DWGCN is developed to refine spatial graph construction for spatial transcriptomics (ST) analysis. The workflow of DWGCN consists of two main stages (Figure 1). In the first stage, a sparse, weighted adjacency matrix is constructed using inverse distance weighting (IDW) (Lu and Wong, 2008) over each spot’s neighbors. In the second stage, this distance-weighted adjacency matrix replaces the conventional degree-normalized adjacency matrix in GCN-based spatial clustering frameworks. The performance of DWGCN was systematically evaluated by comparing baseline models with their DWGCN-enhanced versions across both real and simulated spatial transcriptomic datasets.

Let X ∈ R N × M denote the gene expression matrix, where N denotes the number of spatial spots and M the number of genes.

In a graph convolutional network (GCN), the core operation involves aggregating information from a node’s local neighborhood based on a normalized adjacency matrix. Let Z l ∈ R N × d l denote the node representations at layer l , the layer-wise propagation rule is formally defined in Equation 7 as: Z l = σ A ̃ Z l − 1 W l − 1 (7) where W l − 1 is a learnable weight matrix, σ ( ⋅ ) is a nonlinear activation, and A ̃ ∈ R N × N is the normalized adjacency matrix.

Spatial GCN frameworks typically construct a KNN graph with a binary adjacency matrix A . The degree matrix D is defined as D i i = ∑ j A i j . Applying symmetric degree normalization yields the original normalized adjacency matrix as shown in Equation 8: A ̃ ori = D − 1 2 A D − 1 2 (8)

In DWGCN, the original degree-normalized adjacency matrix A ̃ ori is replaced with the proposed distance-weighted adjacency matrix A ̃ d w (as described in Section 2.2), which assigns larger weights to spatially closer spots. The propagation rule is defined in Equation 9 as: Z l = σ A ̃ d w Z l − 1 W l − 1 (9)

DWGCN replaces the original spatial adjacency in GCN-based frameworks with the distance-weighted adjacency matrix A ̃ d w while leaving all model architectures, losses, and hyperparameters unchanged. This design isolates the effect of graph construction, enabling fair and architecture-independent benchmarking of distance-aware connectivity.

Beyond providing a controlled comparison setting, the resulting A ̃ d w functions as a generalizable graph foundation that can be readily incorporated into diverse spatial transcriptomic workflows. Although broadly applicable to tasks such as spatial domain identification, trajectory inference, visualization, and denoising, here we focus specifically on evaluating its impact on spatial domain identification across representative GCN-based models.

We benchmarked DWGCN against four graph-based deep learning frameworks for spatial domain identification (Supplementary Table S1): SEDR (Xu et al., 2024), GraphST (Long et al., 2023), SpaNCMG (Si et al., 2024), and SpaGIC (Liu et al., 2024). For each baseline method, we followed the official implementations and default parameter settings unless otherwise specified. In the DWGCN-enhanced models, the original degree-normalized adjacency matrix A ̃ ori was replaced with the distance-weighted adjacency matrix A ̃ d w (with k = 12 and p = 2), and all other hyperparameters were kept unchanged to ensure a fair comparison.

Benchmarking was conducted on eight datasets, including four real and four simulated collections. The real-world collection comprised four publicly available spatial transcriptomics datasets with curated domain annotations (Supplementary Table S2): 12 slices of human dorsolateral prefrontal cortex (DLPFC) (Pardo et al., 2022), one mouse brain sagittal anterior section (Mouse_Brain) (Ji et al., 2020), one human breast cancer Block A Section 1 (Human_Breast) (Ji et al., 2020), and three mouse embryos (Mouse_Embryos) of E9.5-stage (Chen et al., 2022). In addition, four in silico datasets were generated using simSRT (Zhu et al., 2023) to model tissues containing 3, 5, 8, and 10 spatial domains (referred to as the cluster_3 to cluster_10 datasets; Supplementary Table S3). Each simulated dataset contained eight independently generated samples to account for stochastic variability and ensure statistical robustness. Details of data preprocessing and normalization are provided in the Supplementary Material.

Clustering accuracy was evaluated using three complementary metrics: Adjusted Rand Index (ARI) (Hubert and Arabie, 1985), Normalized Mutual Information (NMI) (Strehl and Ghosh, 2002), and Homogeneity (Rosenberg and Hirschberg, 2007). For each sample, both the baseline and DWGCN-enhanced models were run 20 times to account for randomness in model training. Performance improvement was evaluated using paired run-wise differences between the two models. For each sample s and run r , the improvement was computed as Δ v a l u e s , r = v a l u e s , r d w − v a l u e s , r ori , where v a l u e s , r d w and v a l u e s , r ori denote the metric values of the DWGCN-enhanced and baseline models, respectively. For summary reporting, run-wise differences were averaged within each sample.

