Document Open Access Logo

Ancestries and Descendants in a Random DAG

Author Fabian Burghart

PDF

File

Thumbnail PDF
PDF
LIPIcs.AofA.2026.7.pdf
  • Filesize: 0.67 MB
  • 14 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Markov processes
Keywords
  • Random DAG
  • descendants
  • Markov process
  • Urn model
  • Limit theorems

Metrics

Abstract

We consider a random recursive DAG G_n on the vertex set [n] where every vertex i ≥ 2 has out-degree d, with the targets chosen uniformly at random among the earlier i-1 vertices. For this model, we propose a novel way to investigate the descendants of n (which have recently been studied in a paper by Janson) through what we call ancestry processes. The ancestor process a_i(n) of a vertex i is defined as the number of ancestors of i in G_n, and is closely related to the evolutions of multi-draw Pólya urns. Results on the descendants can then be obtained via asymptotic results on functionals of the ancestry processes, generally leading to technical integral expressions. We employ this method to make progress on two open problems posed by Janson, as well as to provide an alternative proof of a first-moment result contained in his work. We further prove limit theorems for the ancestor processes a_i(n) depending on i.

Cite As Get BibTex

Fabian Burghart. Ancestries and Descendants in a Random DAG. In 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 381, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.AofA.2026.7

Author Details

Fabian Burghart
  • Institute for Discrete Mathematics, TU Graz, Steyrergasse 30/II, 8010 Graz, Austria

Funding

This research was funded in part by the Austrian Science Fund (FWF) [10.55776/F1002].

Acknowledgements

The author wishes to thank Svante Janson for many helpful comments, including a shortcut to the proof of Theorem 12.

References

  1. S. Arya, M.J. Golin, and K. Mehlhorn. On the expected depth of random circuits. Combinatorics, Probability, and Computing, 8(3):209-227, 1999. URL: https://doi.org/10.1017/S096354839900382X.
  2. J. Björklund, C. Holmgren, S. Janson, and T.Y.Y. Lo. Approximation of subgraph counts in the uniform attachment model. Combinatorics, Probability, and Computing, 34(1):90-114, 2025. URL: https://doi.org/10.1017/S0963548324000294.
  3. N. Broutin and O. Fawzi. Longest path distance in random circuits. Combinatorics, Probability, and Computing, 21(6):856-881, 2012. URL: https://doi.org/10.1017/S0963548312000260.
  4. L. Devroye and S. Janson. Long and short paths in uniform random recursive dags. Arkiv för Matematik, 49:61-77, 2011. URL: https://doi.org/10.1007/s11512-009-0118-0.
  5. L. Devroye and J. Lu. The strong convergence of maximal degrees in uniform random recursive trees and dags. Random Structures and Algorithms, 7(1):1-14, 1995. URL: https://doi.org/10.1002/rsa.3240070102.
  6. M. Drmota. Random Trees. Springer, 2009. URL: https://doi.org/10.1007/978-3-211-75357-6.
  7. S. Janson. The number of descendants in a random directed acyclic graph. Random Structures and Algorithms, 64(3):768-803, 2023. URL: https://doi.org/10.1002/rsa.21195.
  8. S. Janson and T.Y.Y. Lo. The number of descendants in a preferential attachment graph. arxiv:2412.13975, 2024. URL: https://doi.org/10.48550/arXiv.2412.13975.
  9. D.E. Knuth. The Art of Computer Programming. Section 7.2.2.3 (Preliminary draft), 2024. URL: http://cs.standford.edu/~knuth/fasc7a.ps.gz [cited February 2026].
  10. M. Kuba and H.M. Mahmoud. Two-color balanced affine urn models with multiple drawings. Advances in Applied Mathematics, 90:1-26, 2017. URL: https://doi.org/10.1016/j.aam.2017.04.004.
  11. M. Kuba and H. Sulzbach. On martingale tail sums in affine two-color urn models with multiple drawings. Journal of Applied Probability, 54(1):96-117, 2017. URL: https://doi.org/10.1017/jpr.2016.89.
  12. M. Kuba and S. Wagner. On the distribution of depths in increasing trees. Electronic Journal of Combinatorics, 17:R137, 2010. URL: https://doi.org/10.37236/409.
  13. N. Lasmar, C. Mailler, and O. Selmi. Multiple drawing multi-colour urns by stochastic approximation. Journal of Applied Probability, 55(1):254-281, 2018. URL: https://doi.org/10.1017/jpr.2018.16.
  14. H. Mahmoud and T. Tsukiji. On the internal structure of random recursive circuits. Journal of Computational and Applied Mathematics, 142(1):155-171, 2002. URL: https://doi.org/10.1016/S0377-0427(01)00466-6.
  15. C. Mailler and R. Steiner. Multi-drawing Pólya urns via labelled random DAGs. arxiv:2508.20592, 2025. URL: https://doi.org/10.48550/arXiv.2508.20592.
  16. J.A. Moler, F. Plo, M. San Miguel, and H. Urmeneta. Asymptotics in random recursive circuits. Journal of Mathematical Sciences, 196:70-74, 2014. URL: https://doi.org/10.1007/s10958-013-1637-6.
  17. T. Tsukiji and H. Mahmoud. A limit law for outputs in random recursive circuits. Algorithmica, 31(2):403-412, 2001. URL: https://doi.org/10.1007/s00453-001-0044-4.
  18. T. Tsukiji and F. Xhafa. On the depth of randomly generated circuits. In Algorithms - ESA '96, volume 1136 of Lecture Notes in Computer Science, pages 208-220. Springer, 1996. URL: https://doi.org/10.1007/3-540-61680-2_57.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact