Scalable Topology-Preserving Graph Coarsening with Graph Collapse

TL;DR

This work introduces Scalable Topology-Preserving Graph Coarsening (STPGC), a framework that preserves topological features during graph coarsening by extending strong collapse and edge collapse from algebraic topology to graphs. It comprises three operators—GStrongCollapse, GEdgeCollapse, and NeighborhoodConing—plus an ApproximateCoarsening stage based on $r$-relaxed dominance to reach practical coarsening ratios while maintaining the GNN receptive field through homotopy-equivalent reductions. The approach demonstrates superior node-classification performance and up to 37x speedups over the prior topology-preserving method GEC, along with favorable memory usage and Betti-number preservation on diverse datasets. STPGC offers a scalable, topology-aware alternative for accelerating GNNs, with potential for broader applications in domains requiring topology-conscious graph simplification.

Abstract

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.

Figures (10)

• Figure 1: Examples of graph strong collapse and graph edge collapse. The dominated nodes and edges are shown in red.
• Figure 2: Examples of neighborhood coning. The inserted edges and newly created dominated nodes are shown in red.
• Figure 3: Memory overhead of STPGC and baseline methods.
• Figure 4: Betti number preserved by different methods.
• Figure 5: Impact of parameters on accuracy and runtime.

Theorems & Definitions (14)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Example 2.4
• Lemma 2.5
• Definition 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Definition 3.5