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August 2020 Influence of the seed in affine preferential attachment trees
David Corlin Marchand, Ioan Manolescu
Bernoulli 26(3): 1665-1705 (August 2020). DOI: 10.3150/19-BEJ1152

Abstract

We study randomly growing trees governed by the affine preferential attachment rule. Starting with a seed tree $S$, vertices are attached one by one, each linked by an edge to a random vertex of the current tree, chosen with a probability proportional to an affine function of its degree. This yields a one-parameter family of preferential attachment trees $(T_{n}^{S})_{n\geq |S|}$, of which the linear model is a particular case. Depending on the choice of the parameter, the power-laws governing the degrees in $T_{n}^{S}$ have different exponents.

We study the problem of the asymptotic influence of the seed $S$ on the law of $T_{n}^{S}$. We show that, for any two distinct seeds $S$ and $S'$, the laws of $T_{n}^{S}$ and $T_{n}^{S'}$ remain at uniformly positive total-variation distance as $n$ increases.

This is a continuation of Curien et al. (J. Éc. Polytech. Math. 2 (2015) 1–34), which in turn was inspired by a conjecture of Bubeck et al. (IEEE Trans. Netw. Sci. Eng. 2 (2015) 30–39). The technique developed here is more robust than previous ones and is likely to help in the study of more general attachment mechanisms.

Citation

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David Corlin Marchand. Ioan Manolescu. "Influence of the seed in affine preferential attachment trees." Bernoulli 26 (3) 1665 - 1705, August 2020. https://doi.org/10.3150/19-BEJ1152

Information

Received: 1 January 2019; Published: August 2020
First available in Project Euclid: 27 April 2020

zbMATH: 07193939
MathSciNet: MR4091088
Digital Object Identifier: 10.3150/19-BEJ1152

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 3 • August 2020
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