A Unifying Framework for Spectrum-Preserving Graph Sparsification and Coarsening
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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 ∞, respectively. In this work, we provide a unifying framework that captures both of these operations, allowing one to simultaneously coarsen and sparsify a graph, while preserving its large-scale structure. Using synthetic models and real-world datasets, we validate our algorithm and compare it with existing methods for graph coarsening and sparsification. While modern spectral schemes focus on the Laplacian (indeed, an important operator), our framework preserves its inverse, allowing one to quantitatively compare the effect of edge deletion with the (now finite) effect of edge contraction.
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