The statistical significance of the paired differences was assessed using the paired Wilcoxon signed-rank test (Bauer, 1972). Multiple-testing correction was performed using the Benjamini–Hochberg procedure (Benjamini and Hochberg, 1995), and the resulting adjusted p-values are reported as FDR values. Unless otherwise stated, differences were considered statistically significant at FDR < 0.05 . In addition, effect sizes for the paired comparison were quantified using Cliff’s Delta ( δ ) , which measures the probability that a randomly selected value from the DWGCN-enhanced method exceeds that from the baseline (Cliff, 2014).

Analyses were conducted at both the sample and dataset levels. Sample-level analyses enabled direct paired comparison within each biological replicate, controlling for intra-dataset variability. Dataset-level analyses aggregated sample-level statistics to evaluate whether DWGCN yielded consistent, statistically significant improvements across heterogeneous biological and technical contexts. Together, this multi-level paired statistical design provided a robust and interpretable assessment of the performance enhancement introduced by DWGCN.

To examine how DWGCN reshapes the local aggregation structure, we compared the normalized edge-weight profiles generated by DWGCN ( k = 12) with those produced by a degree-normalized KNN adjacency constructed using symmetric normalization. For reference, degree normalization is representative of the normalization schemes widely adopted in standard GCNs, and is therefore used here as a conventional baseline. As summarized in Supplementary Table S4, the conventional adjacency construction produces nearly uniform weights (self-loop ≈ 0.077 ; neighbor weights ≈ 0.064 –0.077), indicating that spatial distance exerts minimal influence on the aggregation process. Because conventional KNN adjacency assigns all connected neighbors an identical weight before normalization, degree normalization cannot recover the lost geometric information. This near-uniform weighting reflects the smoothing-dominant behavior of conventional GCN propagation, where symmetric normalization reduces geometric distinctions among neighbors.

DWGCN, in contrast, introduces distance-aware weighting that becomes progressively more localized as the distance exponent p increases. When p = 0 , DWGCN produces uniform weights within each node’s KNN neighborhood, although the resulting graph is naturally asymmetric due to directional relative distances. For moderate exponents ( p ≈ 0.5 –1), the self-loop weight increases, and closer neighbors consistently receive higher weights than farther ones, establishing meaningful spatial discrimination while still retaining contributions from all neighbors. For large exponents ( p ≥ 4 ) , the distribution becomes highly concentrated: the self-loop weight dominates, and distant neighbors contribute only marginally (on the order of 1 0 − 3 or lower). The parameter p continuously controls the strength of spatial smoothing, ranging from uniform averaging ( p = 0 ) to distance-sensitive, structure-preserving aggregation.

Rather than seeking the optimal hyperparameters, which can vary across datasets and downstream tasks, we adopt a representative and stable configuration ( k = 12 , p = 2 ) to illustrate DWGCN’s intended behavior. At p = 2 , the resulting weight profile (self-loop ≈ 0.36 ; nearest neighbors ≈ 0.09 ; distant neighbors decreasing smoothly to ≈ 0.03 ) reflects a balanced regime in which self-information is strengthened and spatially proximate neighbors are emphasized, while farther neighbors still retain non-negligible influence.

Across four real spatial transcriptomics datasets, incorporating DWGCN consistently improved clustering accuracy under diverse model architectures and biological contexts. As summarized in Figure 2A, the DWGCN-enhanced versions outperformed their baseline counterparts on all three evaluation metrics (Adjusted Rand Index (ARI), Normalized Mutual Information (NMI), and Homogeneity), yielding average performance gains of 0.038 (9.09%), 0.032 (5.44%), and 0.038 (6.30%), respectively. Visual inspection of spatial clustering patterns further corroborated these trends: DWGCN-enhanced models produced clearer laminar structures on representative real datasets (Supplementary Figure S1).

At the dataset level, the accuracy improvement provided by DWGCN was consistent across all four datasets, demonstrating stable performance gains regardless of tissue type or experimental condition (Figure 2B). Across four datasets and three clustering metrics, a total of 48 paired comparisons were performed between each baseline model and its DWGCN-enhanced version. Among them, 27 cases (56.3%) exhibited significant enhancement, whereas only three (6.3%) showed significant decreases (Supplementary Table S5). Effect-size analysis showed consistent trends, with positive large effect sizes in 23 (47.9%) cases and negative large effect sizes in four (8.3%) cases (Supplementary Table S6).

