superiority of cross-view consistency learning over contrastive learning, which is further validated in detail in Section 3.3 and Appendix H.
3.3 ABLATION STUDY
In this section, we analyze the impact of various components of our proposed method.
Comparison of different model variants. We first define different model variants as: (i) M1; solely adopt the local Laplacian smoothing filter to extract node representations (i.e., X!) for clus- tering; (ii) M2: solely adopt the global one (i.e., Xe) for clustering; (iii) M3: adopt local low-rank embeddings (i.e., Z by compactness learning based on X for clustering; (iv) M4: adopt global low-rank embeddings (i.e., Z²) by compactness learning based on X for clustering; (v) M2; move compactness learning from our full model CoCo. The results are summarized in Figure 2.
Comparing M3 with M1 and M2 with M2 mapping raw node features to low-rank em- beddings improves the performance in both cases, indicating the effectiveness of our low-rank representations in learning better cluster assignments. Similarly, when com- paring Ms with our CoCo, removing the low-rank mapping also leads to performance degradation, further emphasizing the neces- sity of compactness learning. In addition, comparing Ms, and CoCo with M,,M4, we observe significant performance difference between the two groups, possibly because consistency learning effectively inte- grates semantic knowledge from both local and global perspectives, enabling more dis- criminative and robust representations compared to any single viewpoint. Below, we further consider the comparison between the consistency loss and other surrogate losses.
200
9A.3
000
ALC
KMI
ACC
AXU
Cora
EAT
7oo
auo
50.3
400
50
ACC
(b) AMAP
(d) CAT
Figure 2: The ablation experimental results.
Influence of Consistency Learning. To investigate the advantages of our pro- posed consistency learming, we compare two widely used loss functions, Mean Squared Error (MSE) and contrastive loss InfoNCE (Chen et al., 2020b), on all datasets. The comparative results are pre- sented in Table 2 and Appendix H. can be observed that InfoNCE performs worse than MSE in most cases. This could be po- tentially attributed to InfoNCE's reliance on extensive negative instance sampling in
Table 2:
The comparative results of consistency learn- ing loss u.s MSE and InfoNCE.
Dataset
Loss
ACC
NMI
ARI
F1
MSE
77,8110.67
60.3110.89
57.81a1.12 12
73.89u1.00 .00
Cora
InfoNCE
75.57=1.16
58.08+1.44
54.69=1.85
72.58=1
Consistency
79.36-0.69
60.7110.39
58.7681.47
77.9540.72
MSR
77.62=0.44
67 66
58.51+0.97
71 77
AMAP
InfoNCE
77.25=0.33
67.1240
58.24e0.57
71 71.8930.33
Consistency
79.27a0.70
68.8511.55
60.94.1.51
72.36x1.15
MSE
57.18:0.74
28, 4519.66662
25.65=1.11
56 96166..77 73
UAT
InfoNCE
56.72+0.23
27 674674044 42
21.01+00:5
56. 56.39H0.3 39
Consistency
59.6820.36
30.12:0.51
29.4620.47
58.03.0.44
contrastive learning, which may introduce more complexity and sensitivity to tlyper-paramen tun- ing compared to the direct regression-based optimization of MSE. Moreover, the performance of consistency learning significantly outperforms the other two loss functions, which fully demon- strates the effectiveness of semantic enhancement brought by aligning the similarity distributions of local and global perspectives in consistency learning.
3.4 VISUALIZATION ANALYSIS
To visually verify the validity of our proposed CoCo, we plot the projected distributions (t-SNE) of the learned embeddings Cora and AMAP (Van der Maaten & Hinton, 2008), compared with six
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SDON
AutoSSL
ProGCL
CCGC
GraphLearner
MAGI
CoCo (Ours)
Figure 3: The i-SNE results comparing our CoCo with competitive baselines on two datasets. The first row and second row correspond to Cora and AMAP, respectively.
baselines. The visualization results are displayed in Figure 3. It can be observed that CoCo exhibits lower degree inter-cluster confusion Cora and better separability among different clusters on AMAP, showcasing CoCo's ability to learn more discriminative representations and effective cluster assignments compared with competitive methods.
3.5 SENSITIVITY ANALYSIS
Here, we study the sensitivity of key hyper-parameters: the subspace dimension K, the anchor number M, and the teleport probability a. The results are presented in Figure 4 and Appendix I. We also perform robust analysis with varying (in Equation 1) in Appendix
Effeet of K. As shown in the first row of Figure 4, when K ranges from 32 to 128, the clus- tering performance under four metrics on two datasets increases slowly. However, as K fur- ther increases, the performance of the model starts to decline. This is because incorporating too nigh-dimensional subspace introduces redun- dant irrelevant underlying structural information. which hinders the compactness of node embeddings.
Effect of M. The second row of Figure reports the impact of the number of anchor samples M in the queue. It can be observed that when M is very small (M=64), the model's performance is poorer. This because for each node, there not enough chor samples in its vicinity to adequately represent the node's 's neighborhood structure. As M increases, the model maintains relatively high performance and remains stable. This allows the model better cap- ture the neighborhood structure information of nodes and enhances consistency learning.
Effeet of In the third row of Figure 4, we report impact of the teleport probability a in graph dif- fusion matrix S. It can be observed that the model performance remains relatively stable when a is set to 0.1 0.2. However, the performance begins to decline, with a particu- larly noticeable drop at a 0.8. This is because larger a leads to more "localized" diffusion scope (prioritizing the node's own information). causing the two branches to capture increasingly similar information and thus failing to provide additional benefits. contrast, a smaller a results in more "global" diffusion (integrating information across the entire graph), which is more critical for capturing richer and complementary knowledge.
Soas
continues to increase,
(a) Cora
(b) AMAP
Figure 4: Sensitivity experimental results.
3.6 ROBUSTNESS ANALYSIS ON NOISY GRAPHS
To further assess the effectiveness and robustness of CoCo, we construct noisy graphs from BAT under three noise settings: (1) attribute noise by adding Gaussian noise (000, 5); (2) edge noise by
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Table 4: Comparison on heterophilic graphs.
Table 5: Comparison on homophilic graphs.
GraphLeurner
MAGI
CoCo (Ours)
DGCN
CoCo (Ours)
Dataset
Dataset
ACC
ACC
NMI
ACC
NMI
ACC
NMI
ACC
NMI
Cornell
43.00
14.17
34,26
7.63
59.68
24.34
Cora AMAP
72.89 76.06
56.82
79.36
60.71
Wisconsin
46.06
19.38
33.39
8.40
56.39
17.04
65.36
79.27
68.85
CuCa (Durs)
CoCa (Ohes)
79.
