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Giant descendant trees, matchings, and independent sets in age-biased attachment graphs

Part of: Combinatorial probability Graph theory

Published online by Cambridge University Press:  25 April 2022

Hüseyin Acan Open the ORCID record for Hüseyin Acan [Opens in a new window] ,
Alan Frieze  and
Boris Pittel
Hüseyin Acan*
Affiliation:
Drexel University
Alan Frieze*
Affiliation:
Carnegie Mellon University
Boris Pittel*
Affiliation:
Ohio State University
*
*Postal address: Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA. Email: [email protected]
**Postal address: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA. Email: [email protected]
***Postal address: Department of Mathematics, Ohio State University, Columbus, OH 43210, USA. Email: [email protected]
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Abstract

We study two models of an age-biased graph process: the $\delta$ -version of the preferential attachment graph model (PAM) and the uniform attachment graph model (UAM), with m attachments for each of the incoming vertices. We show that almost surely the scaled size of a breadth-first (descendant) tree rooted at a fixed vertex converges, for $m=1$ , to a limit whose distribution is a mixture of two beta distributions and a single beta distribution respectively, and that for $m>1$ the limit is 1. We also analyze the likely performance of two greedy (online) algorithms, for a large matching set and a large independent set, and determine – for each model and each greedy algorithm – both a limiting fraction of vertices involved and an almost sure convergence rate.

Keywords

Randompreferential attachment graphasymptotics

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C05: Trees 60C05: Combinatorial probability 05C30: Enumeration in graph theory
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 2 , June 2022 , pp. 299 - 324
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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