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

We are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barabasi-Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel & Racz (9) concerning the influence of the seed graph on the asymptotic behavior of such trees. Separately we study the geometric struc- ture of nodes of large degrees in a plane version of Barabasi-Albert trees via their associated looptrees. As the number of nodes grows, we show that these looptrees, appropriately rescaled, converge in the Gromov-Hausdor 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 Hausdor dimension 2. Resume (Limites d'echelle et ontogenese des arbres construits par attachement preferentiel) Nous nous interessons au comportement asymptotique d'arbres aleatoires construits par attachement preferentiel lineaire, qui sont aussi connus dans la litterature sous le nom d'arbres de Barabasi-Albert ou encore arbres plans recursifs. Nous validons une conjecture de Bubeck, Mossel & Racz relative a l'influence de l'arbre initial sur le comportement asymptotique de ces arbres. Separement, nous etudions la structure geometrique des sommets de grand degre dans la version planaire des arbres de Barabasi-Albert en considerant leurs « arbres a boucles ». Lorsque le nombre de sommets croit, nous prouvons que ces arbres a boucles, convenablement mis a l'echelle, convergent au sens de Gromov-Hausdor vers un espace metrique compact aleatoire, que nous appelons « l'arbre a boucles brownien ». Ce dernier est construit comme un espace quotient de l'arbre continu brownien d'Aldous, et nous prouvons que sa dimension de Hausdor vaut 2 presque surement.

[1]  Arcwise Isometries,et al.  A Course in Metric Geometry , 2001 .

[2]  W. Woess.  Random walks on infinite graphs and groups, by Wolfgang Woess, Cambridge Tracts , 2001 .

[3]  Daniel J. Ford.  Probabilities on cladograms: introduction to the alpha model , 2005, math/0511246.

[4]  J. L. Gall,et al.  Random trees and applications , 2005 .

[5]  B'en'edicte Haas,et al.  Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees , 2010, 1003.3632.

[6]  L. Addario-Berry,et al.  The continuum limit of critical random graphs , 2009, 0903.4730.

[7]  Remco van der Hofstad,et al.  Random Graphs and Complex Networks , 2016, Cambridge Series in Statistical and Probabilistic Mathematics.

[8]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[9]  Elchanan Mossel,et al.  On the Influence of the Seed Graph in the Preferential Attachment Model , 2014, IEEE Transactions on Network Science and Engineering.

[10]  I. Kortchemski,et al.  Percolation on random triangulations and stable looptrees , 2013, 1307.6818.

[11]  J. Szymanski.  On a Nonuniform Random Recursive Tree , 1987 .

[12]  Gr'egory Miermont,et al.  Tessellations of random maps of arbitrary genus , 2007, 0712.3688.

[13]  J. Pitman.  Combinatorial Stochastic Processes , 2006 .

[14]  Nicolas Pouyanne,et al.  Smoothing Equations for Large Pólya Urns , 2013, 1302.1412.

[15]  Pertti Mattila,et al.  Geometry of sets and measures in Euclidean spaces , 1995 .

[16]  Steffen Dereich,et al.  Random networks with sublinear preferential attachment: The giant component , 2010, 1007.0899.

[17]  David Aldous,et al.  The Continuum Random Tree III , 1991 .

[18]  David Aldous.  RECURSIVE SELF-SIMILARITY FOR RANDOM TREES, RANDOM TRIANGULATIONS AND BROWNIAN EXCURSION , 1994 .

[19]  I. Kortchemski,et al.  Random stable looptrees , 2013, 1304.1044.

[20]  N. Curien,et al.  The stable trees are nested , 2012, 1207.5418.

[21]  Amin Saberi,et al.  Asymptotic behavior and distributional limits of preferential attachment graphs , 2014, 1401.2792.

[22]  Tamás F. Móri,et al.  The Maximum Degree of the Barabási–Albert Random Tree , 2005, Combinatorics, Probability and Computing.

[23]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[24]  S. Evans.  Probability and Real Trees , 2008 .

[25]  Jean-Luc Rémy,et al.  Un Procédé Itératif de Dénombrement D'arbres Binaires et Son Application A Leur Génération Aléatoire , 1985, RAIRO Theor. Informatics Appl..

[26]  Douglas Rizzolo.  Scaling limits of Markov branching trees and Galton-Watson trees conditioned on the number of vertices with out-degree in a given set , 2011, 1105.2528.

[27]  P. Mattila.  Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability , 1995 .

[28]  Jean-Franccois Le Gall,et al.  Random geometry on the sphere , 2014, 1403.7943.

[29]  Béla Bollobás,et al.  The degree sequence of a scale‐free random graph process , 2001, Random Struct. Algorithms.

[30]  T. F. Móri.  On random trees , 2002 .