Concentration of hitting times in Erdős-Rényi graphs

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-05-12 DOI:10.1002/jgt.23119

Andrea Ottolini, Stefan Steinerberger

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引用次数: 0

Abstract

We consider Erdős-Rényi graphs $G (n, p)$ for $0 < p < 1$ fixed and $n \to \infty$ and study the expected number of steps, $H_{w v}$ , that a random walk started in $w$ needs to first arrive in $v$ . A natural guess is that an Erdős-Rényi random graph is so homogeneous that it does not really distinguish between vertices and $H_{w v} = (1 + o (1)) n$ . Löwe-Terveer established a CLT for the Mean Starting Hitting Time suggesting $H_{w v} = n \pm O (\sqrt{n})$ . We prove the existence of a strong concentration phenomenon: $H_{w v}$ is given, up to a very small error of size $≲ {(\log n)}^{3 ∕ 2} ∕ \sqrt{n}$ , by an explicit simple formula involving only the total number of edges $∣ E ∣$ , the degree $deg (v)$ and the distance $d (v, w)$ .

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厄尔多斯-雷尼图中命中时间的浓度

我们考虑了固定的和的厄尔多斯-雷尼图，并研究了从开始的随机漫步到达。一个自然的猜测是，Erdős-Rényi 随机图是如此同质，以至于它并不真正区分顶点和。Löwe-Terveer 建立了平均起始击球时间的 CLT，表明 .我们证明了一种强集中现象的存在：它是由一个只涉及边的总数、度和距离的显式简单公式给出的，误差很小。

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .