Graph coarsening reduces the size of a graph while preserving certain properties. Most existing methods preserve either spectral or spatial characteristics. Recent research has shown that preserving topological features helps maintain the predictive performance of graph neural networks (GNNs) trained on the coarsened graph but suffers from exponential time complexity. To address these problems, we propose Scalable Topology-Preserving Graph Coarsening (STPGC) by introducing the concepts of graph strong collapse and graph edge collapse extended from algebraic topology. STPGC comprises three new algorithms, GStrongCollapse, GEdgeCollapse, and NeighborhoodConing based on these two concepts, which eliminate dominated nodes and edges while rigorously preserving topological features. We further prove that STPGC preserves the GNN receptive field and develop approximate algorithms to accelerate GNN training. Experiments on node classification with GNNs demonstrate the efficiency and effectiveness of STPGC.

翻译：图粗化在减小图规模的同时保留特定性质。现有方法大多保持图的谱特征或空间特征。近期研究表明，保持拓扑特征有助于维持在图粗化图上训练的图神经网络（GNN）的预测性能，但存在指数级时间复杂度的缺陷。为解决这些问题，我们提出可扩展的拓扑保持图粗化（STPGC）方法，引入从代数拓扑扩展而来的图强坍缩与图边坍缩概念。STPGC基于这两个概念包含三种新算法：GStrongCollapse、GEdgeCollapse及NeighborhoodConing，这些算法在严格保持拓扑特征的同时消除受支配节点与边。我们进一步证明STPGC能够保持GNN的感知域，并开发了加速GNN训练的近似算法。基于GNN的节点分类实验验证了STPGC的高效性与有效性。