The longest edge of the one-dimensional soft random geometric graph with boundaries

IF 0.5 4区数学 Q4 STATISTICS & PROBABILITY

Stochastic Models Pub Date : 2023-09-22 DOI:10.1080/15326349.2023.2256825

Arnaud Rousselle, Ercan Sönmez

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引用次数: 1

Abstract

AbstractThe object of study is a soft random geometric graph with vertices given by a Poisson point process on a line and edges between vertices present with probability that has a polynomial decay in the distance between them. Various aspects of such models related to connectivity structures have been studied extensively. In this article, we study the random graph from the perspective of extreme value theory and focus on the occurrence of single long edges. The model we investigate has non-periodic boundary and is parameterized by a positive constant α, which is the power for the polynomial decay of the probabilities determining the presence of an edge. As a main result, we provide a precise description of the magnitude of the longest edge in terms of asymptotic behavior in distribution. Thereby we illustrate a crucial dependence on the power α and we recover a phase transition which coincides with exactly the same phases in Benjamini and Berger[ Citation2].Keywords: Extreme value theorymaximum edge-lengthPoisson approximationrandom graphssoft random geometric graphMSC: Primary: 05C8060G70Secondary: 60F0505C8282B21 Disclosure statementNo potential conflict of interest was reported by the authors.Additional informationFundingThe IMB receives support from the EIPHI Graduate School (contract ANR-17-EURE-0002).

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具有边界的一维软随机几何图的最长边

摘要本文研究的对象是一个软随机几何图，其顶点由泊松点过程在一条线上给出，顶点之间的边以距离的多项式衰减概率存在。这些模型与连接结构相关的各个方面已经得到了广泛的研究。本文从极值理论的角度对随机图进行了研究，重点研究了单长边的出现。我们研究的模型具有非周期边界，并由一个正常数α参数化，这是确定边缘存在的概率的多项式衰减的幂。作为一个主要的结果，我们提供了在分布的渐近行为的最长边的大小的精确描述。因此，我们说明了对功率α的关键依赖，并且我们恢复了与Benjamini和Berger完全相同的相一致的相变[Citation2]。关键词:极值理论最大边长泊松近似随机图形软随机几何图形msc:一级:058060g70二级:60F0505C8282B21披露声明作者未报告潜在利益冲突。IMB得到EIPHI研究生院的支持(合同anr -17- eur -0002)。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Stochastic Models 数学-统计学与概率论

CiteScore

1.30

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Stochastic Models publishes papers discussing the theory and applications of probability as they arise in the modeling of phenomena in the natural sciences, social sciences and technology. It presents novel contributions to mathematical theory, using structural, analytical, algorithmic or experimental approaches. In an interdisciplinary context, it discusses practical applications of stochastic models to diverse areas such as biology, computer science, telecommunications modeling, inventories and dams, reliability, storage, queueing theory, mathematical finance and operations research.