{
  "abstract": "We study a natural synchronous greedy matching process in Erdős-Rényi random graphs $\\mathbb{G}(n,p)$ and give a rigorous analysis of this parallel stochastic dynamics. The process evolves as follows: in each round, every unmatched vertex proposes to a uniformly random neighbour, and mutually proposing pairs are matched and removed from the graph, after which the process repeats on the residual graph. We consider the density regime $p=Θ(\\log^{1+ε} n/n)$ for any constant $ε>0$, which lies just above the threshold for the existence of a perfect matching. We show that, with high probability, for $Θ(\\log^{1+ε} n)$ rounds, every unmatched vertex keeps a constant fraction of its initial degree in the residual graph, simultaneously for all vertices and all rounds. By then, the process has matched a constant fraction of the vertices. Our analysis tracks fine-grained invariants that control the evolution of individual vertex degrees, rather than directly tracking the global degree distribution. At the local level, we analyze the per-round degree loss of a vertex via a neighbourhood-exposure martingale. We show that structural sparsity of $\\mathbb{G}(n,p)$ yields $O(1)$ variance and thus the per-round degree loss is strongly concentrated around its conditional expectation. At the global level, these local concentration bounds control the increments of a martingale tracking cumulative degree loss over many rounds, allowing us to show that vertex degrees remain stable for $Ω(\\log^{1+ε} n)$ rounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25507",
  "author_total": 2,
  "authors": [
    {
      "name": "Vishvajeet N"
    },
    {
      "name": "Shrinidhi Teganahally Sridhara"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f9bdcj559evb7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25507",
  "fulltext_url": null,
  "id": "f9bdcj559evb7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25507",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7223049677",
  "pdf_url": "https://arxiv.org/pdf/2610.25507",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f9bdcj559evb7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25507",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25507"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Degree Stability of Parallel Greedy Matching in Random Graphs",
  "title_text": "Degree Stability of Parallel Greedy Matching in Random Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/f9bdcj559evb7",
  "venue": "arXiv",
  "versions": []
}