CaCa
ACC
73
DirioNei
ACC
MAGI
ACC
MALE
Gnghieame
26-
MAGI
Oogel
75.0
OCGC
CuptLamer
CCGC
GapiLann
76.5
Di6.Ne
765
Dik
5000
Time (mns spoch)
Memory
Time (mslepoch)
Memory (MiB)
(a) Cora-Runtime
(b) Cura Memory
(c) AMAP Runtime
AMAP Menoryy
Figure 5: Comparisons of average training time per epoch and memory cost.
randomly adding/deleting edges with probabil- ity 0.3; and (3) both attribule and edge noise. The clustering accuracy results in Table 3 show that, although noise degrades performance for all methods (cf. Table 1), our CoCo still consis- tently outperforms GraphLearner and MAGI. It highlights the strong redundancy deduundancy-rduuction reduction and noise-resistant ability of compactness learning, as well as the role of consistency learning in captur- ing richer semantic information to ensure robustness when facing noise.
Table 3: Comparison of accuracy on noisy BAT.
Noise Type
Attribute
Edge
Attribute Edge
GraphLearner MAGI
52.52
36.79
38.63
36.26
38.70
35.19
CoCo (Ours)
64.12
55.57
48.78
3.7 PERFORMANCE COMPARISON ON HETEROPHILIC GRAPHS
To further validate the applicability of our method, we compare the proposed CoCo with competitive baselines on the heterophilic graphs Cornell and Wisconsin (Pei et al., 2020), and we also compare CoCo on homophilic graphs with method tailored for homophilic graphs, i.c., DGCN (Pan & Kang, 2023). From Table 4, our CoCo mostly outperforms the recent competitive baselines Graphl and MAGI heterophilic graphs, likely due its robustness against cross-class edges via compact- ness and consistency learning. From Table 5, CoCo also surpasses DGCN on homophilic graphs, further confirming the effectiveness and applicability of our CoCo.
3.8 TIME AND SPACE COMPLEXITY COMPARISON
In this part, we compare the time and space complexities of our CoCo with several latest and com- petitive methods, i.e., CCGC, Dink-Net, GraphLearner, MAGI, in Figure 5. The results show comparison of clustering accuracy (ACC, 1) and average runtime/memory cost (L) among different methods. Our CoCo achieves best clustering performance on Cora and AMAP while maintaining relatively low time and space complexity, which further demonstrates the efficiency and scalability of our approach. Detailed theoretical analysis and comparison are provided in Appendix E.
Besides, to explore the impact of EM iterations on efficiency, we here we examine how differ- ent iteration counts affect performance (using NMI as an example) and efficiency (EM time per iteration/ total time per iteration). with the results presented Table 6. From the results, we can see that although the proportion of EM iteration time in the total runtime gradually in- creases with the number of iterations, the per- formance stabilizes after 10 iterations, indicat- ing that the model has already converged. Further increasing the number of EM iterations yields no additional benefits. Therefore, it can be concluded that the EM iterative algorithm accounts for only about 4% of the total training time, demonstrating remarkably high efficiency.
Table 6: Impact of EM iterations.
Cora Efficiency
AMAP Efficiency
Tterations
NMI
NM1
1
47, 14.097
1.62%
62.96.12.16 16
2.23%
5
59.81
2.93%
68.50-11.25
3.27%
10
60.71:1..99
4.28%
68. 1.55
4.24%
20
60.69.1..57
6.69%
68 68.87.11.8
6.16%
30
60.70:10.60
8.46%
68.81111.0
8.42%

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Published as a conference paper at ICLR 2026
3.9 EVALUATION ON NODE CLASSIHICATION TASK
We further analyze the generaliza- tion ability of our CoCo to verify whether the learned node represen- tations perform well on downstream tasks beyond graph clustering task, taking node classification as an exam- ple. We evaluate its performance un- der the predicted accuracy metric us- ing widely adopted datasets including WikiCS (Mernyei & Cangea, 2020), Computers (McAuley et al., 2015), Photo (McAuley et al., 2015). Coau- thor CS (Sinha al., 2015), and Coauthor Physics (Sinha et al., 2015), and compare it with several compet- itive state-of-the-ar methods, i.c., GCN (Kipf& & Welling, 2017), node2vec (Grover & Leskovec, 2016), DeepWalk (Perozzi et al., 2014), DGI (Velickovic al.. 2019), GMI (Peng et al., 2020),
Table 7: Performance comparison on node classification task (OOM denotes Out of Memory).
Dataset
WiliCS
Computers
Photo
Counthor CS Coauthor Physics
17..1900.12
86.51+0.54
92.42+0.22
93.03+0.31
95.65+0.16
node2vee DeepWalk
71,79=0.05
34.39+0.08
89 89.67+0.12
85.08=0.03
91.19+0.04
74.3520.06
85.6820:06
89.44a0.11
84.6120.22
91.7720.15
DGI
25.35=0014 83.95+0.47
91.61+0.22
92.15+0.63
+0,51+0.52
74.85=0.08
#2.21=0.31
90.68-0.17
OOM
MVGRI
77.52=0.08 87,52+0:11
91.74+0.07
92.1 |+0. 12
95. 05.33+00.0
GCA
78.30=0.62
88,49=0.51
92.99+0.27
92.76+0.
0OM
GRACE
78.2540.65
88.1550.43
92 52a0.32
92.60 11
OOM
CCA-SSG
77.88:0.41 87.01=0.41
92 5940.25
92.77=0.11
95,16:0,10
BGRL
79.36+0.53
88.35+0.32
92.87 07.877.227
91.72+0.21
95,43+0.09
GTCA
79.58+0.65
89.15+0:57
92.97+0.31
92.33+0. 18
96.24+0.07
CoCo
80.066+0.45 89.62=0.43 93.34+0.37
93,07H0.13
95.52+0.08
MVGRL (Hassani & Khasahmadi, 2020), GCA (Zhu et al., 2020), GRACE (Zhu et al., 2021),
CCA-SSG (Zhang et al., 2021), BGRL (Thakoor et al., 2022) and GTCA (Liang et al., 2025a).
