Graph Coarsening with Preserved Spectral Properties

Abstract: Large-scale graphs are widely used to represent object relationships in many real world applications. The occurrence of large-scale graphs presents significant computational challenges to process, analyze, and extract information. Graph coarsening techniques are commonly used to reduce the computational load while attempting to maintain the basic structural properties of the original graph. As there is no consensus on the specific graph properties preserved by coarse graphs, how to measure the differences between original and coarse graphs remains a key challenge. In this work, we introduce a new perspective regarding the graph coarsening based on concepts from spectral graph theory. We propose and justify new distance functions that characterize the differences between original and coarse graphs. We show that the proposed spectral distance naturally captures the structural differences in the graph coarsening process. In addition, we provide efficient graph coarsening algorithms to generate graphs which provably preserve the spectral properties from original graphs. Experiments show that our proposed algorithms consistently achieve better results compared to previous graph coarsening methods on graph classification and block recovery tasks.