Topology-preserving Graph Coarsening: An Elementary Collapse-based Approach

Graph coarsening techniques aim at simplifying the graph structure while preserving key properties in the resulting coarsened graph, have been widely used in graph partitioning and graph neural networks (GNNs). Existing graph coarsening techniques mainly focus on preserving cuts or graph spectrums. In this paper, we propose a new method that focuses on preserving graph topological features. In particular, we develop a novel graph coarsening approach, called Graph Elementary Collapse (GEC), by extending the concept of elementary collapse in algebraic topology to graph analysis. With this novel method, we can ensure a kind of equivalence relationship called homotopy equivalence of the graph during the coarsening process, thereby preserving numerous topological properties, including connectivity, rings, and voids. To enhance the scalability, we also propose several carefully-designed optimization techniques to reduce the time and memory consumption of our approach. Extensive experiments on several real-world datasets demonstrate the effectiveness and efficiency of our proposed method across various GNN prediction tasks.