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On the Probability That a Random Digraph Is Acyclic

Authors Dimbinaina Ralaivaosaona, Vonjy Rasendrahasina, Stephan Wagner

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Author Details

Dimbinaina Ralaivaosaona
  • Stellenbosch University, South Africa
Vonjy Rasendrahasina
  • ENS Université d'Antananarivo, Madagascar
Stephan Wagner
  • Uppsala University, Sweden
  • Stellenbosch University, South Africa

Funding

  • Ralaivaosaona, Dimbinaina: Supported by Division for Research Development (DRD) of Stellenbosch University and the National Research Foundation (NRF) of South Africa.
  • Wagner, Stephan: Supported by the Knut and Alice Wallenberg Foundation.

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 AsGet BibTex

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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References

  1. E. Bender and R. W. Robinson. The asymptotic number of acyclic digraphs, ii. J. Comb. Theory, Ser. B, 44:363-369, 1988. Google Scholar
  2. E. A. Bender, L. B. Richmond, R. W. Robinson, and N. C. Wormald. The asymptotic number of acyclic digraphs. i. Combinatorica, 6(1):15-22, 1986. Google Scholar
  3. M. Bloznelis, F Göetze, and J. Jaworski. Birth of a strongly connected giant in an inhomogeneous random digraph. Journal of Applied Probability, 49(3):601-611, 2012. Google Scholar
  4. C. Cooper and A. Frieze. The size of the largest strongly connected component of a random digraph with a given degree sequence. Comb. Probab. Comput., 13(3):319-337, 2004. Google Scholar
  5. C. Cooper, A. Frieze, and M. Molloy. Hamilton cycles in random regular digraphs. Combinatorics, Probability and Computing, 3(1):39-49, 1994. Google Scholar
  6. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth. On the Lambert W function. Adv. Comput. Math., 5(4):329-359, 1996. Google Scholar
  7. M. Coulson. The critical window in random digraphs. Preprint, 2019. URL: http://arxiv.org/abs/1905.00624.
  8. É. de Panafieu and S. Dovgal. Counting directed acyclic and elementary digraphs. Preprint, 2020. Google Scholar
  9. K. Dutta and C. R. Subramanian. On induced acyclic subgraphs in sparse random digraphs. Electronic Notes in Discrete Mathematics, 38:319-324, 2011. Google Scholar
  10. K. Dutta and C. R. Subramanian. Induced acyclic tournaments in random digraphs: sharp concentration, thresholds and algorithms. Discuss. Math. Graph Theory, 34(3):467-495, 2014. Google Scholar
  11. K. Dutta and C. R. Subramanian. Improved bounds on induced acyclic subgraphs in random digraphs. SIAM J. Discrete Math., 30(3):1848-1865, 2016. Google Scholar
  12. A. Ferber, G. Kronenberg, and E. Long. Packing, counting and covering hamilton cycles in random directed graphs. Electronic Notes in Discrete Mathematics, 49:813-819, 2015. Google Scholar
  13. A. Ferber, R. Nenadov, A. Noever, U. Peter, and N. Škorić. Robust hamiltonicity of random directed graphs. Journal of Combinatorial Theory, Series B, 126:1-23, 2017. Google Scholar
  14. P. Flajolet, B. Salvy, and G. Schaeffer. Airy phenomena and analytic combinatorics of connected graphs. Electr. J. Comb., 11(1), 2004. Google Scholar
  15. A. Frieze and M. Karoński. Introduction to Random Graphs. Cambridge University Press, 2015. Google Scholar
  16. P. J. Grabner and B. Steinsky. Asymptotic behaviour of the poles of a special generating function for acyclic digraphs. Aequationes mathematicae, 70(3):268-278, 2005. Google Scholar
  17. F. Harary and E. Palmer. Graphical enumeration. Academic Press, New-York, 1973. Google Scholar
  18. D. Hefetz, A. Steger, and B. Sudakov. Random directed graphs are robustly hamiltonian. Random Structures & Algorithms, 49(2):345-362, 2016. Google Scholar
  19. S. Janson, T. Łuczak, and A. Ruciński. Random Graphs. Wiley-Interscience, 2000. Google Scholar
  20. R. M. Karp. The transitive closure of a random digraph. Random Structures & Algorithms, 1(1):73-93, 1990. Google Scholar
  21. M. Krivelevich, E. Lubetzky, and B. Sudakov. Longest cycles in sparse random digraphs. Random Structures & Algorithms, 43(1):1-15, 2013. Google Scholar
  22. E. N. Laguerre. Memoire sur la théorie des equations numeriques. Journal de mathématiques pures et appliquées 3e série, 9:99-146, 1983. Google Scholar
  23. V. A. Liskovec. On the enumeration of strongly connected directed graphs. Dokl. Akad. Nauk BSSR, 17:1077-1080, 1163, 1973. Google Scholar
  24. V. A. Liskovec. On the number of maximal vertices of a random acyclic digraph. Theory of Probability & Its Applications, 20(2):401-409, 1976. Google Scholar
  25. T. Łuczak. The phase transition in the evolution of random digraphs. Journal of Graph Theory, 14:217-223, 1990. Google Scholar
  26. T. Łuczak and T. Seierstad. The critical behavior of random digraphs. Random Structures & Algorithms, 35:271-293, 2009. Google Scholar
  27. K. Mahler. On a special functional equation. Journal of the London Mathematical Society, s1-15(2):115-123, 1940. Google Scholar
  28. F. W. Olver, D. W. Lozier, Ronald F. Boisvert, and C. W. Clark. NIST Handbook of Mathematical Functions. Cambridge University Press, USA, 1st edition, 2010. Google Scholar
  29. G. Pólya and I. Schur. Uber zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 144:89-113, 1914. Google Scholar
  30. C. R. Subramanian. Finding induced acyclic subgraphs in random digraphs. The Electronic Journal of Combinatorics, 10:#R46, 2003. Google Scholar
  31. R. W. Robinson. Counting labelled acyclic digraphs. In New Directions in Graph Theory (Ed. F. Harary). New York: Academic Press, 1973. Google Scholar
  32. R. W. Robinson. Counting unlabeled acyclic digraphs. In Charles H. C. Little, editor, Combinatorial Mathematics V, pages 28-43, Berlin, Heidelberg, 1977. Springer Berlin Heidelberg. Google Scholar
  33. J. Spencer and C. R. Subramanian. On the size of induced acyclic subgraphs in random digraphs. Discrete Mathematics and Theoretical Computer Science, 10(2):47-54, 2008. Google Scholar
  34. R. P. Stanley. Acyclic orientations of graphs. Discrete Mathematics, 5(2):171-178, 1973. Google Scholar
  35. The Sage Developers. SageMath, the Sage Mathematics Software System (Version 8.9), 2019. https://www.sagemath.org. Google Scholar
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