{"paper":{"title":"On Bollob\\'as-Riordan random pairing model of preferential attachment graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Boris Pittel","submitted_at":"2018-11-27T01:16:13Z","abstract_excerpt":"Bollob\\'as-Riordan random pairing model of a preferential attachment graph $G_m^n$ is studied. Let $\\{W_j\\}_{j\\le mn+1}$ be the process of sums of independent exponentials with mean $1$. We prove that the degrees of the first $\\nu_m^n:=n^{\\frac{m}{m+2}-\\epsilon}$ vertices are jointly, and uniformly, asymptotic to $\\{2(mn)^{1/2}\\bigl(W^{1/2}_{mj}-W^{1/2}_{m(j-1)}\\bigr)\\}_{j\\in [t]}$, and that with high probability (whp) the smallest of these degrees is $n^{\\frac{\\epsilon(m+2)}{2m}}$, at least. In contrast, the degrees of vertices below the top by any fraction of $n$ are whp of $O(\\log n)$ order"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.10764","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"}