Scaling limits and influence of the seed graph in preferential attachment trees
Journal de l’École polytechnique - Mathématiques, Volume 2  (2015), p. 1-34

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 [9] 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.

Nous nous intéressons au comportement asymptotique d’arbres aléatoires construits par attachement préférentiel linéaire, qui sont aussi connus dans la littérature sous le nom d’arbres de Barabási-Albert ou encore arbres plans récursifs. Nous validons une conjecture de Bubeck, Mossel & Rácz relative à l’influence de l’arbre initial sur le comportement asymptotique de ces arbres. Séparément, nous étudions la structure géométrique des sommets de grand degré dans la version planaire des arbres de Barabási-Albert en considérant leurs « arbres à boucles ». Lorsque le nombre de sommets croît, nous prouvons que ces arbres à boucles, convenablement mis à l’échelle, convergent au sens de Gromov-Hausdorff vers un espace métrique compact aléatoire, que nous appelons « l’arbre à boucles brownien ». Ce dernier est construit comme un espace quotient de l’arbre continu brownien d’Aldous, et nous prouvons que sa dimension de Hausdorff vaut 2 presque sûrement.

DOI : https://doi.org/10.5802/jep.15
Classification:  05C80,  60J80,  05C05,  60G42
Keywords: Preferential attachment model, Brownian tree, Looptree, Poisson boundary
@article{JEP_2015__2__1_0,
     author = {Curien, Nicolas and Duquesne, Thomas and Kortchemski, Igor and Manolescu, Ioan},
     title = {Scaling limits and influence of the seed graph in preferential attachment trees},
     journal = {Journal de l'\'Ecole polytechnique - Math\'ematiques},
     publisher = {Ecole polytechnique},
     volume = {2},
     year = {2015},
     pages = {1-34},
     doi = {10.5802/jep.15},
     language = {en},
     url = {http://www.numdam.org/item/JEP_2015__2__1_0}
}
Curien, Nicolas; Duquesne, Thomas; Kortchemski, Igor; Manolescu, Ioan. Scaling limits and influence of the seed graph in preferential attachment trees. Journal de l’École polytechnique - Mathématiques, Volume 2 (2015) , pp. 1-34. doi : 10.5802/jep.15. http://www.numdam.org/item/JEP_2015__2__1_0/

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