Specifically, we first perform unsupervised pre-training using the proposed framework, followed by supervised fine-tuning on a labeled dataset. As shown in Table 7, our proposed CoCo outperforms all bascline methods across all datasets, demonstrating the strong generalizability of our learned node representations. This superior performance can be attributed to our method's 's ability to capture structural information from multiple perspectives and learn low-rank node representations that align more closely with the data distribution, effectively supporting various downstream tasks.
4 CONCLUSION
In this paper, we propose novel approach named CoCo for deep graph clustering. CoCo first encodes the attribute and topology information from local and global views. Then CoCo exploits low-rank embeddings via GMM to remove noise and redundancy, thereby uncovering the intrinsic structure of nodes. Based on the compact low-rank embeddings, CoCo also performs consistency learning of node similarities to enrich semantics information. Experiments under different data types (homophilic, heterophilic, and noisy graphs) and different tasks (graph clustering mainly and node classificaion) show that our CoCo outperforms the state-of-the-art methods. In addition, the comparison of spatiotemporal complexity demonstrates the efficiency of our method, For future work, we aim to extend our model to temporal graph clustering and single-cell genomics clustering, facilitating the analysis of dynamic structures and accurate grouping of cells by genetic profiles.
ACKNOWLEDGMENTS
This work is supported in part by the National Natural Science Foundation of China under Grants 62306014, 12501344 344 and 12131001, the Fundamental Research Funds for the Central Universi- ties, LPMC, and KLMDASR, the Postdoctoral Fellowship Program (Grade A) of CPSF under Grant BX20250376 and BX20240239, the China Postdoctoral Science Foundation under Grant 2024M762201, the Sichuan Science and Technology Program under Grant 2025ZNSFSC1506 and 2025ZNSFSCO808, and the Sichuan University Interdisciplinary Innovation Fund.
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A EXPECTATION-MAXIIIZZATIO ALGORITHM FOR GAUSSIAN MIXTURE
MODEL
The Gaussian mixture model (GMM) (Richardson & Green, 1997) is a linear superposition of mul- tiple Gaussian components. Assuming GMM consists of K Gaussians, given the observed data X (X1,-.., xN) the likelihood function is given by
N
K
p X
0
II
Σ
THN (xi Ext
where = (KK Pk. E& Mx Σx) is the probability density function of the k-th Gaus- sian, and тı is the prior probability. We can employ the masiattiooommmiiziaiiiionnoo (EM) algorithm (Dempster et al., 1977) to fit GMM, in which the latent variables Y = (ysx) E N are introduced to indicate whether the i-th sample belongs to the k-th Gaussian by yik-
In the E step, EM computes the posterior probability:
Th N(x; peg 23%
7(yak) p(yx 1|x;, 0old)
Ni N N(x; (xilpegh >3glit 2== nj
and the posterior expectation:
Q(0,
((sx))(og + LogN(x: En)).
1=
In the M step, EM maximizes )ol)) to achieve the updated parameters Oue:
N Di= T(yx)
1
new 1
)x; DOw
.V ((vck)
N
1
new k
(x; - (x: - 14% )
Σi 7(11x)
B PROOFS
B.1 PROOF OF THEOREM 1.
Proof. The smoothness of graph signal x, defined on the vertices of the graph, can be characterized by the Rayleigh quotient based on the normalized graph Laplacian matrix L (Horn Johnson, 2012), i..., x Lx/xT x, which expresses the normalized variance score of x. The smaller this value is, the smoother the signal is. Denote by ÜP eigenvector matrix and eigenvalue matrix of [ = 1,g). Based on the eigendecomposition of Lp, the signal x can be decomposed into:
.T and AP = diag(Af...... Aby) the
Th
Σ Due (Fourier inverse transform),
where c is the coefficient vector. Then the filtered signal by Equation is x² Em R. and the corresponding Rayleigh quotient is:
(1 T ΣN X//
T -1 /
To filter high-frequency noise, JP should be set to ensure the low-pass property of the filter (for any t), also the smoothness. On the one hand, any S i < N, which leads to IP > Åpm.. On the other hand, - Xt/k# kP should become smaller as XP becomes larger to ensure that the filter captures low-frequency signal. So the value of AP should be small and the optimal JP is Ap which completes the proof.
\ k2 should non-negative. Hence, kp 2 X for
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B.2 PROOF OF REMARK 1.
Proof. By the conditional probability formula, we have:
log p ("ew) log pl Z "[) log (osp((Pu Y >"en).
Hence, we further have:
log p X mew) )
p( PYY Aule) logp uon )YY
p P(Y X ol6 log Y new )dY
P Y A log Y 2 A )dY
Q(A new A H(A new Aall).
If Q(A"<M Aoh) 2 Q(A0l A and H H(A Aold) - H(Aold X old) hold, we can conclude that:
log logp(Z/Au> uow) 2 log (o
In the E step, maximized the Q-function, which implies that Q(A new .Aold) 2 Q(Auld Aoou). As for the H-function, we have:
H(X now Aaol) ( olo Aold)
= -/ pl pY log
p( Y Z, A new
p( Y Z,, Auid) dY
Zlog / p(Y|Z, dY log 1 = 0,
which follows from Jensen inequality, i.e., E(log X) S log E(X). The proof completed.
B.3 PROOF OF THEOREM 2.
Proof. After the subspace training algorithm converges, according to Equation 3 and summing
k, we have:
K
Σ A ((bn))
Σ
1y
5=1
1=1
Then the following equality also holds:
Σ
Σ
= Σ
(Wgsk)
Zig-
implies that ΣØ Zgj Σ Pij. which follows from that Σk (33x) and Equation holds. This guarantees that the total mass of each row in Z preserved, meaning that the aggregated information associated with individual remains unchanged after reconstruction. Also, it ensures that the individual-level signal encoded in the original embedding is retained.
