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The containment profile of hyper-recursive trees

Part of: Combinatorial probability Limit theorems Operations research and management science

Published online by Cambridge University Press:  17 January 2022

Joshua Sparks ,
Srinivasan Balaji  and
Hosam Mahmoud
Joshua Sparks*
Affiliation:
The George Washington University
Srinivasan Balaji*
Affiliation:
The George Washington University
Hosam Mahmoud*
Affiliation:
The George Washington University
*
*Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA.
*Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA.
*Postal address: Department of Statistics, The George Washington University, Washington, DC 20052, USA.
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Abstract

We investigate vertex levels of containment in a random hypergraph grown in the spirit of a recursive tree. We consider a local profile tracking the evolution of the containment of a particular vertex over time, and a global profile concerned with counts of the number of vertices of a particular containment level.

For the local containment profile, we obtain the exact mean, variance, and probability distribution in terms of standard combinatorial quantities such as generalized harmonic numbers and Stirling numbers of the first kind. Asymptotically, we observe phases: the early vertices have an asymptotically normal distribution, intermediate vertices have a Poisson distribution, and late vertices have a degenerate distribution.

As for the global containment profile, we establish an asymptotically normal distribution for the number of vertices at the smallest containment level as well as their covariances with the number of vertices at the second smallest containment level and the variances of these numbers. The orders in the variance–covariance matrix establish concentration laws.

Keywords

Hypergraphlimit lawphase transitionmartingale

MSC classification

Primary: 90B15: Network models, stochastic
Secondary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 1 , March 2022 , pp. 278 - 296
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

David, F. and Barton, E. (1962). Combinatorial Chance. Charles Griffin, London.10.2307/2551259CrossRefGoogle Scholar
Drmota, M. (2009). Random Trees: An Interplay Between Combinatorics and Probability. Springer, New York.10.1007/978-3-211-75357-6CrossRefGoogle Scholar
Feng, Q., Mahmoud, H. and Panholzer, A. (2008). Phase changes in subtree varieties in random recursive trees and binary search trees. SIAM J. Discrete Math. 22, 160–184.CrossRefGoogle Scholar
Graham, R., Knuth, D. and Patashnik, O. (1994). Concrete Mathematics. Addison-Wesley, Reading, MA.Google Scholar
Gut, A. (2013). Probability: A Graduate Course, 2nd edn. Springer.10.1007/978-1-4614-4708-5CrossRefGoogle Scholar
Hall, P. and Heyde, C. (1980). Martingale Limit Theory and its Application. Academic Press, New York.Google Scholar
Hofri, M. and Mahmoud, H. (2018). Algorithmics of Nonuniformity: Tools and Paradigms. CRC Press, Boca Raton, FL.10.1201/9781315368306CrossRefGoogle Scholar
Frieze, A. M. and Karoński, M. (2015). Introduction to Random Graphs. Cambridge University Press.Google Scholar
Kendall, M., Stuart, A. and Ord, K. (1987). Advanced Theory of Statistics, Vol. I, Distribution Theory. Oxford University Press.Google Scholar
Lyon, M. and Mahmoud, H. (2020). Trees grown under young-age preferential attachment. J. Appl. Prob. 57, 911–927.CrossRefGoogle Scholar
Mahmoud, H. (2019). Local and global degree profiles of randomly grown self-similar hooking networks under uniform and preferential attachment. Adv. Appl. Math. 111, 101930.10.1016/j.aam.2019.07.006CrossRefGoogle Scholar
Mahmoud, H. and Smythe, R. (1992). Asymptotic joint normality of outdegrees of nodes in random recursive trees. Random Structures Algorithms 3, 255–266.10.1002/rsa.3240030305CrossRefGoogle Scholar
Ross, N. (2011). Fundamentals of Stein’s method. Prob. Surveys 8, 210–293.10.1214/11-PS182CrossRefGoogle Scholar
Slutsky, E. (1925). Über stochastische Asymptoten und Grenzwerte. Metron 5, 3–89.Google Scholar
Smythe, R. and Mahmoud, H. (1996). A survey of recursive trees. Theory Prob. Math. Statist. 51, 1–29 (appeared in Ukrainian in 1994).Google Scholar
Sparks, J., Balaji, S. and Mahmoud, H. (2021). The containment profile of hyperrecursive trees. Available at arXiv:2101.06359.Google Scholar
Tricomi, F. and Erdélyi, A. (1951). The asymptotic expansion of a ratio of gamma functions. Pacific J. Math. 1, 133–142.CrossRefGoogle Scholar
Williams, D. (1991). Probability with Martingales. Cambridge University Press, New York.10.1017/CBO9780511813658CrossRefGoogle Scholar