The number of descendants in a random directed acyclic graph

IF 0.9 3区数学 Q4 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Random Structures & Algorithms Pub Date : 2023-11-07 DOI:10.1002/rsa.21195

Svante Janson

引用次数: 0

Abstract

Abstract We consider a well‐known model of random directed acyclic graphs of order , obtained by recursively adding vertices, where each new vertex has a fixed outdegree and the endpoints of the edges from it are chosen uniformly at random among previously existing vertices. Our main results concern the number of vertices that are descendants of . We show that converges in distribution; the limit distribution is, up to a constant factor, given by the th root of a Gamma distributed variable with distribution . When , the limit distribution can also be described as a chi distribution . We also show convergence of moments, and find thus the asymptotics of the mean and higher moments.

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随机有向无环图中后代的数目

我们考虑了一个众所周知的有序随机有向无环图模型，该模型通过递归添加顶点获得，其中每个新顶点具有固定的出界度，并且其边缘的端点在先前存在的顶点中随机选择。我们的主要结果与的后代顶点的数量有关。我们证明它在分布上是收敛的;极限分布是，直到一个常数因子，由一个具有分布的分布变量的根号给出。时，极限分布也可以描述为chi分布。我们还证明了矩的收敛性，并由此找到了均值矩和高矩的渐近性。

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

Random Structures & Algorithms 数学-计算机：软件工程

CiteScore

2.50

自引率

10.00%

发文量

审稿时长

>12 weeks

期刊介绍： It is the aim of this journal to meet two main objectives: to cover the latest research on discrete random structures, and to present applications of such research to problems in combinatorics and computer science. The goal is to provide a natural home for a significant body of current research, and a useful forum for ideas on future studies in randomness. Results concerning random graphs, hypergraphs, matroids, trees, mappings, permutations, matrices, sets and orders, as well as stochastic graph processes and networks are presented with particular emphasis on the use of probabilistic methods in combinatorics as developed by Paul Erdõs. The journal focuses on probabilistic algorithms, average case analysis of deterministic algorithms, and applications of probabilistic methods to cryptography, data structures, searching and sorting. The journal also devotes space to such areas of probability theory as percolation, random walks and combinatorial aspects of probability.