{
  "abstract": "We study an asynchronous consensus dynamics on $N$ agents: at each step a uniformly chosen agent replaces its state by $f(Y_1,\\dots,Y_r)$, where $f$ is a fixed monotone aggregation rule and $Y_1,\\dots,Y_r$ are the states of $r$ agents sampled uniformly with replacement. Let $T$ be the first time at which all agents agree. Let $f:\\{0,1\\}^r\\to\\{0,1\\}$ be monotone and non-constant. The expected consensus time is governed by the two \\emph{endpoint degrees} $D_0(f)=\\#\\{i:f(e_i)=1\\}$ and $D_1(f)=\\#\\{i:f(\\mathbf{1}-e_i)=0\\}$, where $e_i$ is the $i$th standard basis vector and $\\mathbf{1}$ the all-ones vector. If $D_0(f)\\ne 1$ and $D_1(f)\\ne 1$, then $E[T]=Θ\\bigl(N(1+\\log d_f)\\bigr)$, where $d_f$ counts the agents that must change state before the nearest attracting consensus is reached, so that $E[T]=O(N\\log N)$ uniformly over initial states. If $f$ is a dictator, $E[T]$ is given by a voter-model formula and equals $Θ\\bigl(N^2 Ent(p_0)\\bigr)$ with absolute constants, uniformly over the initial state, where $Ent$ is the binary entropy and $p_0$ the initial fraction of agents in state $1$. Otherwise exactly one of $D_0(f),D_1(f)$ equals $1$ and the other equals $0$. The worst-case expected consensus time is then $Θ(N^{2-1/m})$, where $m\\ge 2$ is the least number of coordinates that force the value $1$ in the {\\em residual rule} of $f$, defined in the paper (of its dual, when $D_1(f)=1$).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04468",
  "author_total": 1,
  "authors": [
    {
      "name": "Elchanan Mossel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1detjeh9d5g51.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04468",
  "fulltext_url": null,
  "id": "1detjeh9d5g51",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04468",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "3109|Opinion Dynamics and Social Influence"
    ]
  ],
  "openalex": "W7211871206",
  "pdf_url": "https://arxiv.org/pdf/2609.04468",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1detjeh9d5g51.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04468",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04468"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Consensus times for monotone aggregation dynamics",
  "title_text": "Consensus times for monotone aggregation dynamics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/1detjeh9d5g51",
  "venue": "arXiv",
  "versions": []
}