{"paper":{"title":"Concentration of measure for the number of isolated vertices in the Erd\\H{o}s-R\\'{e}nyi random graph by size bias couplings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Larry Goldstein, Martin Raic, Subhankar Ghosh","submitted_at":"2011-05-31T22:38:38Z","abstract_excerpt":"A concentration of measure result is proved for the number of isolated vertices $Y$ in the Erd\\H{o}s-R\\'{e}nyi random graph model on $n$ edges with edge probability $p$. When $\\mu$ and $\\sigma^2$ denote the mean and variance of $Y$ respectively, $P((Y-\\mu)/\\sigma\\ge t)$ admits a bound of the form $e^{-kt^2}$ for some constant positive $k$ under the assumption $p \\in (0,1)$ and $np\\rightarrow c \\in (0,\\infty)$ as $n \\rightarrow \\infty$. The left tail inequality $$ P(\\frac{Y-\\mu}{\\sigma}\\le -t)&\\le& \\exp(-\\frac{t^2\\sigma^2}{4\\mu}) $$ holds for all $n \\in {2,3,...},p \\in (0,1)$ and $t \\ge 0$. The"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.0048","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"}