Dual-Center Graph Clustering (DCGC)

• The paper introduces DCGC, a deep graph clustering framework that jointly optimizes feature-space and neighbor distribution centers to enhance clustering robustness.
• It leverages stable neighbor distribution signals to guide hard negative mining and mitigate the effects of noisy pseudo-labels in graph data.
• Its two-stage training pipeline, combining contrastive representation learning with dual-target refinement, yields superior accuracy across diverse graph structures.

Searching arXiv for the main DCGC paper and a directly related contrastive graph clustering paper for citation support. Graph clustering is crucial for unraveling intricate data structures, yet it presents significant challenges due to its unsupervised nature. Dual-Center Graph Clustering (DCGC) is a deep graph clustering framework that combines contrastive representation learning and goal-directed clustering refinement, while departing from prior methods by using neighbor distribution as an additional, reliable structural signal (Cheng et al., 18 Jul 2025). Its central premise is that graph clustering should not rely only on a single feature-space center; instead, it should jointly optimize toward two complementary centers: a feature center in embedding space and a neighbor distribution center in a space derived from neighbor class distributions. This dual-center design is intended to make clustering guidance more complete and robust, especially on graphs with mixed homophily or heterophily (Cheng et al., 18 Jul 2025).

1. Conceptual motivation and problem setting

Prior goal-directed deep clustering methods such as SDCN and DFCN typically build a single target distribution from node embeddings alone. In DCGC’s formulation, this is incomplete because feature-space assignments are noisy early on and can be wrong for boundary or hard samples, a single center only reflects embedding proximity while ignoring neighborhood structure, and graph data often contain both homophilic and heterophilic patterns so structure cannot be summarized well by a single feature prototype (Cheng et al., 18 Jul 2025).

DCGC argues that neighbor distributions are stable and intra-class consistent: nodes in the same class tend to have similar distributions over the classes of their neighbors. This makes them useful as a second clustering view. The resulting optimization target is therefore not a single prototype-based refinement in embedding space, but a dual-center refinement coupling feature-space organization with structural regularity expressed through neighbor class composition (Cheng et al., 18 Jul 2025).

This positioning places DCGC within the broader family of goal-directed and contrastive clustering methods, but with a distinct emphasis on neighbor-distribution supervision. A related line of work, Deep Contrastive Graph Learning (DCGL), also uses centroid-based clustering-oriented guidance, though it addresses general data clustering without a reliable prior graph through a pseudo-siamese network, feature-level contrastive learning, local/global graph learning, and cluster-level contrastive learning (Chen et al., 2024). This suggests that DCGC belongs to a broader shift from unguided graph representation learning toward explicitly center- or centroid-guided unsupervised objectives.

2. Neighbor distribution as a structural supervision signal

For node $i$, the neighbor distribution $\mathbf{e}_i$ is defined as the average of the one-hot pseudo-labels of its neighbors: $$\mathbf{e}_i = \frac{1}{|N_i|} \sum_{j \in N_i} \mathbf{y}_j, \quad i=1,2,\ldots,N,$$ where $N_i$ is the neighbor set of node $i$ and $\mathbf{y}_j$ is the one-hot pseudo-label of neighbor $j$ (Cheng et al., 18 Jul 2025). Accordingly, $\mathbf{e}_i$ is a probability-like vector over clusters or classes that summarizes the local class composition around node $i$.

DCGC does not assume ground-truth labels. It first runs K-means on current embeddings $\mathbf{Z}$ to obtain pseudo-labels $\mathbf{e}_i$0, then computes neighbor distributions from these pseudo-labels (Cheng et al., 18 Jul 2025). At the cluster level, it aggregates neighbor distributions for nodes assigned to cluster $\mathbf{e}_i$1 to obtain the class neighbor distribution

$\mathbf{e}_i$2

where $\mathbf{e}_i$3 is the set of nodes predicted to belong to cluster $\mathbf{e}_i$4 (Cheng et al., 18 Jul 2025).

