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The containment profile of hyper-recursive trees

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

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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