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A study of the dynamic of influence through differential equations. (English) Zbl 1248.91086
The paper proposes a continuous dynamical model of influence in which agents make their decisions on a certain issue. The model takes into account the opportunity of the agents to change their initial decision, under some possible influences from other agents in the network. This change of opinion can take place on an iterative manner. The agents could also assign weights to each issue, reflecting the importance given to them. These importance weights could be either positive, negative or zero, corresponding to the stimulation of the agent by the others, the inhibition, or the absence of any relation between the two agents in question. The weighted sum of the opinions that a certain agent receives from of all the other agents defines the exhortation obtained by the agent. The dynamics of the model is represented by a system of ODEs, in which the variables are the inclinations of the agents. The authors have shown that if the majority function is used, then the decision of each agent converges to the inclination of the guru. They also describe necessary and sufficient conditions for an agent to be follower of a coalition, and for a set to be the boss set or the approval set of an agent.
MSC:
91D30 Social networks; opinion dynamics
60H30 Applications of stochastic analysis (to PDEs, etc.)
37N40 Dynamical systems in optimization and economics
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
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References:
[1] C. Asavathiratham, Influence model : a tractable representation of networked Markov chains. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA (2000).
[2] C. Asavathiratham, S. Roy, B. Lesieutre and G. Verghese, The influence model. IEEE Control Syst. Mag.21 (2001) 52-64.
[3] R.L. Berger, A necessary and sufficient condition for reaching a consensus using DeGroot’s method. J. Amer. Statist. Assoc.76 (1981) 415-419. · Zbl 0455.60004
[4] M.H. DeGroot, Reaching a consensus. J. Amer. Statist. Assoc.69 (1974) 118-121. · Zbl 0282.92011
[5] P. DeMarzo, D. Vayanos and J. Zwiebel, Persuasion bias, social influence, and unidimensional opinions. Quart. J. Econ.118 (2003) 909-968. Zbl1069.91093 · Zbl 1069.91093
[6] N.E. Friedkin and E.C. Johnsen, Social influence and opinions. J. Math. Sociol.15 (1990) 193-206. · Zbl 0712.92025
[7] N.E. Friedkin and E.C. Johnsen, Social positions in influence networks. Soc. Networks19 (1997) 209-222.
[8] B. Golub and M.O. Jackson, Naïve learning in social networks and the wisdom of crowds. American Economic Journal : Microeconomics2 (2010) 112-149.
[9] M. Grabisch and A. Rusinowska, Measuring influence in command games. Soc. Choice Welfare33 (2009) 177-209. Zbl1190.91017 · Zbl 1190.91017
[10] M. Grabisch and A. Rusinowska, A model of influence in a social network. Theor. Decis.69 (2010) 69-96. Zbl1232.91579 · Zbl 1232.91579
[11] M. Grabisch and A. Rusinowska, A model of influence with an ordered set of possible actions. Theor. Decis.69 (2010) 635-656. Zbl1232.91176 · Zbl 1232.91176
[12] M. Grabisch and A. Rusinowska, Different approaches to influence based on social networks and simple games, in Collective Decision Making : Views from Social Choice and Game Theory, edited by A. van Deemen and A. Rusinowska. Series Theory and Decision Library C 43, Springer-Verlag, Berlin, Heidelberg (2010) 185-209. · Zbl 1331.91148
[13] M. Grabisch and A. Rusinowska, Influence functions, followers and command games. Games Econ Behav.72 (2011) 123-138. · Zbl 1236.91021
[14] M. Grabisch and A. Rusinowska, A model of influence with a continuum of actions. GATE Working Paper, 2010-04 (2010).
[15] M. Grabisch and A. Rusinowska, Iterating influence between players in a social network. CES Working Paper, 2010.89, ftp://mse.univ-paris1.fr/pub/mse/CES2010/10089.pdf (2011). · Zbl 1232.91579
[16] C. Hoede and R. Bakker, A theory of decisional power. J. Math. Sociol.8 (1982) 309-322. · Zbl 0485.92019
[17] X. Hu and L.S. Shapley, On authority distributions in organizations : equilibrium. Games Econ. Behav.45 (2003) 132-152. · Zbl 1054.91011
[18] X. Hu and L.S. Shapley, On authority distributions in organizations : controls. Games Econ. Behav.45 (2003) 153-170. Zbl1071.91006 · Zbl 1071.91006
[19] M.O. Jackson, Social and Economic Networks. Princeton University Press (2008). · Zbl 1149.91051
[20] M. Koster, I. Lindner and S. Napel, Voting power and social interaction, in SING7 Conference. Palermo (2010).
[21] U. Krause, A discrete nonlinear and nonautonomous model of consensus formation, in Communications in Difference Equations, edited by S. Elaydi, G. Ladas, J. Popenda and J. Rakowski. Gordon and Breach, Amsterdam (2000). · Zbl 0988.39004
[22] J. Lorenz, A stabilization theorem for dynamics of continuous opinions. Physica A355 (2005) 217-223.
[23] E. Maruani, Jeux d’influence dans un réseau social. Mémoire de recherche, Centre d’Economie de la Sorbonne, Université Paris 1 (2010).
[24] A. Rusinowska, Different approaches to influence in social networks. Invited tutorial for the Third International Workshop on Computational Social Choice (COMSOC 2010). Düsseldorf, available at (2010). URIhttp://ccc.cs.uni-duesseldorf.de/COMSOC-2010/slides/invited-rusinowska.pdf
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