The paper emphasizes two properties of neighbor distribution: intra-class consistency and reliability. Nodes from the same class often have similar neighbor distributions, and neighbor distributions are less sensitive to individual pseudo-label errors than raw node-level predictions (Cheng et al., 18 Jul 2025). On that basis, neighbor distribution is used as a supervision signal both in contrastive learning and in clustering refinement. A plausible implication is that DCGC treats local class-composition statistics as a more stable proxy for cluster identity than direct pseudo-label assignments alone.

3. Contrastive representation learning with hard negative mining

DCGC improves contrastive learning by making negative-sample weighting depend on whether two nodes are actually similar in neighborhood structure. Using two MLP views $\mathbf{e}_i$5 and $\mathbf{e}_i$6, it applies an InfoNCE-style loss: $\mathbf{e}_i$7 Here $\mathbf{e}_i$8 is cosine similarity (Cheng et al., 18 Jul 2025).

In standard InfoNCE, all negatives are treated equally. DCGC instead introduces a weighting function

$\mathbf{e}_i$9

where

$$\mathbf{e}_i = \frac{1}{|N_i|} \sum_{j \in N_i} \mathbf{y}_j, \quad i=1,2,\ldots,N,$$0

The weighting therefore compares structure-aware similarity via neighbor distributions with current embedding similarity (Cheng et al., 18 Jul 2025).

The interpretation given is explicit. If two nodes have similar neighbor distributions but their embeddings are not yet similar, they are hard negatives and get larger weight. If two nodes are already consistent with neighbor structure and embedding similarity, they are down-weighted. The confidence threshold $$\mathbf{e}_i = \frac{1}{|N_i|} \sum_{j \in N_i} \mathbf{y}_j, \quad i=1,2,\ldots,N,$$1 acts on neighbor-distribution similarity (Cheng et al., 18 Jul 2025). This mechanism differs from prior hard-negative mining methods such as ProGCL or HSAN, which rely on pseudo-labels or embedding-based heuristics; DCGC’s signal is described as more stable because it uses neighborhood class composition (Cheng et al., 18 Jul 2025).

Within the larger training logic of the model, neighbor distribution is therefore not an auxiliary statistic but a direct supervisory signal for representation learning. During representation learning, it is used to identify hard negatives and improve the quality of embeddings; during clustering refinement, the improved embeddings produce better pseudo-labels, which produce better neighbor distributions, which then define the neighbor-distribution center and structural target (Cheng et al., 18 Jul 2025). This creates a feedback loop linking embeddings, pseudo-labels, neighbor distributions, hard-negative mining, and refined assignments.

4. Dual-center formulation and dual-target refinement

After pretraining, DCGC runs K-means on the learned embeddings $$\mathbf{e}_i = \frac{1}{|N_i|} \sum_{j \in N_i} \mathbf{y}_j, \quad i=1,2,\ldots,N,$$2 to obtain the feature-space centers

$$\mathbf{e}_i = \frac{1}{|N_i|} \sum_{j \in N_i} \mathbf{y}_j, \quad i=1,2,\ldots,N,$$3

which are the standard feature-space centers used in many deep clustering methods (Cheng et al., 18 Jul 2025).

DCGC additionally computes a center in the neighbor-distribution space,

$$\mathbf{e}_i = \frac{1}{|N_i|} \sum_{j \in N_i} \mathbf{y}_j, \quad i=1,2,\ldots,N,$$4

where $$\mathbf{e}_i = \frac{1}{|N_i|} \sum_{j \in N_i} \mathbf{y}_j, \quad i=1,2,\ldots,N,$$5 is the class neighbor distribution for cluster $$\mathbf{e}_i = \frac{1}{|N_i|} \sum_{j \in N_i} \mathbf{y}_j, \quad i=1,2,\ldots,N,$$6, initialized from the current pseudo-labels or soft assignments and then updated jointly with training (Cheng et al., 18 Jul 2025). This center represents the typical neighbor composition of nodes in cluster $$\mathbf{e}_i = \frac{1}{|N_i|} \sum_{j \in N_i} \mathbf{y}_j, \quad i=1,2,\ldots,N,$$7, serving as the structural counterpart of the feature center.

