{"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/randomized-near-neighbor-graphs-giant","title":"Randomized Near Neighbor Graphs, Giant Components, and Applications in Data Science","arxiv_id":"1711.04712","date":"2017-11-13","proceeding":null,"authors":["George C. Linderman","Gal Mishne","Yuval Kluger","Stefan Steinerberger"],"abstract":"If we pick $n$ random points uniformly in $[0,1]^d$ and connect each point to\nits $k-$nearest neighbors, then it is well known that there exists a giant\nconnected component with high probability. We prove that in $[0,1]^d$ it\nsuffices to connect every point to $ c_{d,1} \\log{\\log{n}}$ points chosen\nrandomly among its $ c_{d,2} \\log{n}-$nearest neighbors to ensure a giant\ncomponent of size $n - o(n)$ with high probability. This construction yields a\nmuch sparser random graph with $\\sim n \\log\\log{n}$ instead of $\\sim n \\log{n}$\nedges that has comparable connectivity properties. This result has nontrivial\nimplications for problems in data science where an affinity matrix is\nconstructed: instead of picking the $k-$nearest neighbors, one can often pick\n$k' \\ll k$ random points out of the $k-$nearest neighbors without sacrificing\nefficiency. This can massively simplify and accelerate computation, we\nillustrate this with several numerical examples.","url_abs":"http://arxiv.org/abs/1711.04712v1","url_pdf":"http://arxiv.org/pdf/1711.04712v1.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":"randomized-near-neighbor-graphs-giant","repo_url":"https://github.com/KlugerLab/pyFIt-SNE","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"unanswered"}},{"paper_slug":"randomized-near-neighbor-graphs-giant","repo_url":"https://github.com/alec-hoyland/crabsort-FItSNE","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"GPL-3.0"}},{"paper_slug":"randomized-near-neighbor-graphs-giant","repo_url":"https://github.com/klugerlab/FIt-SNE","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1711.04712","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}