An Efficient and Scalable Graph Condensation with Structure-Preserving

Abstract: Graph condensation (GC) is pivotal for enabling Graph Neural Networks (GNNs) deployment in resource-constrained scenarios by compressing large-scale graphs into compact synthetic counterparts. Existing GC methods commonly suffer from computational inefficiency due to coupled optimization as well as encountering poor generalization across GNN architectures. To address these challenges, this study proposes an Efficient and Scalable Graph Condensation with Structure-Preserving (SP-ESGC), which possesses a decoupled design that separates node condensation from graph structure generation. Specifically, it first employs heat kernel feature propagation to generate node representation via spectral graph theory-inspired diffusion. Further, a novel hybrid clustering strategy is designed to extracts discriminative intra-class centroids from the node representation. Finally, a pre-trained edge predictor infers transferable structural patterns from the original graph, ensuring accurate synthetic graph generation. Extensive experiments on real-world graph datasets demonstrate that the proposed SP-ESGC implementes a precise GC with significantly high computational efficiency. Moreover, SP-ESGC also generalizes well across diverse GNN architectures.

• The paper introduces SP-ESGC, a decoupled graph condensation method combining heat-kernel feature propagation, RFF-enhanced spectral clustering, and learned edge prediction without bi-level optimization.
• SP-ESGC achieves competitive or superior accuracy at condensation ratios as low as 0.05%, including 90.7% on Reddit versus 88.0% for GCond, while scaling to graphs with 232,965 nodes and 57 million edges.
• The method reduces condensation time dramatically, processing Reddit in 162.5 seconds versus 2,655.9 seconds for SimGC, while maintaining stable performance across GCN, GAT, GraphSAGE, SGC, and APPNP architectures.

Motivation and problem setting

Graph condensation (GC) compresses a large graph $\mathcal{T}=\{A,X,Y\}$ into a small synthetic graph $\mathcal{S}=\{A',X',Y'\}$ such that GNNs trained on $\mathcal{S}$ approach the accuracy of models trained on the full graph. The paper identifies three persistent deficiencies in existing GC methods. First, most approaches (gradient matching, trajectory matching) couple the optimization of node features, graph structure, and relay GNN parameters, requiring bi-level or tri-level optimization that is slow and hard to scale. Second, condensation objectives tied to a specific relay model generalize poorly across GNN architectures. Third, structural information is often lost or preserved only at high computational cost, as in SGDD. The proposed method, SP-ESGC, addresses these issues through a fully decoupled pipeline: node feature condensation and graph structure generation are handled by separate, training-free or lightly trained modules, eliminating bi-level optimization entirely.

Methodology

Heat kernel feature propagation. Node representations are obtained by solving the heat equation on the graph, $X(t)=e^{-tL}X$, where $L=I-S$ is the symmetric normalized Laplacian. Since exact matrix exponentiation is infeasible at scale, the authors use a truncated Taylor series with truncation coefficient $K$, yielding $H\approx\sum_{k=0}^{K}\frac{(-tL)^k}{k!}X$. This produces smoothed features that integrate local and global neighborhood information prior to any condensation step.

Hybrid clustering for node condensation. For each class, the propagated features are mean-centered and compressed via truncated SVD to a low-rank basis whose rank matches the target number of intra-class clusters. Random Fourier Features then approximate an RBF kernel mapping (justified via Bochner's theorem), embedding the low-rank basis into a nonlinearly discriminative space without constructing the kernel matrix explicitly. Spectral-space clustering is performed on this embedding, and centroids are computed as means of cluster members in the original feature space, stacked across classes to form $X'$.

Edge predictor for structure generation. A parameterized edge predictor $f_\theta$ is pre-trained on the original graph, taking concatenated node-pair features $[x_u;x_v]$ and outputting edge probability. All-pairs inference over the synthetic features yields a probabilistic adjacency matrix, which is sparsified by retaining only entries above a high-quantile threshold $\mathcal{S}=\{A',X',Y'\}$0. Structure generation thus becomes a learnable functional mapping that requires no access to the original adjacency during synthesis — a key source of the method's scalability.

