{"paper":{"title":"A curious gap in one-dimensional geometric random graphs between connectivity and the absence of isolated node","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.SI","math.CO","math.PR"],"primary_cat":"physics.soc-ph","authors_text":"Jun Zhao, Osman Ya\\u{g}an, Virgil Gligor","submitted_at":"2015-02-02T09:03:13Z","abstract_excerpt":"One-dimensional geometric random graphs are constructed by distributing $n$ nodes uniformly and independently on a unit interval and then assigning an undirected edge between any two nodes that have a distance at most $r_n$. These graphs have received much interest and been used in various applications including wireless networks. A threshold of $r_n$ for connectivity is known as $r_n^{*} = \\frac{\\ln n}{n}$ in the literature. In this paper, we prove that a threshold of $r_n$ for the absence of isolated node is $\\frac{\\ln n}{2 n}$ (i.e., a half of the threshold $r_n^{*}$). Our result shows ther"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.00404","kind":"arxiv","version":2},"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"}