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arXiv · 1908.02407

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

Huseyin Acan·Alan Frieze·Boris Pittel
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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 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.

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Huseyin Acan, Alan Frieze, Boris Pittel. 2019-08-07. Giant descendant trees, matchings and independent sets in the age-biased attachment graphs. https://arxiv.org/abs/1908.02407

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