On the Concentration of the Maximum Degree in the Duplication-Divergence Models

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-03-07 DOI:10.1137/23m1592766

Alan M. Frieze, Krzysztof Turowski, Wojciech Szpankowski

引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 988-1006, March 2024.
Abstract. We present a rigorous and precise analysis of the maximum degree and the average degree in a dynamic duplication-divergence graph model introduced by Solé et al. [Adv. Complex Syst., 5 (2002), pp. 43–54] in which the graph grows according to a duplication-divergence mechanism, i.e., by iteratively creating a copy of some node and then randomly alternating the neighborhood of a new node with probability [math]. This model captures the growth of some real-world processes, e.g., biological or social networks. In this paper, we prove that for some [math], the maximum degree and the average degree of a duplication-divergence graph on [math] vertices are asymptotically concentrated with high probability around [math] and [math], respectively, i.e., they are within at most a polylogarithmic factor from these values with probability at least [math] for any constant [math].

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论重复-发散模型中最大度数的集中问题

SIAM 离散数学杂志》第 38 卷第 1 期第 988-1006 页，2024 年 3 月。摘要。我们对 Solé 等人[Adv. Complex Syst.，5 (2002)，pp. 43-54] 提出的动态复制-发散图模型中的最大度和平均度进行了严格而精确的分析。这一模型捕捉到了现实世界中某些过程的增长，如生物或社会网络。在本文中，我们证明了对于某些[math]，[math]顶点上的复制-发散图的最大度和平均度分别以很高的概率渐近地集中在[math]和[math]附近，也就是说，对于任意常数[math]，它们与这些值的距离最多在一个多项式因子之内，概率至少为[math]。

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

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.