Strategic Influence in Social Networks

We consider a model of influence with a set of non-strategic agents and two strategic agents. The non-strategic agents have initial opinions and are linked through a simply connected network. They update their opinions as in the DeGroot model. The two strategic agents have fixed opinions, 1 and 0 respectively, and are characterized by the magnitude of the impact they can exert on non-strategic agents. Each strategic agent forms a link with one non-strategic agent in order to alter the average opinion that eventually emerges in the network. This procedure defines a zero-sum game whose players are the two strategic agents and whose strategy set is the set of non-strategic agents. We focus on the existence and the characterization of equilibria in pure strategy in this setting. Simple examples show that the existence of a pure strategy equilibrium does depend on the structure of the network. The characterization of equilibrium we obtain emphasizes on the one hand the influenceability of target agents and on the other hand their centrality whose natural measure in our context defines a new concept, related to betweenness centrality, that we call intermediacy. We also show that in the case where the two strategic agents have the same impact, symmetric equilibria emerge as natural solutions whereas in the case where the impacts are uneven, the strategic players generally have differentiated equilibrium targets, the high-impacts agent focusing on centrality and the low-impact agent on influenceability.

[1]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[2]  Mark S. Granovetter.  Threshold Models of Collective Behavior , 1978, American Journal of Sociology.

[3]  P. Bonacich.  Power and Centrality: A Family of Measures , 1987, American Journal of Sociology.

[4]  Nikolas Tsakas.  Optimal Influence Under Observational Learning , 2016 .

[5]  Asuman E. Ozdaglar,et al.  Spread of (Mis)Information in Social Networks , 2009, Games Econ. Behav..

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

[7]  M. Degroot.  Reaching a Consensus , 1974 .

[8]  Antoni Calvó-Armengol,et al.  Centre De Referència En Economia Analítica Barcelona Economics Working Paper Series Working Paper Nº 178 Who's Who in Networks. Wanted: the Key Player Who's Who in Networks. Wanted: the Key Player Barcelona Economics Wp Nº 178 , 2022 .

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

[10]  Andrea Galeotti,et al.  Strategic information transmission networks , 2013, J. Econ. Theory.

[11]  Douglas Gale,et al.  Bayesian learning in social networks , 2003, Games Econ. Behav..

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

[13]  T. Schelling.  Micromotives and Macrobehavior , 1978 .

[14]  Jan Lorenz.  Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics background , 2006 .

[15]  Peter Duersch,et al.  Pure Saddle Points and Symmetric Relative Payoff Games , 2010, ArXiv.

[16]  U. Krause.  A DISCRETE NONLINEAR AND NON–AUTONOMOUS MODEL OF CONSENSUS FORMATION , 2007 .

[17]  Rainer Hegselmann,et al.  Opinion dynamics and bounded confidence: models, analysis and simulation , 2002, J. Artif. Soc. Soc. Simul..

[18]  Matthew O. Jackson,et al.  Naïve Learning in Social Networks and the Wisdom of Crowds , 2010 .

[19]  Berno Buechel,et al.  The dynamics of continuous cultural traits in social networks , 2014, J. Econ. Theory.

[20]  Noah E. Friedkin,et al.  Social influence and opinions , 1990 .

[21]  Michel Grabisch,et al.  A model of influence based on aggregation functions , 2013, Math. Soc. Sci..

[22]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[23]  Noah E. Friedkin,et al.  Social positions in influence networks , 1997 .

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

[25]  Asuman E. Ozdaglar,et al.  Opinion Fluctuations and Disagreement in Social Networks , 2010, Math. Oper. Res..

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

[27]  Martin Shubik,et al.  Systems Defense Games: Colonel Blotto, Command and Control , 1981 .

[28]  S. Bikhchandani,et al.  You have printed the following article : A Theory of Fads , Fashion , Custom , and Cultural Change as Informational Cascades , 2007 .

[29]  Michael Kearns,et al.  Competitive contagion in networks , 2019, Games Econ. Behav..

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

[31]  J. Snyder.  Election Goals and the Allocation of Campaign Resources , 1989 .

[32]  A. Galeotti,et al.  Influencing the influencers: a theory of strategic diffusion , 2009 .

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

[34]  Jan Lorenz,et al.  A stabilization theorem for dynamics of continuous opinions , 2005, 0708.2981.

[35]  Béla Bollobás,et al.  Robustness and Vulnerability of Scale-Free Random Graphs , 2004, Internet Math..

[36]  Asuman E. Ozdaglar,et al.  Opinion Dynamics and Learning in Social Networks , 2010, Dyn. Games Appl..

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

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

[39]  Guillaume Deffuant,et al.  Mixing beliefs among interacting agents , 2000, Adv. Complex Syst..

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

[41]  N. Pimlott.  Learning from the neighbours. , 2012, Canadian family physician Medecin de famille canadien.

[42]  Arun G. Chandrasekhar,et al.  The Diffusion of Microfinance , 2012, Science.

[43]  E. Seneta.  Non-negative Matrices and Markov Chains , 2008 .

[44]  Leo Katz,et al.  A new status index derived from sociometric analysis , 1953 .

[45]  P. Bonacich.  Factoring and weighting approaches to status scores and clique identification , 1972 .

[46]  Leonard M. Freeman,et al.  A set of measures of centrality based upon betweenness , 1977 .

[47]  Peter Duersch,et al.  Pure strategy equilibria in symmetric two-player zero-sum games , 2011, Int. J. Game Theory.

[48]  David Kempe,et al.  Price of Anarchy for the N-Player Competitive Cascade Game with Submodular Activation Functions , 2013, WINE.

[49]  Abhijit Banerjee,et al.  Gossip: Identifying Central Individuals in a Social Network , 2014, ArXiv.

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

[51]  Albert-László Barabási,et al.  Error and attack tolerance of complex networks , 2000, Nature.

[52]  Berno Büchel,et al.  Opinion Dynamics and Wisdom under Conformity , 2014 .

[53]  A. Banerjee,et al.  A Simple Model of Herd Behavior , 1992 .

[54]  Nikolas Tsakas,et al.  Diffusion by Imitation: The Importance of Targeting Agents , 2015 .

[55]  Michel Grabisch,et al.  Anonymous Social Influence , 2013, Games Econ. Behav..

[56]  J. Hagenbach,et al.  Strategic Communication Networks , 2009 .

[57]  Mehmet E. Yildiz,et al.  Competitive Targeted Advertising Over Networks , 2016, Oper. Res..

[58]  P. DeMarzo,et al.  Persuasion Bias, Social Influence, and Uni-Dimensional Opinions , 2001 .