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On the Probability That a Random Digraph Is Acyclic

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Acknowledgements

We want to thank the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa, for partially funding Vonjy Rasendrahasina’s visit to Stellenbosch University in December 2018.

Cite As

Dimbinaina Ralaivaosaona, Vonjy Rasendrahasina, and Stephan Wagner. On the Probability That a Random Digraph Is Acyclic. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 25:1-25:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.AofA.2020.25

Abstract

Given a positive integer n and a real number p ∈ [0,1], let D(n,p) denote the random digraph defined in the following way: each of the binom(n,2) possible edges on the vertex set {1,2,3,…,n} is included with probability 2p, where all edges are independent of each other. Thereafter, a direction is chosen independently for each edge, with probability 1/2 for each possible direction. In this paper, we study the probability that a random instance of D(n,p) is acyclic, i.e., that it does not contain a directed cycle. We find precise asymptotic formulas for the probability of a random digraph being acyclic in the sparse regime, i.e., when np = O(1). As an example, for each real number μ, we find an exact analytic expression for φ(μ) = lim_{n→ ∞} n^{1/3} ℙ{D(n,1/n (1+μ n^{-1/3})) is acyclic}.

Subject Classification

ACM Subject Classification
• Mathematics of computing → Random graphs
• Mathematics of computing → Generating functions
Keywords
• Random digraphs
• acyclic digraphs
• asymptotics

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