On a per-sample basis, 75%, 67.65%, and 69.12% of samples exhibited improved performance, approximately 50% reached statistical significance (Supplementary Table S7). In comparison, only 4%–10% of samples exhibited significant declines. The heatmap in Figure 2C further visualizes sample-level performance gains, where color intensity reflects the magnitude of improvement. Sample-wise distributions of ARI, NMI, and Homogeneity also consistently favored DWGCN across methods (Supplementary Figure S2), with performance gains consistently observed across individual samples (Supplementary Figure S3).

To further characterize model-specific behavior, volcano plots highlighted distinct sensitivity patterns among the four representative models (Figure 2D). Overall, DWGCN yielded the greatest enhancement when integrated with SpaNCMG, followed by GraphST, SpaGIC, and SEDR. DWGCN achieved 12 and 11 significant improvements for SpaNCMG and GraphST, respectively, without any observed degradation. SpaNCMG displayed the most stable behavior, showing 100% significant gains based on 12 dataset–metric combinations. GraphST exhibited slightly lower but still consistent improvement with 80%–85% significant improvements and no negative cases. In contrast, SpaGIC exhibited both positive and negative effects, with roughly half of its comparisons showing significant improvements and the remainder showing significant decreases. SEDR showed minimal response to DWGCN, with only marginal decreases in the Human_Breast dataset for NMI and Homogeneity, suggesting that its embedding structure is less influenced by distance-weighted connectivity.

Performance evaluation on simulated datasets revealed an even more pronounced improvement achieved by DWGCN across all tested frameworks (Figure 3). As shown in Figure 3A, the DWGCN-enhanced versions consistently outperformed their baseline counterparts across three metrics. The average gains over the baseline methods were 0.047 (an 11.46% improvement relative to the baseline mean ARI) for ARI, 0.033 (5.66%) for NMI, and 0.036 (5.67%) for Homogeneity. A large majority of samples (84.38%, 89.06%, and 88.28% for ARI, NMI, and Homogeneity) showed improved performance, of which approximately 57.81%–72.66% reached statistical significance (Supplementary Table S10, Supplementary Figure S5). The proportion of samples with decreased performance remained low (10%–15%), and significant decreases were minimal (3%–6%). Representative examples from simulated datasets also illustrated the improved recovery of ground-truth spatial structure under DWGCN (Supplementary Figure S4).

At the dataset level, DWGCN outperformed baseline models in 89.6% (43/48) of pairwise comparisons (Figure 3B). Statistical significance was achieved in 81.3% (39/48) of cases, and 70.8% (24/48) showed large effect sizes (Supplementary Tables S8, S9). In contrast, only 10.4% (5/48) exhibited significant decreases, and 8.3% (4/48) showed no difference. Consistently, the sample-level heatmap in Figure 3C illustrated that 87.2% (335/384) of individual samples showed higher accuracy under DWGCN, with 66.1% achieving significant improvements. The volcano plot in Figure 3D further confirmed this pattern, with most points shifted toward the positive-effect side, indicating predominantly beneficial and frequently significant improvements. Together, these results demonstrate that the observed enhancement is both statistically significant and a large-effect improvement across dataset- and sample-level analyses.

Notably, the performance benefit of DWGCN became more pronounced on datasets with higher clustering complexity. As shown in Figure 3A, clustering performance declined as the number of clusters increased, reflecting the increasing difficulty of resolving finer-grained spatial boundaries as cluster numbers grow. However, DWGCN-enhanced models maintained higher clustering accuracy and demonstrated progressively stronger benefits as the number of spatial domains increased from 3 to 10 (Supplementary Figure S6). At low complexity (3 clusters), improvements were modest and occasionally unstable, with slight declines observed for GraphST and SpaNCMG. In contrast, at higher complexity (5–10 clusters), all frameworks achieved consistent and substantial performance gains. Effect size analysis confirmed this trend: the average Cliff’s Delta increased steadily from 0.272 (small effect) at cluster_3 setting, to 0.575 (large effect) at cluster_5, 0.760 (large effect) at cluster_8, and 0.816 (large effect) at cluster_10 (Supplementary Table S10).

Collectively, these findings indicate that DWGCN not only improves clustering accuracy but also enhances model robustness and scalability under increasingly complex spatial structures, underscoring its strong potential to generalize across diverse spatial transcriptomics settings.