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arXiv · 1406.1758

Scaling limits and influence of the seed graph in preferential attachment trees

Nicolas Curien·Thomas Duquesne·Igor Kortchemski·Ioan Manolescu
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Abstract

We are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barabási-Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel \& Rácz concerning the influence of the seed graph on the asymptotic behavior of such trees. Separately we study the geometric structure of nodes of large degrees in a plane version of Barabási-Albert trees via their associated looptrees. As the number of nodes grows, we show that these looptrees, appropriately rescaled, converge in the Gromov-Hausdorff sense towards a random compact metric space which we call the Brownian looptree. The latter is constructed as a quotient space of Aldous' Brownian Continuum Random Tree and is shown to have almost sure Hausdorff dimension $2$.

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Nicolas Curien, Thomas Duquesne, Igor Kortchemski, Ioan Manolescu. 2014-06-06. Scaling limits and influence of the seed graph in preferential attachment trees. https://arxiv.org/abs/1406.1758

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