Feature-space soft assignments are computed with Student’s $$\mathbf{e}_i = \frac{1}{|N_i|} \sum_{j \in N_i} \mathbf{y}_j, \quad i=1,2,\ldots,N,$$8-distribution kernel: $$\mathbf{e}_i = \frac{1}{|N_i|} \sum_{j \in N_i} \mathbf{y}_j, \quad i=1,2,\ldots,N,$$9 These are then sharpened into a target distribution: $N_i$0 The paper describes this as the standard DEC-style target refinement (Cheng et al., 18 Jul 2025).

For the structural branch, DCGC replaces pseudo-labels with the soft assignment $N_i$1 to compute neighbor distributions $N_i$2, then defines assignment to the neighbor-distribution center as

$N_i$3

followed by sharpening: $N_i$4 The dual-target distribution thus consists of $N_i$5 for feature centers and $N_i$6 for neighbor-distribution centers (Cheng et al., 18 Jul 2025).

The clustering objective aligns both assignment types to their respective targets via a KL-type loss: $N_i$7 Here $N_i$8 balances the two terms (Cheng et al., 18 Jul 2025). The key distinction from single-center methods is that DCGC does not only refine assignments in embedding space; it also refines assignments in the structural neighbor-distribution space. The two targets are intended to be complementary rather than redundant.

5. Training pipeline and optimization procedure

DCGC has two stages. In Stage 1, pretraining is performed through representation learning. The method first builds an adaptive filterbank over the graph: $N_i$9 This combines low-pass filtering, high-pass filtering, and identity features (Cheng et al., 18 Jul 2025).

The filtered representation $i$0 is then fed into two unshared MLPs to form two views: $i$1 K-means is run on $i$2 to get pseudo-labels, after which neighbor distributions are computed and used to mine hard negatives and guide the contrastive loss (Cheng et al., 18 Jul 2025). The pretraining objective is

$i$3

where the reconstruction loss is

$i$4

with $i$5 (Cheng et al., 18 Jul 2025).

In Stage 2, dual-center optimization is performed. Pretrained $i$6 is used to initialize feature centers $i$7 and neighbor-distribution centers $i$8. The model computes $i$9 and $\mathbf{y}_j$0 using $\mathbf{y}_j$1 and $\mathbf{y}_j$2, updates target distributions $\mathbf{y}_j$3 and $\mathbf{y}_j$4 by sharpening every $\mathbf{y}_j$5 epochs, and minimizes the total loss

$\mathbf{y}_j$6

The resulting joint objective learns discriminative embeddings, preserves graph structure, and refines clustering assignments using both feature and neighbor structure (Cheng et al., 18 Jul 2025).

A plausible implication is that the two-stage design allows the model to avoid immediate dependence on unstable clustering targets, while still coupling contrastive learning to downstream cluster refinement. The paper explicitly states that contrastive learning is no longer an isolated pretext task; it is made directly useful for clustering through dual-center refinement (Cheng et al., 18 Jul 2025).

6. Empirical behavior, ablations, and comparative context

DCGC is evaluated on 10 datasets spanning high homophily, low or moderate homophily, and heterophilic settings: CORA, CITE, AMAP, UAT, EAT, BAT, Texas, Cornell, Wisconsin, and Squirrel (Cheng et al., 18 Jul 2025). The paper reports that DCGC outperforms all baselines across metrics including ACC, NMI, ARI, and F1. Specific reported examples include 78.68 ACC on CORA, 79.90 ACC on AMAP, 64.54 ACC and 37.13 NMI on Wisconsin, and 33.57 ACC and 7.96 NMI on Squirrel (Cheng et al., 18 Jul 2025).

The reported gains are especially large on heterophilic graphs, where feature-only methods struggle and structural signals become more valuable (Cheng et al., 18 Jul 2025). This suggests that the structural branch is not merely a regularizer for homophilic settings but an important source of information when local neighborhoods do not align cleanly with feature similarity.