Empirical results

Experiments cover five datasets (Cora, Citeseer, Ogbn-arxiv, Flickr, Reddit), spanning transductive and inductive settings, with up to 232,965 nodes and 57 million edges; all experiments run on a single RTX 3050 GPU.

Accuracy. SP-ESGC achieves best or second-best accuracy in the majority of the fifteen dataset/ratio configurations against eight baselines including GCond, SGDD, SimGC, GC-SNTK, and SFGC. Notably, it exceeds whole-dataset training accuracy on Cora (82.6% vs. 81.2% at $\mathcal{S}=\{A',X',Y'\}$1=1.30%) and matches it on Flickr (47.2% at $\mathcal{S}=\{A',X',Y'\}$2=0.10%). On Reddit at $\mathcal{S}=\{A',X',Y'\}$3=0.05%, SP-ESGC reaches 90.7% versus 88.0% for GCond, while GC-SNTK fails with out-of-memory errors — evidence that kernel-based alternatives do not scale to graphs of this size whereas SP-ESGC does.

Efficiency. The runtime results are the paper's strongest quantitative claim. At moderate ratios, SP-ESGC condenses Reddit in 162.5 seconds versus 2655.9 s for SimGC (the second fastest baseline) — roughly one-sixteenth of the time — and versus 378,220.9 s for SGDD. On Ogbn-arxiv it requires 143.4 s versus 28,897.8 s for GC-SNTK. These speedups stem directly from the decoupled design: no iterative GNN training occurs during condensation.

Generalization. Across GCN, SGC, GAT, GraphSAGE, and APPNP evaluated on condensed graphs, SP-ESGC maintains stable accuracy, whereas GCond degrades noticeably on attention-based architectures such as GAT. This supports the claim that decoupling condensation from a relay model improves cross-architecture transfer.

Ablations. Removing heat kernel propagation causes the largest degradation on Cora (78.6% vs. 82.7%), confirming its role in supplying stable global-information priors. Replacing the edge predictor with cosine-similarity edge construction hurts Flickr substantially (45.8% vs. 47.2%), indicating that feature similarity alone is insufficient for topology generation. Substituting plain K-means for the spectral clustering strategy costs roughly 1–1.4 points on Cora, Citeseer, and Reddit, validating the nonlinear clustering design.

Limitations and open questions

The paper concedes several constraints implicitly rather than explicitly. The edge predictor assumes that feature-similarity patterns learned on the original graph transfer to synthetic nodes; this assumption may weaken under very aggressive condensation ratios or domain shift, and the paper does not test cross-dataset transfer of the predictor. The hyperparameters $\mathcal{S}=\{A',X',Y'\}$4, $\mathcal{S}=\{A',X',Y'\}$5, $\mathcal{S}=\{A',X',Y'\}$6, and the threshold quantile $\mathcal{S}=\{A',X',Y'\}$7 are not subjected to sensitivity analysis, leaving their robustness unquantified. Evaluation is limited to node classification; extension to link prediction or graph classification tasks remains untested. Finally, since structure generation depends on a pre-trained model, the total runtime includes predictor pretraining, and the paper does not analyze how pretraining cost scales beyond the datasets studied here.

Conclusion

SP-ESGC offers a decoupled, largely training-free alternative to optimization-heavy graph condensation. Its combination of heat-kernel-smoothed features, RFF-augmented spectral clustering, and a pre-trained edge predictor delivers state-of-the-art or near-state-of-the-art accuracy at condensation ratios as low as 0.05%, with order-of-magnitude runtime reductions on large graphs and stable performance across heterogeneous GNN architectures. The main open questions concern sensitivity to its hyperparameters, the transferability of the learned edge predictor, and applicability beyond node classification.