In addition, the Q-function defined in the proof of Remark can be rewritten as:
K ((Mx) log
|z., - A
Q(A, A =
exp
(2TT) a
20
k=1
D K
D
K
||z
Σ
(1 log
Σ
(Urk)
(2t)) 0
20
j=1
j=1 k=1
N
K
=
log (2
2
g
+
P(Use = 1/2.,j. X. x) - Ast
20
k=
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which is maximized in the M step. Obviously, continuously maximizing Q-function throughout the entire iteration process is equivalent to minimizing the objective:
N
K
plus = 1 1/2 A. (2 - 2a))
(8)
Again, when converging, due to that Exs ((brk) = 1 and Equation holds, the optimized objec- tive in Equation can be written as:
N ΣΣ
(yga) -225)11))
As for the term Σ {VgR)) ) we have:
K
K
D
Σ
X2 j=1
(gra))
K
EJ= (Ugk)22s
(by Equation 3)
dek
k=
D
(Ugk))
Σ
j=1
Thus, we can obtain that the objective in Equation is minimized to
D
21j)
Σ
Σ 1
2
1=
original total variation
alignment with reconstruction
The first term, E., Zij- measures the total variation (energy) of the original embedding, while Σij ui quantifies how much of this variation is preserved in the reconstructed representation. Notably, compared with the self-energy Eiy(Et))² the cross-term 2, directly captures how well the reconstructed embedding aligns with the original signal rather than merely measuring its own magnitude. Hence, maximizing the cross-cores cors-correl term corresponds to keeping the dominant variations present in Z, and the above minimized objective brings reconstruction Z that preserves as much of the original total variation as possible while suppressing redundant or noisy components, which completes the proof.
C MORE DISCUSSIONS ON THE PROPOSED CoCo
GMM v.s. Other optional Rank Reduction Methods. In order to obtain low-rank representations, many methods can achieve this goal. However, the advantage of GMM lies its linear modeling complexity, with time and space complexities of ONN DK) and O(ND), respectively, while PCA has time and space complexities of O(N DD2 and O(ND+ D²). When the number of features is high, the complexity of PCA is also high. Compared to K-means, since GMM is probabilistic model and belongs to soft clustering algorithms, is more robust to outliers and can be applied to wider range of data distributions. Compared to methods like autoencoders for dimensionality reduction, GMM has fewer training parameters and lower time and space complexities. Therefore, paper chooses GMM to achieve low-rank representation learning. Of course, other better ways to conduct low-rank space training can be further explored.
Consistency Learning r.s. Contrastive Learning. Contrastive learning (Chen et al., 2020b; You et al., 2020) shares similar ideas with our consistency learning, utilizing the similarity of data rep- resentations under different data augmentations to bring similar samples closer and push dissimilar samples apart. However, there are significant differences between contrastive learning and our con- sistency learning. First, the proposed consistency learning aims to uncover relationship information
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among non-iid nodes, thereby better reflecting the underlying structural distribution of nodes in the representation. Second, we consider consistency learning at the relational information level from both local and global perspectives, emphasizing alignment learning. This enables representations from two perspectives to interact, guide each other, and learn, thereby obtaining more semantically rich node representations to better serve graph clustering. Third, contrastive learning involves neg- ative samples, whereas our consistency learning does not. In summary, contrastive learning focuses on sample discriminability at the instance level, whereas our consistency learning considers bridge the gap between the considered local and global views at the relational distribution level. This is higher-order, more global approach to capturing semantic knowledge in graph data.
Besides. in Table of the experiments section and Appendix H, we also compare the performance of our consistency learning loss with the contrastive learning loss (InfoNCE) (Chen et al., 2020b; You et al., 2020). It's clear that our consistency learning outperforms contrastive learning by a con- siderable margin, which fully demonstrates that our distribution- based consistency is more capable of capturing the intrinsic dependencies between nodes at a higher semantic level.
D TRAINING PROCEDURE
The training procedure of the whole framework is shown in Algorithm I below.
Algorithm The Optimization Framework CoCo
Input: Graph 9 {V, E,X, X, A); Cluster number C;
Output: Clustering result c;
Obtain the local- and global-view filtered attribute matrix X! and X in Equation 1;
2: Initialize the trainable parameters in MLP; and MLP2;
3: while not converge do
4: Update Z/ and Z by encoding X! and X in the unshared MLP encoders;
5: Update the optimal subspace Á by performing iterations through Equation 3;
6: Update the reconstructed embeddings Z and Z by Equation and Equation 5;
7: Update the similarity distributions p' and q' for i I,
8: Calculate the loss L in Equation 6;
9: Perform backpropagation and update the entire network in CoCo by minimizing L;
10: end while
11: Derive clustering result by applying K-means the fused representations from Equation 7:
E COMPLEXITY ANALYSIS
Given a graph dataset comprising N nodes and E edges. The sparse diffu- sion matrix S can be approximately calculated in linear time and space, ie., O(N) (Gasteiger et al., 2019). Assume the number of non-zero el- ements in the sparse diffusion matrix S is denoted as mnz(S), the compu- tational complexity of graph convolu- tional filters is (S)= N). For each dataset, they only need to be computed once at the beginning of training. In each epoch, GMM training and feature reconstruction have linear time complexity of D(NDK) and space complex- ity of O(N Consistency learning has time complexity of O(N MD) and space complexity of M(N In summary, the preprocessing time and space complexities of CoCo based sparsifi- cation are O(N) and OEE), respectively. In each training epoch, the time complexity of CoCo is O(NDK + NM and the space complexity of CoCo is O(ND NM).
Table 8: Theoretical analysis of complexity
Method
Time
Space
MVGRL.
O(N2D)
O(N2)
RGC
O(N2D)
O(N2)
CCGC
O(N2D)
O(N2)
Dink-Net
O(NCD)
O(NC I ND)
GraphLearner
O(N'D)
O(N2)
MAGI
O(ND²)
O + E N2)
CoCo (Ours)
OINDK + NMD)
OIND + NM)
Comparison between our CoCo and several recent and competitive baselines is presented in the Table 8. It can be observed that, comparing with the quadratic time complexity of MVGRL, RGC,
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CCGC and GraphLeamnr. our CoCo scales linearly with the number of samples, making it more suitable for large-scale datasets where computational efficiency is crucial.
F DETAILS OF EXPERIMENTAL SETUP
Datasets. Following Yang et al. (2023), we assess the performance of our CoCo with five widely used benchmark datasets for deep graph clustering, i.c., Cora (Sen et al., 2008), AMAP (Shchur et al., 2018), BAT (Liu et al., 2023c), EAT (Liu et al., 2023c), and UAT (Liu et al., 2023c).
