{"paper":{"title":"Asymptotic distribution of the numbers of vertices and arcs of the giant strong component in sparse random digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Boris Pittel, Daniel Poole","submitted_at":"2014-05-15T21:50:28Z","abstract_excerpt":"Two models of a random digraph on $n$ vertices, $D(n,\\text{Prob}(\\text{arc})=p)$ and $D(n,\\text{number of arcs}=m)$ are studied. In 1990, Karp for $D(n,p)$ and independently T. \\L uczak for $D(n,m=cn)$ proved that for $c>1$, with probability tending to 1, there is an unique strong component of size of order $n$. Karp showed, in fact, that the giant component has likely size asymptotic to $n\\theta^2$, where $\\theta=\\theta(c)$ is the unique positive root of $1-\\theta=e^{-c \\theta}$. In this paper we prove that, for both random digraphs, the joint distribution of the number of vertices and number"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.4022","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"}