{"paper":{"title":"How to Realize a Graph on Random Points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Alexander Russell, Saad Quader","submitted_at":"2018-04-23T19:41:44Z","abstract_excerpt":"We are given an integer $d$, a graph $G=(V,E)$, and a uniformly random embedding $f : V \\rightarrow \\{0,1\\}^d$ of the vertices. We are interested in the probability that $G$ can be \"realized\" by a scaled Euclidean norm on $\\mathbb{R}^d$, in the sense that there exists a non-negative scaling $w \\in \\mathbb{R}^d$ and a real threshold $\\theta > 0$ so that \\[\n  (u,v) \\in E \\qquad \\text{if and only if} \\qquad \\Vert f(u) - f(v) \\Vert_w^2 < \\theta\\,, \\] where $\\| x \\|_w^2 = \\sum_i w_i x_i^2$.\n  These constraints are similar to those found in the Euclidean minimum spanning tree (EMST) realization prob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.08680","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}