{"paper":{"title":"The phase transition in the multi-type binomial random graph $G(\\mathbf{n},P)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Ang\\'elica Pach\\'on, Christoph Koch, Mihyun Kang","submitted_at":"2014-07-17T16:25:30Z","abstract_excerpt":"We determine the asymptotic size of the largest component in the $2$-type binomial random graph $G(\\mathbf{n},P)$ near criticality using a refined branching process approach. In $G(\\mathbf{n},P)$ every vertex has one of two types, the vector $\\mathbf{n}$ describes the number of vertices of each type, and any edge $\\{u,v\\}$ is present independently with a probability that is given by an entry of the probability matrix $P$ according to the types of $u$ and $v.$ We prove that in the weakly supercritical regime, i.e. if the distance to the critical point of the phase transition is given by an $\\va"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6248","kind":"arxiv","version":3},"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"}