Competing first passage percolation on random regular graphs

We consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on N vertices. The processes are allowed to spread with different rates, start from vertex subsets of different sizes or at different times. We obtain tight results regarding the sizes of the vertex sets occupied by each process, showing that in the generic situation one process will occupy i¾?1Nα vertices, for some 0<α<1. The value of α is calculated in terms of the relative rates of the processes, as well as the sizes of the initial vertex sets and the possible time advantage of one process. The motivation for this work comes from the study of viral marketing on social networks. The described processes can be viewed as two competing products spreading through a social network random regular graph. Considering the processes which grow at different rates corresponding to different attraction levels of the two products or starting at different times the first to market advantage allows to model aspects of real competition. The results obtained can be interpreted as one of the two products taking the lion share of the market. We compare these results to the same process run on d dimensional grids where we show that in the generic situation the two products will have a linear fraction of the market each. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 534-583, 2017

[1]  Shishir Bharathi,et al.  Competitive Influence Maximization in Social Networks , 2007, WINE.

[2]  S. Griffis.  EDITOR , 1997, Journal of Navigation.

[3]  Elchanan Mossel,et al.  Coexistence in Preferential Attachment Networks , 2013, Combinatorics, Probability and Computing.

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

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

[6]  First passage percolation on the Erd\H{o}s-R\'enyi random graph , 2010 .

[7]  Éva Tardos,et al.  Influential Nodes in a Diffusion Model for Social Networks , 2005, ICALP.

[8]  T. Liggett.  Interacting Particle Systems , 1985 .

[9]  Christopher Hoffman.  Geodesics in first passage percolation , 2005 .

[10]  T. Liggett,et al.  Stochastic Interacting Systems: Contact, Voter and Exclusion Processes , 1999 .

[11]  Robin Pemantle,et al.  First passage percolation and a model for competing spatial growth , 1997 .

[12]  Svante Janson,et al.  One, Two and Three Times log n/n for Paths in a Complete Graph with Random Weights , 1999, Combinatorics, Probability and Computing.

[13]  O. Haggstrom,et al.  Nonmonotonic Coexistence Regions for the Two-Type Richardson Model on Graphs , 2006, 1509.06972.

[14]  Olivier Garet,et al.  Coexistence in two-type first-passage percolation models , 2003 .

[15]  P. Donnelly,et al.  Finite particle systems and infection models , 1983, Mathematical Proceedings of the Cambridge Philosophical Society.

[16]  Olle Häggström,et al.  The Initial Configuration is Irrelevant for the Possibility of Mutual Unbounded Growth in the Two-Type Richardson Model , 2006, Comb. Probab. Comput..

[17]  Svante Janson,et al.  Limit theorems for triangular urn schemes , 2006 .

[18]  Amin Saberi,et al.  On the spread of viruses on the internet , 2005, SODA '05.

[19]  H. Kesten.  Aspects of first passage percolation , 1986 .

[20]  P. Clifford,et al.  A model for spatial conflict , 1973 .

[21]  Piet Van Mieghem,et al.  The Flooding Time in Random Graphs , 2002 .

[22]  J. T. Cox,et al.  Some Limit Theorems for Percolation Processes with Necessary and Sufficient Conditions , 1981 .

[23]  Matthew Richardson,et al.  Mining the network value of customers , 2001, KDD '01.

[24]  G. Hooghiemstra,et al.  First passage percolation on random graphs with finite mean degrees , 2009, 0903.5136.

[25]  Shankar Bhamidi,et al.  Extreme value theory, Poisson-Dirichlet distributions, and first passage percolation on random networks , 2010, Advances in Applied Probability.

[26]  W. T. Gowers,et al.  RANDOM GRAPHS (Wiley Interscience Series in Discrete Mathematics and Optimization) , 2001 .

[27]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[28]  Peter H. Reingen,et al.  Social Ties and Word-of-Mouth Referral Behavior , 1987 .

[29]  Shankar Bhamidi,et al.  First passage percolation on locally treelike networks. I. Dense random graphs , 2008 .

[30]  Remco van der Hofstad,et al.  First Passage Percolation on the Erdős–Rényi Random Graph , 2010, Combinatorics, Probability and Computing.

[31]  Shankar Bhamidia,et al.  First passage percolation on locally treelike networks , 2009 .

[32]  Béla Bollobás,et al.  A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs , 1980, Eur. J. Comb..

[33]  D. Richardson.  Random growth in a tessellation , 1973, Mathematical Proceedings of the Cambridge Philosophical Society.

[34]  R. Holley,et al.  Ergodic Theorems for Weakly Interacting Infinite Systems and the Voter Model , 1975 .

[35]  D. Freedman.  On Tail Probabilities for Martingales , 1975 .

[36]  Olle Häggström,et al.  Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model , 2000 .

[37]  Stanley Wasserman,et al.  Social Network Analysis: Methods and Applications , 1994, Structural analysis in the social sciences.

[38]  Matthew Richardson,et al.  Mining knowledge-sharing sites for viral marketing , 2002, KDD.

[39]  R. Durrett,et al.  Two phase transitions for the contact process on small worlds , 2005, math/0501481.

[40]  Christopher Hoffman.  Coexistence for Richardson type competing spatial growth models , 2004 .

[41]  Éva Tardos,et al.  Maximizing the Spread of Influence through a Social Network , 2015, Theory Comput..

[42]  David Griffeath,et al.  The basic contact processes , 1981 .

[43]  Maury Bramson,et al.  Asymptotics for interacting particle systems onZd , 1980 .

[44]  Sandip Roy,et al.  The influence model , 2001 .

[45]  R. Durrett.  Lecture notes on particle systems and percolation , 1988 .

[46]  First-passage competition with different speeds: positive density for both species is impossible , 2006, math/0608667.

[47]  宮沢 政清,et al.  P. Bremaud 著, Markov Chains, (Gibbs fields, Monte Carlo simulation and Queues), Springer-Verlag, 1999年 , 2000 .

[48]  Olle Haggstrom,et al.  The two-type Richardson model with unbounded initial configurations , 2007, 0710.5602.

[49]  Jacob Goldenberg,et al.  Using Complex Systems Analysis to Advance Marketing Theory Development , 2001 .

[50]  鈴木 増雄.  Time-Dependent Statistics of the Ising Model , 1965 .

[51]  Elchanan Mossel,et al.  Submodularity of Influence in Social Networks: From Local to Global , 2010, SIAM J. Comput..