Baseline Methods. To comprehensively assess the effectiveness of our proposed CoCo, we benchmark it against leading sfate-of-the-art methods, including autoencoder-based deep graph clustering/seffssupevvvedd learning methods: DEC (Xie et al., 2016), IDEC (Guo et al., 2017), DAEGC (Wang et al., 2019). ARGA (Pan et al., 2019), SDCN (Bo et al., 2020), DFCN (Tu al., 2021); and contrastive deep graph buustrring/sel-ssueerve
learning methods: AGE (Cui et al., 2020), MVGRL (Hassani & Khasahmadi, 2020), GDCL (Zhao et al., 2021), AutoSSL (Jin et al., 2022b), AGC-DRR (Gong et al., 2022), AFGRL (Lee et al., 2022), GDCL (Zhao et al., 2021), ProGCL (Xia et al., 2022), RGC (Liu et al., 2023a), Dink-Net (Liu et al., 2023b), CCGC (Yang et al., 2023), GraphLearner (Yang et al., 2024), and MAGI (Liu et al., 2024a).
Evaluation Metries and Implementation Details. We adopt four benchmark metrics following Bo et al. (2020) for evaluation: Accuracy (ACC), Normalized Mutual Information (NMI), Average Rand Index (ARI). and Macro Fi-score (F1). Larger values imply better clustering results. For each method, we present the mean and standard deviation of the four metrics across 10 runs. The proposed model implemented with PyTorch and all experiments are carried out on NVIDIA GeForce RTX 4090. The hidden dimension d is set to 500 for AMAP/UAT, and 1500 for other datasets. We set the total number of training epochs to 1000 and the number of filter layers to 3. The dimension of the subspace K is set to 64 and the number of the selected anchor samples M is set to 128. The parameters of temperature r and teleport probability a are set to 0.02 and 0.2, respectively.
G ADDITIONAL MAIN EXPERIMENTAL RESULTS
Table 9: Clustering performance on five benchmark datasets (mean + standard deviation). The top two results for each method are marked in bold and underline, respectively.
Dataset MMtric]
DARGC
ARGA
MYGRL
GDCL
ACC-DRR
RGC
Dink-Net
|CaCo (Ours)
ACC
146.5000 26 51.61=1 02 70-43:0.36
04 25 1.11.3 70.4743770 70 70.83:0.47 47 40.62
77.11=0.10
79.36:0.69
Cora
NMI
ARI
23,5440 34
15.13+042 22,07e1.55 53 49:63-0043
26.31=1.22
52 1.-000069
43 47.71=0.33 48 18.0540.72 72 48, 18,70e3 94 50,10e2 14 14800+164 49.46-2,72
51 52 57 $7.5 55.57+1.54 54 56. 30,0000022 36 18 74H447 57 57.60-1.36
50 7ettl 1D
16.16)
69.71
58.76+1.47
39.2360.17 17
47.17E1.11
68.27=0.57 69 (9.27:0.39
19.28 67.15&1.86
86 52.88:0.97 31,2340.57
74.7640110
77.95:0.72
ACC
05.22440 27.6240.00
08
9620.23
75.9810.68
41,0711311 12 43.7510.75 78 76.81x1.45
79.1140:43
79.27.0.70
AMAP
NMI
ARI
57,35+0.05 37,83+0.88
18,5940.04 19.24e0.07
58. 12=0.22 24 44 18s4.41
25-0.45
36.2.76
55.89+1.54
65.3840.61
18:7742.34
30.28+3.00
34
37.32+0
21.57H0.51
00.15+1.56
66.5441.24
69:6140.36
59.58-0039
59.76=01.6
67.70H0.30
60.94+1.51
68.85+1
F1
46.7160.15 12
47.20e60.1
69.87=0.54
64 1,95
71.74a0.93
32.8885.50
50 38.3750.29
7100340.64
72.86.0032
72.36el 15
ACC
[42,0962.2] 21
39.6240 87 52.67=0.00
0) 5640.80
56 66.6860076
76
37, 77.5664
32 45, 45,4260
54 47.79e0.02
54 20.00.1.33
78.8540.99
14.10+1.99
12.80s11.74
21.43=0.35
39.0990.54
36.04a1.54
31.70a0.42
19.9140.24
51.580083
28.33=0.32
55.00:0.87
ARI
07,9911.21
07. 17.85111.3
18.0820029 29 42 1.22..211
26 1.11
13:4510.03
19.3310.57
14 14.5910.13
12.16-1.35
23.0020.54
33.5221115
F]
42.6312.35 35
10.11
52.2310.03
67 1.15
55,0710.880
29 29.64.10449
39.9410.57
42.3310..1
53.2220.17
7..56.1.01
ACC
36.4711
60 35.5611.34
34 16.8910.15
13
47 17.26.0..3
32.88.1071
33.46.0 18 37.3720.11
50.78.0.17
58.8710.49
EAT
04.9611
(4463600
06:77:0.06
23.74x0090
1.72x11.0 08
13.22:0 33 07 00 37.7740.13
ARI
03.60+11.7
03.19-0.76
03-000
20+0.50
16.57+0.46
14.68+1.30
04.31+0.20
04 8840.9
30.16-0.15
188740.66
27.91+1
$4.84+1 18
35.52+1
50 34.72=0.16
52. $2.7500.07
45 45.54400.0
25, 25.3540.75
25.0220021
38 38.2070.17
48.2840.19
58.0662.64
ACC
#5.61a1.81
$1 16. 16.9010.17
52. $2.2990.19
19.3 19.11. 52. 52.3710.12
11.1611.38
48, 16.7010.06
42.6110.31
57.6510.09
09.6810.36
UAT
NMI
16.6312.30 39 17,8410.35 35 21.3310.44 44 25 4410.31
13.14+1 16.34H0.44
20.50-0.51
16-57-0.31
20 20.39H0..0
23 63.44.0666 66.10.66666
17.12+1.44 46 21.76=0.0
21.5310.94
25. 1.100000
09. 20.50H1.25
11.15ati.24
19.89=1.30
28.7950.35
2644746114 14
25.28.00.15
29.46:9.47
30.12.0.51
46,22+1.51
46.51=0.17
50.33=0.64
50.26+0.16
50.15+0.73
30 .9.45+2. 19 45.69+0
35. 1841323
54,5240.10
58.0350034
Apart from the comparisons with a range of competitive baselines in Table 1, we carry out more extensive comparative experiments herein. Table 9 presents a performance comparison between our proposed CoCo and several other graph clustering or unsupervised learning methods. It can be seen that RGC (Liu et al., 2023a) outperforms our model on certain metrics in AMAP and EAT, possibly because RGC can automatically learn the number of clusters, which may, to some extent, better approximate the true data distribution. However, we still achieve the best performance on most datasets. It can be attributed to the fact that our proposed CoCo captures complementary se- mantic information from both local and global views, enabling more comprehensive exploration
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of the graph's characteristics. Moreover, CoCo learns low-rank embeddings, which effectively removes redundancies and noise. Finally, consistency learning enriches the semantics of node rep- resentations from both local and global perspectives. Hence, different modules complement each other to facilitate better clustering assignments.
