Seed阈值与度数方差在度数增长的异质自助渗透中的作用

中文摘要 AI 辅助

我们研究具有独立顶点阈值的自助渗透，阈值取值为1和2，其中阈值取1的概率为$(1-c/d)/d$，其中$c>0$为固定常数。种子集在给定确定性大小的集合中均匀选取，与图和阈值独立。我们在固定的相对边际上证明阈值陈述。对于具有指定非负整数度数（和为偶数）且平均度数为$d$的均匀简单图，假设$\nmax_i|d_i-d|\le C\sqrt d$且$c+1-v_n\ge\kappa>0$，其中$v_n=\operatorname{Var}(D)/d$，$C$和$\kappa$为固定常数。当$d\to\infty$且$d=o(n^{1/7})$时，主导种子规模为$n(c+1-v_n)^2/(2d^4)$，无需$v_n$的极限存在。在此规模以下和以上的固定相对边际处，最终活跃集的大小分别以高概率为$O(n/d^3)$和$n-o(n)$。对于$G(n,d/n)$，当$d\to\infty$且$d^5/n\to0$时，有单独结果，给出规模$nc^2/(2d^4)$。因此，正则图和独立边图在相同的渐近平均度数下具有不同的系数。局部探索估计给出显式的非活跃余量，并在指定模型中经受简单性条件的约束。我们还量化了静态包含转移的精度障碍，并计算确定性响应势垒修正，而未识别出收缩的随机临界窗口。

英文摘要

We study bootstrap percolation with independent vertex thresholds taking values one and two, with threshold-one probability $(1-c/d)/d$ for fixed $c>0$. The seed set is chosen uniformly among sets of a prescribed deterministic size, independently of the graph and thresholds. We prove threshold statements at fixed relative margins. For uniform simple graphs with prescribed nonnegative integer degrees of even sum and exact mean $d$, assume $\max_i|d_i-d|\le C\sqrt d$ and $c+1-v_n\geκ>0$, where $v_n=\operatorname{Var}(D)/d$ and $C,κ$ are fixed. When $d\to\infty$ and $d=o(n^{1/7})$, the leading seed scale is $n(c+1-v_n)^2/(2d^4)$, without requiring a limit of $v_n$. At fixed relative margins below and above this scale, the final active set has size $O(n/d^3)$ and $n-o(n)$, respectively, with high probability. A separate result for $G(n,d/n)$ holds when $d\to\infty$ and $d^5/n\to0$, and gives scale $nc^2/(2d^4)$. Thus regular and independent-edge graphs have different coefficients at the same asymptotic mean degree. Local exploration estimates yield explicit inactive remainders and survive conditioning on simplicity in the prescribed model. We also quantify the precision obstruction to static inclusion transfer and compute deterministic response-barrier corrections, without identifying a shrinking random critical window.

发表机构

• Yandex

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26 pages