{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-unifying-framework-for-spectrum-preserving","title":"A Unifying Framework for Spectrum-Preserving Graph Sparsification and Coarsening","arxiv_id":null,"date":"2019-12-01","proceeding":"NeurIPS 2019 12","authors":["Gecia Bravo Hermsdorff","Lee Gunderson"],"abstract":"How might one ``reduce'' a graph? \nThat is, generate a smaller graph that preserves the global structure at the expense of discarding local details?  \nThere 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.  \nInterestingly, for a planar graph, edge deletion corresponds to edge contraction in its planar dual (and more generally, for a graphical matroid and its dual).  \nMoreover, 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.  \nIn 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.  \nThe 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 $\\mat{L}^\\dagger$, which remains finite in this limit.  \nMotivated by this insight, we provide a probabilistic algorithm that reduces graphs while preserving $\\mat{L}^\\dagger$, using an unbiased procedure that minimizes its variance. \nWe 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.","url_abs":"http://papers.nips.cc/paper/8989-a-unifying-framework-for-spectrum-preserving-graph-sparsification-and-coarsening","url_pdf":"http://papers.nips.cc/paper/8989-a-unifying-framework-for-spectrum-preserving-graph-sparsification-and-coarsening.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"a-unifying-framework-for-spectrum-preserving","repo_url":"https://github.com/Gecia/A-Unifying-Framework-for-Spectrum-Preserving-Graph-Sparsification-and-Coarsening","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[{"task_slug":"graph-similarity","task_name":"Graph Similarity"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}