{
  "abstract": "We study a growing random graph in which each new vertex chooses a uniformly random parent, joins that parent with probability $r\\in(0,1]$, and independently joins each of the parent's current neighbors with probability $p\\in(0,1)$. Let $p_c$ be the unique solution of $p_c e^{p_c}=1$. For $0<p<p_c$, we prove that the empirical degree distribution converges in probability to the stationary distribution $π$ of an explicit reset-and-growth Markov chain. Its upper tail satisfies $π((y,\\infty))\\sim C_{p,r}y^{-κ}$, where $C_{p,r}>0$ and $κ>0$ solves $p^κ+pκ=1$. This confirms predictions from physics (Lambiotte, Krapivsky, Bhat and Redner, \\textit{Physical Review Letters}, 2016). We further show that for every fixed radius $\\ell$ and $p<p_c$, the empirical distribution of induced rooted $\\ell$-neighborhoods converges to the invariant law of a graph-valued reset-and-growth Markov process. We give an explicit construction of the limiting graph using a marked ancestral genealogy. Its root component is almost surely finite for $r<1$, and infinite with exactly one end for $r=1$. Moreover, for every $0<p<1$, both the fixed-vertex degrees and the maximum degree grow on the scale $t^p$, and the vertex of maximum degree is eventually unique and persistent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00123",
  "author_total": 4,
  "authors": [
    {
      "name": "Partha S. Dey"
    },
    {
      "name": "Neeladri Maitra"
    },
    {
      "name": "Jaya M. Manthripragada"
    },
    {
      "name": "Grigory Terlov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bsve3amkc540t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00123",
  "fulltext_url": null,
  "id": "bsve3amkc540t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00123",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00123",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bsve3amkc540t.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00123",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00123"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Uniform duplication-divergence random graphs I: Degrees and Local Limits",
  "title_text": "Uniform duplication-divergence random graphs I: Degrees and Local Limits",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/bsve3amkc540t",
  "venue": "arXiv",
  "versions": []
}