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arXiv 2003.08856math.PRmath.CO
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增长优先连接图中的距离演化

Distance evolutions in growing preferential attachment graphs

Joost Jorritsma, Júlia Komjáthy

备注：
42 pages, 4 figures. Revised version with corrected typos and more elaborate proofs. Includes correction of an error in Theorem 2.5 that required a shift of indices in the summation
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AI总结：

本文研究优先连接随机图中两个固定顶点间距离随图增长的演化过程，证明距离演化以高概率保持在特定对数函数周围的狭窄带状区域内，并将结果推广到加权距离情形。

摘要：

我们研究动态增长的随机图模型中两个固定顶点之间的图距离和加权距离的演化。更准确地说，我们考虑幂律指数 $τ\in(2,3)$ 的优先连接模型，当图有 $t$ 个顶点时均匀随机采样两个顶点 $u_t,v_t$，并研究随着周围图的增长这两个固定顶点之间图距离的演化。这产生了一个关于 $t'\geq t$ 的离散时间随机过程，称为距离演化。我们证明，在函数 $4\frac{\log\log(t)-\log(\log(t'/t)\vee1)}{|\log(τ-2)|}\vee 2$ 周围存在一个狭窄的带状区域，当 $t$ 趋于无穷时，距离演化以高概率不会离开该区域。我们将结果推广到加权距离，其中每条边都配备一个非负随机变量 $L$ 的独立同分布副本。

英文摘要：

We study the evolution of the graph distance and weighted distance between two fixed vertices in dynamically growing random graph models. More precisely, we consider preferential attachment models with power-law exponent $τ\in(2,3)$, sample two vertices $u_t,v_t$ uniformly at random when the graph has $t$ vertices, and study the evolution of the graph distance between these two fixed vertices as the surrounding graph grows. This yields a discrete-time stochastic process in $t'\geq t$, called the distance evolution. We show that there is a tight strip around the function $4\frac{\log\log(t)-\log(\log(t'/t)\vee1)}{|\log(τ-2)|}\vee 2$ that the distance evolution never leaves with high probability as $t$ tends to infinity. We extend our results to weighted distances, where every edge is equipped with an i.i.d. copy of a non-negative random variable $L$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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