Upper Tail Behavior of the Number of Triangles in Random Graphs with Constant Average Degree

IF 1 2区 数学 Q1 MATHEMATICS
Shirshendu Ganguly, Ella Hiesmayr, Kyeongsik Nam
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引用次数: 0

Abstract

Let N be the number of triangles in an Erdős–Rényi graph \({\mathcal {G}}(n,p)\) on n vertices with edge density \(p=d/n,\) where \(d>0\) is a fixed constant. It is well known that N weakly converges to the Poisson distribution with mean \({d^3}/{6}\) as \(n\rightarrow \infty \). We address the upper tail problem for N, namely, we investigate how fast k must grow, so that \({\mathbb {P}}(N\ge k)\) is not well approximated anymore by the tail of the corresponding Poisson variable. Proving that the tail exhibits a sharp phase transition, we essentially show that the upper tail is governed by Poisson behavior only when \(k^{1/3} \log k< (\frac{3}{\sqrt{2}} - {o(1)})^{2/3} \log n\) (sub-critical regime) as well as pin down the tail behavior when \(k^{1/3} \log k> (\frac{3}{\sqrt{2}} + {o(1)})^{2/3} \log n\) (super-critical regime). We further prove a structure theorem, showing that the sub-critical upper tail behavior is dictated by the appearance of almost k vertex-disjoint triangles whereas in the supercritical regime, the excess triangles arise from a clique like structure of size approximately \((6k)^{1/3}\). This settles the long-standing upper-tail problem in this case, answering a question of Aldous, complementing a long sequence of works, spanning multiple decades and culminating in Harel et al. (Duke Math J 171(10):2089–2192, 2022), which analyzed the problem only in the regime \(p\gg \frac{1}{n}.\) The proofs rely on several novel graph theoretical results which could have other applications.

平均度数恒定的随机图中三角形数量的上端行为
设 N 是一个厄尔多斯-雷尼图(Erdős-Rényi graph)中三角形的数量,该图有 n 个顶点,边密度为(p=d/n,),其中(d>0)是一个固定常数。众所周知,N弱收敛于均值为\({d^3}/{6}\)的泊松分布。我们要解决 N 的上尾问题,即研究 k 必须增长多快才能使 \({\mathbb {P}}(N\ge k)\)不再被相应泊松变量的尾部很好地近似。为了证明尾部会出现急剧的相变,我们从本质上证明,只有当 \(k^{1/3}\log k< (\frac{3}\{sqrt{2}}- {o(1)})^{2/3}\log k> (\frac{3}\{sqrt{2}}+ {o(1)})^{2/3}\log n\) (超临界机制)。我们进一步证明了一个结构定理,表明亚临界上尾行为是由近 k 个顶点相交三角形的出现决定的,而在超临界机制中,多余的三角形来自于一个类似于小集团的结构,其大小约为\((6k)^{1/3}\)。在这种情况下,解决了长期存在的上尾问题,回答了奥尔德斯的一个问题,补充了横跨数十年的一系列工作,并在哈雷尔等人(Duke Math J 171(10):2089-2192,2022)的研究中达到了顶峰,该研究仅在 \(p\gg \frac{1}{n}.\) 机制下分析了该问题。
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
Combinatorica
Combinatorica 数学-数学
CiteScore
1.90
自引率
0.00%
发文量
45
审稿时长
>12 weeks
期刊介绍: COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are - Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups). - Combinatorial optimization. - Combinatorial aspects of geometry and number theory. - Algorithms in combinatorics and related fields. - Computational complexity theory. - Randomization and explicit construction in combinatorics and algorithms.
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