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The location of high-degree vertices in weighted recursive graphs with bounded random weights

Part of: Graph theory

Published online by Cambridge University Press:  19 January 2024

Bas Lodewijks Open the ORCID record for Bas Lodewijks [Opens in a new window]
Bas Lodewijks*
Affiliation:
University of Augsburg
*
*Postal address: Lehrstuhl für Stochastik und ihre Anwendungen, Universität Augsburg, Universitätsstr. 14, 86159 Augsburg, Germany. Email address: bas.lodewijks@uni-a.de
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Abstract

We study the asymptotic growth rate of the labels of high-degree vertices in weighted recursive graphs (WRGs) when the weights are independent, identically distributed, almost surely bounded random variables, and as a result confirm a conjecture by Lodewijks and Ortgiese (‘The maximal degree in random recursive graphs with random weights’, preprint, 2020). WRGs are a generalisation of the random recursive tree and directed acyclic graph models, in which vertices are assigned vertex-weights and where new vertices attach to $m\in\mathbb{N}$ predecessors, each selected independently with a probability proportional to the vertex-weight of the predecessor. Prior work established the asymptotic growth rate of the maximum degree of the WRG model, and here we show that there exists a critical exponent $\mu_m$ such that the typical label size of the maximum-degree vertex equals $n^{\mu_m(1+o(1))}$ almost surely as n, the size of the graph, tends to infinity. These results extend results on the asymptotic behaviour of the location of the maximum degree, formerly only known for the random recursive tree model, to the more general weighted multigraph case of the WRG model. Moreover, for the weighted recursive tree model, that is, the WRG model with $m=1$, we prove the joint convergence of the rescaled degree and label of high-degree vertices under additional assumptions on the vertex-weight distribution, and also extend results on the growth rate of the maximum degree obtained by Eslava, Lodewijks, and Ortgiese (Stoch. Process. Appl. 158, 2023).

Keywords

Weighted recursive graphweighted recursive treemaximum degreehigh degreerandom environmentpersistencevertex depthmarked point processes

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C07: Vertex degrees
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 59
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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