Coexistence in Preferential Attachment Networks

We introduce a new model of competition on growing networks. This extends the preferential attachment model, with the key property that node choices evolve simultaneously with the network. When a new node joins the network, it chooses neighbours by preferential attachment, and selects its type based on the number of initial neighbours of each type. The model is analysed in detail, and in particular, we determine the possible proportions of the various types in the limit of large networks. An important qualitative feature we find is that, in contrast to many current theoretical models, often several competitors will coexist. This matches empirical observations in many real-world networks.

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

[2]  Thilo Gross,et al.  Adaptive coevolutionary networks: a review , 2007, Journal of The Royal Society Interface.

[3]  Stochastic Orders , 2008 .

[4]  Glenn Ellison,et al.  Rules of Thumb for Social Learning , 1993, Journal of Political Economy.

[5]  Jure Leskovec,et al.  The dynamics of viral marketing , 2005, EC '06.

[6]  Christos Faloutsos,et al.  Interacting viruses in networks: can both survive? , 2012, KDD.

[7]  D. North.  Competing Technologies , Increasing Returns , and Lock-In by Historical Events , 1994 .

[8]  R. Pemantle,et al.  A time-dependent version of Pólya's urn , 1990 .

[9]  M. Newman.  Threshold effects for two pathogens spreading on a network. , 2005, Physical review letters.

[10]  M. Hirsch,et al.  Asymptotic pseudotrajectories and chain recurrent flows, with applications , 1996 .

[11]  Robin Pemantle.  When are touchpoints limits for generalized Pólya urns , 1991 .

[12]  Christos Faloutsos,et al.  Winner takes all: competing viruses or ideas on fair-play networks , 2012, WWW.

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

[14]  Glenn Ellison,et al.  Word-of-Mouth Communication and Social Learning , 1995 .

[15]  R. May,et al.  How Viruses Spread Among Computers and People , 2001, Science.

[16]  H. Ohtsuki,et al.  A simple rule for the evolution of cooperation on graphs and social networks , 2006, Nature.

[17]  Karen Wright,et al.  The shape of things to go , 1990 .

[18]  W. Brian Arthur,et al.  Path-dependent processes and the emergence of macro-structure , 1987 .

[19]  Duncan J Watts,et al.  A simple model of global cascades on random networks , 2002, Proceedings of the National Academy of Sciences of the United States of America.

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

[21]  R. May,et al.  Infection dynamics on scale-free networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Jun Chen,et al.  A generalized Pólya's urn with graph based interactions , 2012, Random Struct. Algorithms.

[23]  R. Pemantle,et al.  Nonconvergence to Unstable Points in Urn Models and Stochastic Approximations , 1990 .

[24]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[25]  Chrysanthos Dellarocas,et al.  The Digitization of Word-of-Mouth: Promise and Challenges of Online Feedback Mechanisms , 2003, Manag. Sci..

[26]  S. Redner.  How popular is your paper? An empirical study of the citation distribution , 1998, cond-mat/9804163.

[27]  W. Arthur,et al.  Complexity and the economy , 2014, Science.

[28]  B Skyrms,et al.  A dynamic model of social network formation. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[29]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and an Introduction to Chaos , 2003 .

[30]  Drew Fudenberg,et al.  Word-of-mouth learning , 2004, Games Econ. Behav..

[31]  Marc Lelarge.  Diffusion and cascading behavior in random networks , 2012, Games Econ. Behav..

[32]  Béla Bollobás,et al.  The Diameter of a Scale-Free Random Graph , 2004, Comb..

[33]  J. Andrew Petersen,et al.  Driving Profitability by Encouraging Customer Referrals: Who, When, and How , 2010 .

[34]  W. Arthur,et al.  Increasing Returns and Path Dependence in the Economy , 1996 .

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

[36]  Mikhail Borisovich Nevelʹson,et al.  Stochastic Approximation and Recursive Estimation , 1976 .

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

[38]  P. Kaye.  Infectious diseases of humans: Dynamics and control , 1993 .

[39]  Robert P. Leone,et al.  How valuable is word of mouth? , 2007, Harvard business review.

[40]  N. Ling.  The Mathematical Theory of Infectious Diseases and its applications , 1978 .

[41]  Marc Lipsitch,et al.  No coexistence for free: neutral null models for multistrain pathogens. , 2009, Epidemics.

[42]  W. Arthur.  ‘Silicon Valley’ locational clusters: when do increasing returns imply monopoly? , 1990 .

[43]  Yuri Ermoliev,et al.  Strong laws for a class of path-dependent stochastic processes with applications , 1986 .

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

[45]  J. Arndt.  Role of Product-Related Conversations in the Diffusion of a New Product , 1967 .

[46]  F. Buttle.  Word of mouth: understanding and managing referral marketing , 1998 .

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

[48]  Jacob Goldenberg,et al.  Talk of the Network: A Complex Systems Look at the Underlying Process of Word-of-Mouth , 2001 .

[49]  Yuval Peres,et al.  Competing first passage percolation on random regular graphs , 2011, Random Struct. Algorithms.

[50]  José Carlos Goulart de Siqueira,et al.  Differential Equations , 1991, Nature.

[51]  P. Herr,et al.  Effects of Word-of-Mouth and Product-Attribute Information on Persuasion: An Accessibility-Diagnosticity Perspective , 1991 .

[52]  Jari Saramäki,et al.  Temporal Networks , 2011, Encyclopedia of Social Network Analysis and Mining.

[53]  Mark E. J. Newman,et al.  Competing epidemics on complex networks , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  Alexander Grey,et al.  The Mathematical Theory of Infectious Diseases and Its Applications , 1977 .

[55]  William D. Sudderth,et al.  A Strong Law for Some Generalized Urn Processes , 1980 .

[56]  Naoki Masuda,et al.  Epidemic dynamics of two species of interacting particles on scale-free networks. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[57]  W. Arthur.  Positive feedbacks in the economy , 1990 .

[58]  Remco van der Hofstad,et al.  The winner takes it all , 2013, 1306.6467.

[59]  M. Newman.  Spread of epidemic disease on networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.