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Trees grown under young-age preferential attachment

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

Published online by Cambridge University Press:  04 September 2020

Merritt R. Lyon  and
Hosam M. Mahmoud
Merritt R. Lyon*
Affiliation:
The George Washington University
Hosam M. Mahmoud*
Affiliation:
The George Washington University
*
*Postal address: Department of Statistics, The George Washington University, Washington, DC20052, USA.
*Postal address: Department of Statistics, The George Washington University, Washington, DC20052, USA.
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Abstract

We introduce a class of non-uniform random recursive trees grown with an attachment preference for young age. Via the Chen–Stein method of Poisson approximation, we find that the outdegree of a node is characterized in the limit by ‘perturbed’ Poisson laws, and the perturbation diminishes as the node index increases. As the perturbation is attenuated, a pure Poisson limit ultimately emerges in later phases. Moreover, we derive asymptotics for the proportion of leaves and show that the limiting fraction is less than one half. Finally, we study the insertion depth in a random tree in this class. For the insertion depth, we find the exact probability distribution, involving Stirling numbers, and consequently we find the exact and asymptotic mean and variance. Under appropriate normalization, we derive a concentration law and a limiting normal distribution. Some of these results contrast with their counterparts in the uniform attachment model, and some are similar.

Keywords

Random treerecursive treeoutdegreeinsertion depthtree leavespreferential attachmentPoisson approximationChen–Stein methodphase transitionsmall worldStirling numbers

MSC classification

Primary: 05C05: Trees 05C82: Small world graphs, complex networks 90B15: Network models, stochastic
Secondary: 60C05: Combinatorial probability 60F25: $L^p$-limit theorems 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 3 , September 2020 , pp. 911 - 927
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Copyright
© Applied Probability Trust 2020

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