H ADDITIONAL RESULTS ON INFLUENCE OF CONSISTENCY LEARNING
Table 10: The comparative analysis of consistency learning across all datasets.
Dataset
Loss
ACC
NMI
ARI
MSE
77.84+0.67
60.31
73.89#1.09 47.81=1.42
Cora
InfoNCE
75.57+1.16
58.03+1
54.69+1.85
1..58+1.81
Consistency
79.36+4.69
60.7120.59
58.7621.47
77.9520.72
MSE
77.62:0.44
67.6810.76
58.5120.97
718420.77
AMAP
InfoNCE
77.25+0.33
67.12+0.46
58.24+0.57
7189+0.53
Consistency
79,27+0070
68.85&1.55
72.36=1.15 6.594s1.51
MSE 76.87=0.60
13.30&1.18
50.85=1.07
76.32=0.71
BAT
InfoNCE
78.70+0.53
53,99+0.74
51.85+1.14
78.03=0,51
Consistency
78.8520.91
55,0010.87
53.52&1.11
7..5611.01
MSE
57.4240.45
32.8010.86
26.27+0.74
37.8310.38
EAT
InfoNCE
55.26+1125
30,34+0.96
23.76=00944
55.53=0,94
Consistency
58.87=40.99
34.10=1.26
27.91#1.52
58.06+2.64
MSE
57.18+0.74
28.43+0.62
25.65=1.11
56.96+0.73
UAT
InfoNCE
56.72+0.23
27.67+0.42
25.01+0.45
56. 39:099
Consistency
59.68160.36
30.1240.51
29.4620.47
58.0320.34
To further comprehensively verify the effectiveness of our proposed consistency learning strategy. conduct additional experiments two more datasets beyond Table 2. As clearly shown in Table 10, the reported results consistently show that our consistency learning significantly outperforms both MSE and InfoNCE across all evaluated datasets. This superior overall performance strongly highlights the ability of our method to effectively capture high-order semantic similarity relations the powerful distribution-based alignment. The reason why the two compared metrics perform poorly may be that MSE only focuses on point-wise errors and ignores inter-sample correlations, while InfoNCE prone to instability due to the influence negative sample selection. Overall, these additional results provide broader and more empirical basis for our conclusions, further reinforcing the superiority, and robustness of our proposed consistency learning approach.
I ADDITIONAL SENSITIVITY ANALYSIS
Here we extend the sensitivity analysis of the key hyper-parameters K, M and to three additional datasets. The results in Figure 6 show that the clustering performance on all datasets increases slowly K ranges from 32 to 128, but starts to decline when K becomes too large. This is con- sistent with the observations on the two datasets in Figure 4 of the main text, indicating that a too tigh-diiensiooal subspace introduces redundant irrelevant information, which hinders the com- pactness of node embeddings. For the number of anchor samples M, the model's performance is poor when M is very small, due the insufficient anchor samples represent the neighborhood structure of nodes. As M increases, the performance improves and remains stable, which aligns with the findings in the main text. For the teleport probability a, the model performance is stable at low values but deteriorates with further increases, showing a marked decline at a=0.8. This trend aligns with findings in the main text. The degradation occurs because higher a results in more localized diffusion process, which over-prioritizes the node's own information. Conversely, a smaller a promotes global integration of graph information, which is essential for learning di- verse and complementary representations. These consistent trends across different datasets further validate the robustness of our model with respect to the three hyper-parameters.
J GRAPH CONVOLUTIONAL FILTER v.s. GCN
To show that our proposed disentangling of graph convolutional filters (GCF) and weight matrices can enhance the robustness the model, we conduct comparative experiments between GCF+MLP
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ANE
(a) Corn
(b) AMAP
(c) BAT
(d) EAT
(c) UAT
Figure 6: The sensitivity analysis of three hyper-parameters on all five datasets.
and GCN under the CoCo framework as the number convolutional layers
6231
(i.e., the parameter t in GCF, Equa- tion 1) increased. The results are shown in Figure 7. It can be observed that GCF+MLP consistently outper- forms GCN in terms of clustering performance across different convo- lutional layers. We also note that the performance of GCF+MLP remains relatively stable as the number of lay- ers (1 in Equation 1) increases, whereas GCN tends to first improve and then deteriorate with the addition of more layers most cases. This evidences the high performance and robustness brought about by disentangling the GCF from the weight matrices within our CoCo framework.
001
NMI
$3.5
50.1
53.5
$40
Layers
Layers
(a) Cora
(b) AMAP
Figure 7: Robustness analysis.
K
RELATED WORK
K.1 DEEP GRAPH CLUSTERING
Over the years, substantial research efforts have been devoted to advancing deep graph cluster- ing, with the goal of partitioning nodes in graph into cohesive and well-connected clusters in an end-to-end framework (Liu et al., 2022b). Deep learning breakthroughs have enabled the devel- opment of graph algorithms that leverage both graph structures and node attributes tackle this inherently complex task (Shi et al., 2019; Cheng et al., 2021; Yang et al., 2023; Yi et al., 2023; Liu al., 2024b; Trivedi et al., 2024; Guo et al., 2025; Ren al., 2025; Wang al., 2025b). Specif- ically, SDCN (Bo et al., 2020) and DFCN (Tu et al., 2021) provide a foundation for integrating graph structure into clustering objectives, laying the groundwork for more recent models. Several advanced methods have since been proposed, each targeting specific challenges in graph cluster- ing. AGCN (Peng et al., 2021) designs dynamic attention mechanism to adaptively aggregate the multi-scale features, while MCGC (Pan & Kang, 2021) learns consensus graph by filtering out high-frequency noise, preserving graph geometric features. GLCC (Ju et al., 2023) the first to extend clustering tasks to the graph-level scenario using the concept of contrastive learning. To ad- dress the challenges of targe-scale graph data, Dink-Net (Liu et al., 2023b) integrates representation learning with clustering optimization through dilation-shrink mechanisms and uses mini-batches for scalability, To eliminate the need for a predefined cluster number, RGC (Liu et al., 2023a) employs reinforcement learning jointly learn representations and adaptively adjust cluster numbers for bet-
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ter cohesion and separation. To generalize graph clustering to heterophilic scenarios, DGCN (Pan & Kang. 2023) and DMGC (Guo et al., 2025) reconstructs homophilic and heterophilic graphs with mixed filter that captures multi-frequenc features. Additionally, GCLR (Trivedi et al., 2024) and MARK (Fu et al., 2025) leverage large language models to enhance text-attributed graph cluster- ing. Despite these advancements, common challenge remains: the inability to capture long-range dependencies in graphs and the neglect of inherent redundant information and noise in data. These factors are crucial to capturing cohesive clusters complex graph structures. By learning robust and compact node representations in our method CoCo, it is promising to uncover meaningful patterns and communities within the graph.
