A Unifying Framework for Spectrum-Preserving Graph Sparsification and Coarsening

How might one "reduce" a graph? That is, generate a smaller graph that preserves the global structure at the expense of discarding local details? There has been extensive work on both graph sparsification (removing edges) and graph coarsening (merging nodes, often by edge contraction); however, these operations are currently treated separately. Interestingly, for a planar graph, edge deletion corresponds to edge contraction in its planar dual (and more generally, for a graphical matroid and its dual). Moreover, with respect to the dynamics induced by the graph Laplacian (e.g., diffusion), deletion and contraction are physical manifestations of two reciprocal limits: edge weights of $0$ and $\infty$, respectively. In this work, we provide a unifying framework that captures both of these operations, allowing one to simultaneously sparsify and coarsen a graph while preserving its large-scale structure. The limit of infinite edge weight is rarely considered, as many classical notions of graph similarity diverge. However, its algebraic, geometric, and physical interpretations are reflected in the Laplacian pseudoinverse $L^\dagger$, which remains finite in this limit. Motivated by this insight, we provide a probabilistic algorithm that reduces graphs while preserving $L^\dagger$, using an unbiased procedure that minimizes its variance. We compare our algorithm with several existing sparsification and coarsening algorithms using real-world datasets, and demonstrate that it more accurately preserves the large-scale structure.