Sublinear but never superlinear preferential attachment by local network growth

We investigate a class of network growth rules that are based on a redirection algorithm wherein new nodes are added to a network by linking to a randomly chosen target node with some probability 1 − r or linking to the parent node of the target node with probability r. For fixed 0 < r < 1, the redirection algorithm is equivalent to linear preferential attachment. We show that when r is a decaying function of the degree of the parent of the initial target, the redirection algorithm produces sublinear preferential attachment network growth. We also argue that no local redirection algorithm can produce superlinear preferential attachment.

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

[2]  Sanjeev Khanna,et al.  The Power of Local Information in Social Networks , 2012, WINE.

[3]  K. Pearson.  Biometrika , 1902, The American Naturalist.

[4]  J. Machta,et al.  Parallel dynamics and computational complexity of network growth models. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  A. Baskakov,et al.  Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations , 2013 .

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

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

[8]  S. Redner,et al.  A Kinetic View of Statistical Physics , 2010 .

[9]  Michael Golosovsky,et al.  Stochastic dynamical model of a growing network based on self-exciting point process , 2012, Physical review letters.

[10]  Derek de Solla Price,et al.  A general theory of bibliometric and other cumulative advantage processes , 1976, J. Am. Soc. Inf. Sci..

[11]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

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

[13]  Chadi Barakat,et al.  Passive and Active Network Measurement , 2004, Lecture Notes in Computer Science.

[14]  Z. Neda,et al.  Measuring preferential attachment in evolving networks , 2001, cond-mat/0104131.

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

[16]  P. Krapivsky,et al.  Random ancestor trees , 2010, 1004.1690.

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

[18]  Stephanie Forrest,et al.  Email networks and the spread of computer viruses. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

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

[21]  M. Tomassini,et al.  Empirical analysis of the evolution of a scientific collaboration network , 2007 .

[22]  Dmitri V. Krioukov,et al.  Scale-free networks as pre-asymptotic regimes of super-linear preferential attachment , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[24]  E. Levanon,et al.  Preferential attachment in the protein network evolution. , 2003, Physical review letters.

[25]  Alessandro Vespignani,et al.  Dynamical Processes on Complex Networks , 2008 .

[26]  Luis E C Rocha,et al.  Information dynamics shape the sexual networks of Internet-mediated prostitution , 2010, Proceedings of the National Academy of Sciences.

[27]  S. Redner,et al.  Finiteness and fluctuations in growing networks , 2002, cond-mat/0207107.

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

[29]  R. Merton.  The Matthew Effect in Science , 1968, Science.

[30]  Jon M. Kleinberg,et al.  The Web as a Graph: Measurements, Models, and Methods , 1999, COCOON.

[31]  G. Yule,et al.  A Mathematical Theory of Evolution, Based on the Conclusions of Dr. J. C. Willis, F.R.S. , 1925 .

[32]  A. Vázquez.  Growing network with local rules: preferential attachment, clustering hierarchy, and degree correlations. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[34]  Hernán D Rozenfeld,et al.  Designer nets from local strategies. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.