A scaling limit for the degree distribution in sublinear preferential attachment schemes

We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures. In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to ‘non-tree’ evolutions where cycles may develop in the network. A main part of the argument is to show that there is a unique nonnegative solution to an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, through C0-semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

[1]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[2]  B. Tóth,et al.  Erdos-Renyi random graphs + forest fires = self-organized criticality , 2008, 0808.2116.

[3]  Sunder Sethuraman,et al.  Large deviations for the degree structure in preferential attachment schemes , 2011, 1105.6047.

[4]  Guido Caldarelli,et al.  Scale-Free Networks , 2007 .

[5]  J. Norris,et al.  Differential equation approximations for Markov chains , 2007, 0710.3269.

[6]  Codina Cotar,et al.  On a Preferential Attachment and Generalized Pólya's Urn Model , 2012, 1203.5758.

[7]  Krishna B. Athreya.  Preferential Attachment Random Graphs with General Weight Function , 2007, Internet Math..

[8]  S. N. Dorogovtsev,et al.  Structure of growing networks with preferential linking. , 2000, Physical review letters.

[9]  Stefan Bornholdt,et al.  Handbook of Graphs and Networks: From the Genome to the Internet , 2003 .

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

[11]  Heriot-Watt University.  AN OVERVIEW OF SOME STOCHASTIC STABILITY METHODS , 2004 .

[12]  Joel H. Spencer,et al.  Connectivity Transitions in Networks with Super-Linear Preferential Attachment , 2005, Internet Math..

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

[14]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

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

[16]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[17]  S. Foss,et al.  AN OVERVIEW OF SOME STOCHASTIC STABILITY METHODS( Network Design, Control and Optimization) , 2004 .

[18]  H. Simon,et al.  ON A CLASS OF SKEW DISTRIBUTION FUNCTIONS , 1955 .

[19]  S. Redner,et al.  Organization of growing random networks. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Mark E. J. Newman,et al.  Structure and Dynamics of Networks , 2009 .

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

[22]  Editors , 1986, Brain Research Bulletin.

[23]  Albert-László Barabási,et al.  Evolution of Networks: From Biological Nets to the Internet and WWW , 2004 .

[24]  G. Webb.  Theory of Nonlinear Age-Dependent Population Dynamics , 1985 .

[25]  Alan M. Frieze,et al.  Balls and bins models with feedback , 2002, SODA '02.

[26]  N. Wormald.  Differential Equations for Random Processes and Random Graphs , 1995 .

[27]  Alan Baker,et al.  DIOPHANTINE APPROXIMATION (Lecture Notes in Mathematics, 785) , 1981 .

[28]  Horst R. Thieme,et al.  Infinite ODE Systems Modeling Size-Structured Metapopulations, Macroparasitic Diseases, and Prion Proliferation , 2008 .

[29]  R. Durrett.  Random Graph Dynamics: References , 2006 .

[30]  Philippe Robert.  Stochastic Networks and Queues , 2003 .

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

[32]  V. Roychowdhury,et al.  Re-inventing Willis , 2006, physics/0601192.

[33]  K. B. Athreya,et al.  Growth of preferential attachment random graphs via continuous-time branching processes , 2007 .

[34]  Nathan Ross,et al.  Degree asymptotics with rates for preferential attachment random graphs , 2011, 1108.5236.

[35]  Reuven Cohen,et al.  Complex Networks: Structure, Robustness and Function , 2010 .

[36]  F. Chung,et al.  Complex Graphs and Networks , 2006 .

[37]  Maia Martcheva,et al.  Kolmogorov’s Differential Equations and Positive Semigroups on First Moment Sequence Spaces , 2006, Journal of mathematical biology.

[38]  F. Chung,et al.  Generalizations of Polya's urn Problem , 2003 .

[39]  Maximilian Wimmer,et al.  A LAW OF LARGE NUMBERS AND CENTRAL LIMIT THEOREM FOR THE LEAVES IN A RANDOM GRAPH MODEL , 2006 .

[40]  Nathan Ross,et al.  Power Laws in Preferential Attachment Graphs and Stein's Method for the Negative Binomial Distribution , 2013, Advances in Applied Probability.

[41]  V. Baladi.  Positive transfer operators and decay of correlations , 2000 .

[42]  Albert-László Barabási,et al.  Scale-Free Networks: A Decade and Beyond , 2009, Science.

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

[44]  A. D. Barbour,et al.  Asymptotic behavior of a metapopulation model , 2005 .

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

[46]  Michael Mitzenmacher,et al.  A Brief History of Generative Models for Power Law and Lognormal Distributions , 2004, Internet Math..

[47]  S. Redner,et al.  Connectivity of growing random networks. , 2000, Physical review letters.