K.2 GLOBAL DEPENDENCIES IN GNNS
Despite the success of GNNs in various graph-structured applications, their cesssage-passingg mech- anism (Gilmer et al., 2017) usually restricts them from capturing long-range dependencies. To overcome this, growing body of research has been dedicated to enhancing GNNs with global dependency capabilities, which can be roughly categorized into two lines: implicil global interac- tion and explicit structural augmentation. The first group keeps the original graph unchanged and introduces model-level global interactions, enabling node representations to exchange information globally in feature space (Ying al., 2021; Zhang et al., 2022; Cai al., 2023; Liang et al., 2023; Xing et al., 2024; Yi et al., 2025a). For example, Graphormer (Ying et al., 2021) leverages struc- tural encodings within Transformer framework to effectively capture graph dependencies, while CoBFormer (Xing et al., 2024) adopts bi-level architecture and collaborative training to more effectively capture global dependencies while retaining important local information. The second group focuses directly modifying the graph's connectivity by adding 'shortcuts' to physically shorten the distance between distant nodes, enabling standard GNNs to more ficxibly capture global relationships (Wu et al., 2022; Guo et al., 2024; Ju et al., 2024a; Yi et al., 2025b; Li et al., 2025). Specifically, GraphEdit (Guo et al., 2024) leverages large language models to learn global node-wise dependencies, denoise connections, and enhance graph structure learning through instruction-tune reasoning. In our work, we follow the second paradigm by constructing a graph diffusion matrix capture global dependencies, effectively modeling indirect connections and enabling clusters to accurately represent underlying community structures.
K.3 HOISE-RESILIEN LEARNING IN GNNS
GNNs are highly sensitive to various types of noise (Ju et al., 2024b; 2025), including feature per- turbations and redundant information, which can distort feature distributions and degrade perfor- mance. Existing nosse-resilient learning approaches can be broadly divided into two lines: adver- sarial defense and loss refinement. The first line explicitly defends against malicious worst-case perturbations by simulating or constraining adversarial noise (Feng et al., 2019; Alchihabi et al., 2023; Yuan et al., 2023a). For instance, BVAT (Deng al., 2023) employs batch virtual adversarial training to counteract noise and better capture local graph structures, while GCORN (Abbahad- dou al., 2024) enhances robustness against node feature attacks by enforeing orthonormal weight matrices. The second line enhances robustness by modifying training objectives to handle noisy or redundant inputs (Wang & Yang. 2022; Huo al., 2023; Yuan et al., 2023b; Ju et al., 2026). Specifically, BRGCL (Wang & Yang, 2022) iteratively estimates confident nodes to compute robust cluster prototypes and applies prototypical contrastive learning to learn noise-resistant node rep- resentations, while T2-GNN (Huo et al., 2023) uses a dual teacher-student framework to transfer clean patterns and recover noisy or corrupted node attributes. Unlike adversarial or loss refinement approaches, introduce the idea of low-rank learning to help the model learn compact represen- tations for graph clustering, as it explicitly models the underlying low-dimensional structure, and separate noise and redundancy the data-distribution level, thereby producing more robust and more clustering-frieet friendly representations.
24


Reviews:
[{"review_id": "6fLZCWuwc6", "reviewer": "Reviewer_bn3i", "summary": "This paper introduces a compactness-and-consistency framework designed for deep graph clustering. The contributions are three-fold as follows:\n1.The method jointly captures local and global graph structures via disentangled low-pass filters, effectively alleviating the over-smoothing problem.\n2.Low-rank GMM reconstruction removes redundancy/noise and yields compact embeddings while maintaining linear complexity.\n3.Cross-view consistency learning transfers knowledge at the distribution level, enabling mutual enrichment between different semantic spaces.\nExtensive comparisons with numerous baselines and across diverse data scenarios demonstrate the superiority and robustness of the proposed framework.", "questions": "1.Perform a sensitivity analysis on the parameter $\\alpha$ in the graph diffusion matrix, since different settings of $\\alpha$ correspond to different graph structures.\n2.What role does low-rank learning play in graph clustering, and what specific benefits does it provide in this paper?\n3.How can guarantee that the node representations obtained by feature reconstruction possess the low-rank property?\n4.What are the advantages of consistency learning over contrastive learning? Provide both theoretical and experimental analyses.\nIf the author addresses my concerns and doubts well, I am willing to raise my score.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1.**Theoretically sound**: The proposed framework provides a theoretical analysis of the effective parameter settings for graph filters and the ability of feature reconstruction to preserve semantic information.\n2.**Broad Applicability**: Superior and stable state-of-the-art performance on homophilic, heterophilic, nosiy and large-scale graphs.\n3.**Low Training Cost**: Both theoretically and experimentally, the proposed approach is verified to deliver low runtime and low memory costs.", "weaknesses": "1.How to guarantee the low-rank property?\n2.The effectiveness of consistency learning is insufficiently justified.\n3.Hyper-parameter analysis is inadequate.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "Grn5aeRxy8", "reviewer": "Reviewer_wkWL", "summary": "This paper proposes a conjoint framework named CoCo for deep graph clustering, which captures compactness and consistency in learned node representations by leveraging graph convolutional filters for local-global view feature extraction, GMM-based low-rank embedding for redundancy elimination, and consistency learning for semantic enhancement. Experimental results on multiple benchmark datasets show CoCo outperforms state-of-the-art methods in graph clustering and demonstrates good scalability and robustness.", "questions": "1.Although this paper claims a linear time complexity, could the iteration count of the EM algorithm impact the overall training efficiency?\n\n2.Why is EM-based subspace learning used?\n\n3.Why compact learning can eliminate redundancy and mitigate the impact of noise requires further clarification.\n\n4.When fusing local and global representations, equal weights are used. Why not adopt adaptive weights (e.g., learned via attention) to prioritize more informative views?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "1.The paper is clearly written and logically rigorous, presenting the proposed method in a clear way.\n\n2.The novelty and technical solution of the paper is sound and reasonable, long-range dependence and redundancy/noise are indeed the points that are easily ignored in graph clustering.\n\n3.The paper provides rigorous theoretical analysis and well-defined theoretical framework, Theorems 1-3 provide the optimal filtering parameters and the convergence guarantee for the dimension reduction, respectively.\n\n4.The authors conduct extensive experiments for the proposed framework, showing the superiority of the overall results. The ablation study also verifies the effectiveness of each component, and the analysis on different tasks and different scenarios also illustrates the good generalization of the framework.", "weaknesses": "1.The details regarding compactness learning and the EM algorithm are not elaborated or analyzed with sufficient clarity.