{
  "abstract": "The Ramsey community number $r_k$ is the smallest size at which a network is better described by communities than by none, under a Bayesian detection rule. On the diamond hierarchical lattice we show that $r_k$ is an exact renormalization-group crossing: the block-model sufficient statistics obey a linear map with eigenvalues $\\{bs,b\\}$, the degree-corrected evidence density flows to $\\ln K$ at a community fixed point, and $r_k$ is the generation at which the running evidence clears the detection threshold. Degree correction advances detection by two generations. We derive $r_k(b,s;q)$ in closed form for the whole family. Finally, placing on the lattice the Reichardt--Bornholdt community Hamiltonian -- whose ground state is the partition itself -- we find an exact community-ordered phase: below the ferromagnetic critical temperature the two hubs lock into opposite communities for any resolution $γ>0$, a staggered order that persists as $n\\to\\infty$. Allowing each nested sub-community its own label, the optimal partition is a hierarchy of $q_{\\rm opt}\\sim\\sqrt{n}$ communities, so the number of Potts states that best describes the network grows with the network. This hierarchy orders thermally level by level, through a cascade of first-order transitions whose temperatures fall as $1/\\ln q$, so every stable level persists as $n\\to\\infty$: the emergent partition is detectable, optimal, and thermodynamically ordered.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Physica A 701, 132048 (2026)",
  "arxiv": "2607.05954",
  "author_total": 1,
  "authors": [
    {
      "name": "Alexei Vazquez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/19pfs06t2sxmq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.05954",
  "fulltext_url": null,
  "id": "19pfs06t2sxmq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.05954",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "physics.soc-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2607.05954",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/19pfs06t2sxmq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:11Z",
      "native_id": "2607.05954",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.05954"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "The Ramsey community number as a renormalization-group crossing",
  "title_text": "The Ramsey community number as a renormalization-group crossing",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/19pfs06t2sxmq",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1016/j.physa.2026.132048",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}