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  "slug": "biased-small-world-mixing",
  "title": "Biased random walks on small-world networks",
  "shortTitle": "When drift speeds up a small world",
  "url": "https://evidencepress.org/releases/biased-small-world-mixing/",
  "oneLine": "Candidate proof of the stated AIM mixing bounds for four fixed-weight clock conventions: logarithmic in the denser regime, square-root up to polylogarithmic factors in the sparser regime.",
  "abstract": "A cycle with independent undirected shortcuts and fixed forward and backward weights admits a candidate worst-start mixing theorem under four explicit clocks. At shortcut probability epsilon/n the mixing order is logarithmic; at epsilon n^(-3/2), the bounds run from c sqrt(n) log n to C sqrt(n)(log n)^18. The proof keeps random spacing, physical crossing durations, exceptional starts and parity separate. Exact rational tests corroborate finite identities only.",
  "datePublished": "2026-09-08",
  "dateModified": "2026-09-08",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22663470",
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    "Anonymous"
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    "name": "AIM-PROBABILITY-0043 / Markov mixing Problem 3.2",
    "url": "https://www.unsolvedmath.com"
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  "keywords": [
    "small-world networks",
    "nonreversible mixing",
    "Markov renewal",
    "random graphs",
    "parity"
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    "Worst-start Theta(log n) mixing in the denser bounded-average-degree regime.",
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    {
      "citation": "Addario-Berry and Lei, The mixing time of the Newman-Watts small-world model (2015): reversible lazy benchmark.",
      "url": "https://doi.org/10.1239/aap/1427814580",
      "doi": "10.1239/aap/1427814580"
    },
    {
      "citation": "Gerencsér, Mixing times of Markov chains on a cycle with additional long range connections; arXiv v3, Theorems 23–24.",
      "url": "https://arxiv.org/abs/1401.1692v3",
      "doi": "10.48550/arXiv.1401.1692"
    },
    {
      "citation": "Gerencsér and Hendrickx, Improved mixing rates of directed cycles with additional sparse interconnections (2023).",
      "url": "https://arxiv.org/abs/2307.09949",
      "doi": "10.48550/arXiv.2307.09949"
    },
    {
      "citation": "Feng and Gerencsér, Mixing on the cycle with constant size perturbation (2026): fixed shortcut count.",
      "url": "https://doi.org/10.1214/26-EJP1545",
      "doi": "10.1214/26-EJP1545"
    },
    {
      "citation": "Espuny Díaz, Morris, Perarnau and Serra, Speeding up random walk mixing by starting from a uniform vertex (2024).",
      "url": "https://doi.org/10.1214/24-EJP1091",
      "doi": "10.1214/24-EJP1091"
    },
    {
      "citation": "Janson, The probability that a random multigraph is simple (2009).",
      "url": "https://doi.org/10.1017/S0963548308009644",
      "doi": "10.1017/S0963548308009644"
    }
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