\n\n2.The motivation behind compact learning for redundancy reduction is not sufficiently clear.\n\n3.The fusion of representations from different branches may lack flexibility.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "ghau5CGfV1", "reviewer": "Reviewer_y7YE", "summary": "This paper addresses the limitations of graph clustering, particularly their inability to capture global node relations and their susceptibility to redundancy and noise in learned representations. The authors propose a conjoint framework, CoCo, which learns robust representations from local and global views through graph convolutional filters, encodes them into low-rank compact embeddings to eliminate redundancy and noise, and further introduces a consistency learning strategy to transfer knowledge across views. Extensive experiments on benchmark datasets demonstrate that CoCo consistently outperforms state-of-the-art methods.", "questions": "- For the graph convolutional filter adopted in Eq. 1, how well would it perform if this filter were replaced with GCN?\n- The authors proposed GMM-based compactness learning; what impact would using only PCA dimensionality reduction have?\n- In Eq. 7, the authors only use sum averaging for the semantic representation of the two different branches. Would performance improve if attention weighting were used instead?\n- For the node classification evaluation, clarification is needed on whether the authors applied a classification head to the representation and trained via a supervised loss, or trained with an unsupervised loss before supervised fine-tuning.\n- The calculations of $\\mathbf{p}$ and $\\mathbf{q}$ are based on cosine similarity. Why was this similarity metric chosen instead of other methods? How would the results compare with those from other methods?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- The research topic is important and meaningful in the graph learning field, and appeals to a broad audience.\n- The paper is well-written and easy to follow. Its cutting-edge, rational technical solution addresses long-distance dependence, redundancy, and noise in feature representation, which are often overlooked in the current graph clustering field.\n- The theoretical analysis further reinforces the soundness and validity of the proposed scheme.\n- Extensive experiments and analyses across multiple downstream tasks and over ten baseline methods fully demonstrate the proposed method’s superiority, and effectively validate the study’s stated motivations.", "weaknesses": "- Neither the Introduction section nor the methods selected for comparative experiments have incorporated studies published in 2025.  \n- The main research results lack statistical significance tests.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "t11c4Bs3xJ", "reviewer": "Reviewer_4B3S", "summary": "This paper proposes a novel framework for deep graph clustering that aims to learn node representations which are both compact and consistent. The model addresses two key limitations in existing methods: the inability of standard GNNs to capture global node relationships and the sensitivity to noise and redundancy in graph data. The methodology first extracts node features from both local (original adjacency matrix) and global (graph diffusion matrix) views using disentangled graph convolutional filters. These features are then encoded into a low-rank subspace via a Gaussian mixture model to eliminate redundancy and noise, producing compact embeddings. Finally, a consistency learning strategy aligns the similarity distributions of nodes between the two views using a symmetric KL-divergence loss, enriching the semantics without needing negative samples. Extensive experiments on benchmark datasets show that CoCo outperforms state-of-the-art methods.", "questions": "- In Equation 1, what is the impact of different values of $k$ on model performance? Please provide further experimental explanation.\n- The author's original intention to propose the use of graph convolutional filters is to disentangle the filters and the weight matrix. How to prove its effectiveness?\n- In Section 2.2, why is GMM used for low-rank subspace training? Why is the original embedding representation $\\mathbf{Z}$ added in Equation 5?\n- Since the EM algorithm is independent of the model training, how does its initialization affect the model? What is the additional training cost required?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- The model effectively integrates both local and global structural information to form comprehensive node representations\n- The low-rank compactness learning effectively reduces noise and redundancy, leading to more robust embeddings.\n- The authors provide a detailed theoretical analysis that justifies the rationale behind the proposed technique.\n- It demonstrates strong generalization capability by achieving superior performance across diverse tasks, including both node clustering and classification.\n- The approach is computationally efficient and scalable to large graphs, owing to its linear complexity.", "weaknesses": "- The GMM-based low-rank learning adds model complexity and training steps.\n- The sensitivity analysis of hyperparameters is relatively limited.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "2ykQ0Nalje", "reviewer": "Reviewer_3Q5j", "summary": "This paper introduces CoCo, a conjoint framework for deep graph clustering. Addressing limitations of Graph Neural Networks (GNNs) in capturing global relationships and handling redundancy and noise in graph data, CoCo aims to learn node representations endowed with compactness and consistency. The method first leverages graph convolutional filters to extract node features from both local and global views. These representations are then encoded into low-rank compact embeddings using a Gaussian Mixture Model (GMM) to eliminate redundancy and noise. Finally, a consistency learning strategy is developed to facilitate knowledge transfer between the compact embeddings from the two perspectives, thereby enriching node semantics. Experimental results demonstrate that CoCo outperforms state-of-the-art methods on various benchmark datasets.", "questions": "Theorem 2 seems to be insufficiently proven.\n\nIs the setting of noise only beneficial to the author's method?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The paper clearly identifies existing GNN limitations in capturing global relationships and handling redundancy/noise, proposing targeted solutions. The proposed CoCo framework innovatively combines graph diffusion (for global view), disentangled graph convolutional filters (for feature extraction), GMM-based low-rank compactness learning, and cross-view consistency learning, with these components synergistically improving clustering performance.", "weaknesses": "The paper states that GMM aims to maximize p(Z|\\lambda) and that prior probabilities and covariance matrices are fixed. A more detailed explanation is needed for this simplification, such as why these parameters are fixed and how this impacts the model's generality. The limitations of the existing methods lack support from newer literature. Related work does not correspond to the limitations of the existing works. Theorem 2 requires further explanation. The information loss or over-smoothing risk that low-pass filtering and low-rank approximation may bring during denoising may reduce the practical application of this paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Oral)