{"paper":{"title":"Random Subgraphs in Sparse Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Felix Joos","submitted_at":"2013-12-03T08:53:16Z","abstract_excerpt":"We investigate the threshold probability for connectivity of sparse graphs under weak assumptions. As a corollary this completely solve the problem for Cartesian powers of arbitrary graphs. In detail, let $G$ be a connected graph on $k$ vertices, $G^n$ the $n$-th Cartesian power of $G$, $\\alpha_i$ be the number of vertices of degree $i$ of $G$, $\\lambda$ be a positive real number, and $G^n_p$ be the graph obtained from $G^n$ by deleting every edge independently with probability $1-p$. If $\\sum_{i}\\alpha_i(1-p)^i=\\lambda^{\\frac{1}{n}}$, then $\\lim_{n\\rightarrow \\infty}\\mathbb{P}[G^n_p {\\rm\\ is\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0732","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"}