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  "title": "Small diameter does not control hitting-time robustness",
  "shortTitle": "Why a short diameter does not protect a random walk",
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  "oneLine": "Refutes Zuckerman's Conjecture 1.16 (ITCS 2026): explicit cubic and diameter-three graph families on which relative transition errors of size ε with ε·diameter → 0 still slow random-walk hitting times by unbounded factors, because the short routes are rarely used; with a proved resistance bound of 684 on the cubic family and a two-sided robustness bracket on the diameter-three family.",
  "abstract": "A random walk is robust if an adversary who may change every transition probability by a small relative amount cannot make it much slower to reach its targets. Zuckerman (Markov chain robustness, ITCS 2026) conjectured that the largest safe relative change is at least a constant divided by the graph's diameter (Conjecture 1.16). This paper shows that the conjecture is false, even for graphs in which every vertex has exactly three neighbours. A graph can have short routes that the walk almost never uses; a small fixed bias that steers the walk away from a target accumulates between the rare visits to those routes, the target's stationary probability collapses, and Kac's formula turns that into a long hitting time. For an explicit family of cubic graphs J_b with N vertices and diameter O(log N), every effective resistance is at most 684, so unperturbed hitting times are of order N, and a fixed perturbation of size 1/b, with ε·diameter → 0, multiplies the worst-case hitting time by a factor of order b; the robustness is at most of order N^(-1/3). For explicit 3m-regular graphs G_m of diameter exactly three, the mixing time is at most 50(m+1), so with Zuckerman's mixing-time theorem the robustness lies between constant multiples of N^(-1/2) and N^(-1/4). An exact computation on G_54, with 5,995 vertices and ε·diameter = 0.74, shows a 5.80-fold slowdown. The paper also gives a diameter-two example without regularity, explains why reversible perturbations cannot do this, and shows that Kemeny's constant does not detect the effect. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
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    "Markov chains",
    "random walks",
    "hitting times",
    "robustness",
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    "mixing time",
    "cubic graphs",
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    "Zuckerman's Conjecture 1.16 (ITCS 2026), that hitting-time robustness is Ω(1/diameter), is false: fixed relative perturbations with ε·diameter → 0 multiply the worst-case expected hitting time by unbounded factors (proof).",
    "Cubic graphs J_b on N_b = (2b²+1)(2b⁴+1)−1 vertices with diameter at most 12 log₂ b + O(1): every effective resistance is at most 684 (Theorem 3.2), so t_hit = Θ(N_b); a fixed perturbation of size 1/b amplifies t_hit by Ω(b), and ρ_hit(J_b; C) = O_C(N_b^(-1/3)) (proof).",
    "3m-regular graphs G_m of diameter exactly three on (2m+1)(m+1) vertices: amplification at least b²/27 − o(1) at ε ~ 1/b when m = b⁴; the mixing time is at most 50(m+1) (Proposition 2.6), so with Zuckerman's Theorem 1.10 the robustness lies between Ω(N^(-1/2)) and O_C(N^(-1/4)), for C at least Zuckerman's constant (proof; the bracket is not matching).",
    "Fixed reversible perturbations with ε ≤ c/diameter, c < 1, change t_hit by at most a bounded factor, so the counterexamples need nonreversibility; this recovers Zuckerman's tree theorem with an explicit constant (Proposition 5.1, proof; no novelty claimed).",
    "Finite sizes (computations, not proofs): exact 5.80-fold slowdown on G_54 (5,995 vertices, ε·diameter = 0.74); 8.56-fold on J_2 (exact) and 40.1-fold on J_3 (double precision), both with ε·diameter > 1; Kemeny's constant changes only by factors 0.957 and 0.927."
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  "relatedWorks": [
    {
      "citation": "Zuckerman (2026), Markov chain robustness. 17th Innovations in Theoretical Computer Science Conference (ITCS 2026), LIPIcs 362, 118:1–118:22 (source of Conjecture 1.16).",
      "url": "https://doi.org/10.4230/LIPIcs.ITCS.2026.118",
      "doi": "10.4230/LIPIcs.ITCS.2026.118"
    },
    {
      "citation": "Azar, Broder, Karlin, Linial and Phillips (1996), Biased random walks. Combinatorica 16(1), 1–18.",
      "url": "https://doi.org/10.1007/BF01300124",
      "doi": "10.1007/BF01300124"
    },
    {
      "citation": "Chandra, Raghavan, Ruzzo, Smolensky and Tiwari (1996), The electrical resistance of a graph captures its commute and cover times. Computational Complexity 6(4), 312–340.",
      "url": "https://doi.org/10.1007/BF01270385",
      "doi": "10.1007/BF01270385"
    },
    {
      "citation": "Aldous and Fill (2002, recompiled 2014), Reversible Markov Chains and Random Walks on Graphs (rarely used shortcuts, Section 4.3.5).",
      "url": "https://www.stat.berkeley.edu/~aldous/RWG/book.html",
      "doi": null
    },
    {
      "citation": "Haslegrave, Sauerwald and Sylvester (2022), Time dependent biased random walks. ACM Transactions on Algorithms 18(2), Article 12.",
      "url": "https://doi.org/10.1145/3498848",
      "doi": "10.1145/3498848"
    },
    {
      "citation": "Olesker-Taylor, Sauerwald and Sylvester (2026), Time-biased random walks and robustness of expanders. Proceedings of SODA 2026, 2928–2941.",
      "url": "https://doi.org/10.1137/1.9781611978971.109",
      "doi": "10.1137/1.9781611978971.109"
    },
    {
      "citation": "Ding and Peres (2013), Sensitivity of mixing times. Electronic Communications in Probability 18, paper 88.",
      "url": "https://doi.org/10.1214/ECP.v18-2765",
      "doi": "10.1214/ECP.v18-2765"
    },
    {
      "citation": "Hermon and Kozma (2024), Sensitivity of mixing times of Cayley graphs. Canadian Journal of Mathematics 76(4), 1400–1431.",
      "url": "https://doi.org/10.4153/S0008414X23000421",
      "doi": "10.4153/S0008414X23000421"
    },
    {
      "citation": "Levin and Peres, with Wilmer (2017), Markov Chains and Mixing Times, 2nd edition. American Mathematical Society.",
      "url": "https://doi.org/10.1090/mbk/107",
      "doi": "10.1090/mbk/107"
    },
    {
      "citation": "Lyons and Peres (2016), Probability on Trees and Networks. Cambridge University Press (Thomson's principle).",
      "url": "https://doi.org/10.1017/9781316672815",
      "doi": "10.1017/9781316672815"
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