Robust Analysis of Preferential Attachment Models with Fitness

The preferential attachment network with fitness is a dynamic random graph model. New vertices are introduced consecutively and a new vertex is attached to an old vertex with probability proportional to the degree of the old one multiplied by a random fitness. We concentrate on the typical behaviour of the graph by calculating the fitness distribution of a vertex chosen proportional to its degree. For a particular variant of the model, this analysis was first carried out by Borgs, Chayes, Daskalakis and Roch. However, we present a new method, which is robust in the sense that it does not depend on the exact specification of the attachment law. In particular, we show that a peculiar phenomenon, referred to as Bose–Einstein condensation, can be observed in a wide variety of models. Finally, we also compute the joint degree and fitness distribution of a uniformly chosen vertex.

[1]  Carsten Wiuf,et al.  Convergence properties of the degree distribution of some growing network models , 2006, Bulletin of mathematical biology.

[2]  Jonathan Jordan,et al.  The degree sequences and spectra of scale‐free random graphs , 2006, Random Struct. Algorithms.

[3]  A survey of random processes with reinforcement , 2007, math/0610076.

[4]  Jonathan Jordan,et al.  Geometric preferential attachment in non-uniform metric spaces , 2012, 1208.4938.

[5]  Gerard Hooghiemstra,et al.  A preferential attachment model with random initial degrees , 2007, 0705.4151.

[6]  Mark Newman,et al.  Networks: An Introduction , 2010 .

[7]  Olle Nerman,et al.  On the convergence of supercritical general (C-M-J) branching processes , 1981 .

[8]  Alan M. Frieze,et al.  A general model of web graphs , 2003, Random Struct. Algorithms.

[9]  Bálint Tóth,et al.  Random trees and general branching processes , 2007, Random Struct. Algorithms.

[10]  Jonathan Jordan.  The degree sequences and spectra of scale-free random graphs , 2006 .

[11]  J. Doob.  Stochastic processes , 1953 .

[12]  Svante Janson,et al.  Functional limit theorems for multitype branching processes and generalized Pólya urns , 2004 .

[13]  Shankar Bhamidi,et al.  Universal techniques to analyze preferential attachment trees : Global and Local analysis , 2007 .

[14]  S. Dereich,et al.  Emergence of Condensation in Kingman’s Model of Selection and Mutation , 2012, 1207.6203.

[15]  H. Robbins.  A Stochastic Approximation Method , 1951 .

[16]  Benedek Valkó,et al.  Random trees and general branching processes , 2007 .

[17]  Jonathan Jordan.  Degree sequences of geometric preferential attachment graphs , 2010, Advances in Applied Probability.

[18]  A. Barabasi,et al.  Bose-Einstein condensation in complex networks. , 2000, Physical review letters.

[19]  Christian Borgs,et al.  First to market is not everything: an analysis of preferential attachment with fitness , 2007, STOC '07.

[20]  M. Benaïm.  Dynamics of stochastic approximation algorithms , 1999 .

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

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

[23]  Olle Nerman,et al.  The asymptotic composition of supercritical, multi-type branching populations , 1996 .

[24]  Steffen Dereich,et al.  Random networks with sublinear preferential attachment: Degree evolutions , 2008, 0807.4904.

[25]  M. Émery,et al.  Seminaire de Probabilites XXXIII , 1999 .

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

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