{
  "abstract": "How does the number of sampled opinions affect the time required to reach consensus in distributed consensus dynamics? We study the synchronous $h$-\\textsf{Majority}, in which every node independently samples $h$ nodes uniformly at random with replacement in each round and adopts a most frequent sampled opinion, breaking ties uniformly at random. We show that $h$-\\textsf{Majority} achieves the nearly-optimal trade-off between the sample size and the consensus time, among a broad class of consensus dynamics. In particular, we prove the following results. For every sample size $h \\ge 3$ and for every initial configuration, we prove that the process reaches consensus with high probability within $\\widetilde{O}\\left(1+\\min\\left\\{\\frac{1}{\\min\\{h^2p_{\\max},1\\}},\\frac{\\sqrt{n}}{h}\\right\\}\\right)$ rounds, where $n$ is the number of nodes and $p_{\\max}$ is the largest initial fraction of nodes supporting any single opinion. This is the first bound on the consensus time for $h$-\\textsf{Majority} that holds for every $h\\ge3$ and every initial configuration. When the initially most popular opinion has a sufficiently large lead over every other opinion, we show that all nodes adopt this opinion with high probability within $O(\\log n/\\min\\{h^2p_{\\max},1\\})$ rounds. We establish a matching lower bound up to polylogarithmic factors for any fixed update rule that treats all opinions symmetrically and that selects either a sampled opinion or the node's current opinion.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40562",
  "author_total": 5,
  "authors": [
    {
      "name": "Francesco d'Amore"
    },
    {
      "name": "Joel Rybicki"
    },
    {
      "name": "Nobutaka Shimizu"
    },
    {
      "name": "Takeharu Shiraga"
    },
    {
      "name": "Robin Vacus"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9rfbnhp4frwsb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40562",
  "fulltext_url": null,
  "id": "9rfbnhp4frwsb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40562",
  "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",
      "cs.DC"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40562",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9rfbnhp4frwsb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:35Z",
      "native_id": "2610.40562",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40562"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Nearly-Optimal Time-Sample Trade-off in Consensus Dynamics",
  "title_text": "Nearly-Optimal Time-Sample Trade-off in Consensus Dynamics",
  "type": "preprint",
  "updated_at": "2026-11-04T15:21:29Z",
  "url": "https://research.draconitic.com/paper/9rfbnhp4frwsb",
  "venue": "arXiv",
  "versions": []
}