{"paper":{"title":"Connectivity Threshold of Random Geometric Graphs with Cantor Distributed Vertices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Antar Bandyopadhyay, Farkhondeh Sajadi","submitted_at":"2012-04-03T11:54:57Z","abstract_excerpt":"For connectivity of \\emph{random geometric graphs}, where there is no density for underlying distribution of the vertices, we consider $n$ i.i.d. \\emph{Cantor} distributed points on $[0,1]$. We show that for this random geometric graph, the connectivity threshold $R_{n}$, converges almost surely to a constant $1-2\\phi$ where $0 < \\phi < 1/2$, which for the standard Cantor distribution is 1/3. We also show that $\\| R_n - (1 - 2 \\phi) \\|_1 \\sim 2 \\, C(\\phi) \\, n^{-1/d_{\\phi}}$ where $C(\\phi) > 0$ is a constant and $d_{\\phi} := - {\\log 2}/{\\log \\phi}$ is the \\emph{Hausdorff dimension} of the gene"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.0667","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"}