The ablation analysis distinguishes representation-learning supervision from clustering-refinement design. For representation learning, the paper compares B, B+PL, and B+ND, corresponding respectively to baseline representation learning, pseudo-label supervision, and neighbor distribution supervision. The result reported is that B+ND consistently outperforms B+PL, indicating that neighbor distribution is a more reliable supervisory signal than pseudo-labels (Cheng et al., 18 Jul 2025). For clustering refinement, the comparison includes B, B+ED, B+ND, and Ours, corresponding to no goal-directed optimization, single feature center, single neighbor-distribution center, and dual-center optimization. Both single-center variants improve over no refinement, the two single centers are roughly comparable, and dual-center optimization is best, which is presented as confirmation of complementarity between feature and neighbor-distribution targets (Cheng et al., 18 Jul 2025).

The paper also reports that t-SNE plots show DCGC produces more meaningful cluster separation and improves boundary classification rather than merely maximizing compactness (Cheng et al., 18 Jul 2025). This is consistent with the earlier motivation that boundary or hard samples are especially vulnerable to early feature-space noise in single-center methods.

A related comparative reference is DCGL, which also uses clustering-oriented guidance but does so through sample-to-centroid relations, a pseudo-siamese architecture, local and global graph learning, and centroid-level contrastive alignment (Chen et al., 2024). DCGL is conceptually close in emphasizing two complementary views or branches, centroid-level or center-level supervision, structuring clusters around representative centers, and using cross-view consistency to refine clustering (Chen et al., 2024). The difference stated is that a typical dual-center graph clustering method like DCGC emphasizes two cluster centers or dual prototypes and consistency between them, whereas DCGL extends the center-guided intuition into a local/global graph and feature/cluster contrastive framework (Chen et al., 2024).

7. Advantages, limitations, and interpretive significance

The advantages identified for DCGC are specific. It uses a more reliable signal than pseudo-labels because neighbor distributions are less noisy and more structurally grounded; it works for both homophilic and heterophilic graphs because the adaptive filterbank and neighbor-distribution modeling help across graph types; it improves contrastive learning by mining hard negatives based on structural disagreement; it improves clustering refinement by adding a second target distribution in neighbor-distribution space; it provides end-to-end integration of representation learning and clustering optimization; and it incurs low additional cost, with dual-center optimization complexity stated as $\mathbf{y}_j$7, lower than contrastive learning’s $\mathbf{y}_j$8 when $\mathbf{y}_j$9 (Cheng et al., 18 Jul 2025).

The limitations are equally explicit. Neighbor distributions are derived from pseudo-labels, so the method still depends on the quality of initial clustering. The confidence threshold $j$0 is fixed, and the paper explicitly notes that learning a dynamic confidence mechanism is a future direction. The method uses K-means-based initialization and iterative updates, so it may still be sensitive to initialization in some settings. The paper also mentions future work on large-scale datasets, implying that scalability is not yet fully explored (Cheng et al., 18 Jul 2025).

These limitations delimit the scope of the claimed robustness. DCGC does not eliminate pseudo-label dependence; rather, it attempts to attenuate pseudo-label noise by aggregating class information over neighborhoods and by coupling feature and structural refinement. A plausible implication is that its main contribution lies not in removing the instability of unsupervised targets altogether, but in replacing a single fragile supervision channel with two mutually constraining ones.

In summary, DCGC improves graph clustering by combining representation learning with neighbor-distribution-aware contrastive learning and dual-center clustering refinement in both feature space and neighbor-distribution space (Cheng et al., 18 Jul 2025). Its key innovation is the recognition that neighbor distribution is a stable, class-consistent structural summary that can supervise hard-negative mining and serve as a second clustering center. Within the broader landscape of center-guided and contrastive clustering, this makes DCGC a formulation in which cluster refinement is simultaneously geometric and structural rather than purely embedding-centric.