Preferential attachment without vertex growth: Emergence of the giant component

Janson, Svante; Warnke, Lutz

doi:10.1214/20-aap1610

Cited by 4 publications

(1 citation statement)

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“…Interpolating between Erdős-Rényi and preferential attachment, Pittel [35] considered the birth of a giant component in a graph process G M on a fixed vertex set, when G M+1 is obtained by inserting a new edge between vertices i and j with probability proportional to [ deg (i) + δ] • [ deg (j) + δ], with δ > 0 being fixed. Confirming a conjecture of Pittel [35], Janson and Warnke [25] recently determined the asymptotic size of the giant component in the supercritical phase in this graph model.…”

Section: Introductionsupporting

confidence: 75%

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

2022

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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.

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Section: Introductionsupporting

confidence: 75%