{
  "abstract": "For a graph $G$ of order $N$, an integer $k$ with $0\\leq k< N$ and a real number $p$ with $0\\leq p\\leq 1$, let $p(G,k)$ and $t(G,p)$ be the probabilities that $G$ is disconnected when $k$ vertices are removed randomly and when each vertex is removed independently with probability $p$, respectively. We deduce a formula of $p(G,k)$ for Harary graphs and show that the phase transition on $p(G,k)$ and $t(G,p)$ occurs in the class that has logarithmical degree. We also perform simulations on $p(G,k)$ for the Hypercube and randomly generated graphs of the same scale, which suggests that the regularity is a key structural feature that guarantees nice performance on $p(G,k)$. In particular, the Hypercubes have better performance than regular graphs, on average. Surprisingly, Harary graphs, known to be super-connected, have neither better performance than Hypercubes nor even than the average of regular graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07581",
  "author_total": 2,
  "authors": [
    {
      "name": "Huanshen Jia"
    },
    {
      "name": "Jianguo Qian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7g4jgt7qpyzm3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07581",
  "fulltext_url": null,
  "id": "7g4jgt7qpyzm3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07581",
  "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.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07581",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7g4jgt7qpyzm3.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07581",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07581"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Probability model of connectedness for graphs with logarithmical degree",
  "title_text": "Probability model of connectedness for graphs with logarithmical degree",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/7g4jgt7qpyzm3",
  "venue": "arXiv",
  "versions": []
}