{"paper":{"title":"Random Geometric Graph Diameter in the Unit Ball","license":"","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Catherine Yan, Jeremy L. Martin, Robert B. Ellis","submitted_at":"2005-01-14T04:08:46Z","abstract_excerpt":"The unit ball random geometric graph $G=G^d_p(\\lambda,n)$ has as its vertices $n$ points distributed independently and uniformly in the $d$-dimensional unit ball, with two vertices adjacent if and only if their $l_p$-distance is at most $\\lambda$. Like its cousin the Erdos-Renyi random graph, $G$ has a connectivity threshold: an asymptotic value for $\\lambda$ in terms of $n$, above which $G$ is connected and below which $G$ is disconnected (and in fact has isolated vertices in most cases). In the connected zone, we determine upper and lower bounds for the graph diameter of $G$. Specifically, a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0501214","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"}