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

Curien, Nicolas; Duquesne, Thomas; Kortchemski, Igor; Manolescu, Ioan

doi:10.5802/jep.15

Curien, Nicolas ¹; Duquesne, Thomas ²; Kortchemski, Igor ³; Manolescu, Ioan ⁴

¹ Département de mathématiques, Université Paris-Sud Orsay Bâtiment 425, 91405 Orsay, France
² LPMA, Université Pierre et Marie Curie (Paris 6) Case courrier 188, 4 place Jussieu, 75252 Paris Cedex 05, France
³ Département de Mathématiques et Applications, École Normale Supérieure 45 rue d’Ulm, 75230 Paris Cedex 05, France
⁴ Département de Mathématiques, Université de Genève 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Suisse

Journal de l’École polytechnique — Mathématiques, Volume 2 (2015), pp. 1-34.

Abstract
Résumé

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: 10.5802/jep.15

Classification: 05C80, 60J80, 05C05, 60G42
Keywords: Preferential attachment model, Brownian tree, Looptree, Poisson boundary

Author's affiliations:

Curien, Nicolas ¹; Duquesne, Thomas ²; Kortchemski, Igor ³; Manolescu, Ioan ⁴

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     title = {Scaling limits and influence of the seed graph in preferential attachment trees},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1--34},
     publisher = {Ecole polytechnique},
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     year = {2015},
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     url = {http://www.numdam.org/articles/10.5802/jep.15/}
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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/articles/10.5802/jep.15/

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