A study of the dynamic of influence through differential equations
RAIRO - Operations Research - Recherche Opérationnelle, Tome 46 (2012) no. 1, pp. 83-106.

The paper concerns a model of influence in which agents make their decisions on a certain issue. We assume that each agent is inclined to make a particular decision, but due to a possible influence of the others, his final decision may be different from his initial inclination. Since in reality the influence does not necessarily stop after one step, but may iterate, we present a model which allows us to study the dynamic of influence. An innovative and important element of the model with respect to other studies of this influence framework is the introduction of weights reflecting the importance that one agent gives to the others. These importance weights can be positive, negative or equal to zero, which corresponds to the stimulation of the agent by the ‘weighted' one, the inhibition, or the absence of relation between the two agents in question, respectively. The exhortation obtained by an agent is defined by the weighted sum of the opinions received by all agents, and the updating rule is based on the sign of the exhortation. The use of continuous variables permits the application of differential equations systems to the analysis of the convergence of agents' decisions in long-time. We study the dynamic of some influence functions introduced originally in the discrete model, e.g., the majority and guru influence functions, but the approach allows the study of new concepts, like e.g. the weighted majority function. In the dynamic framework, we 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. equations to the influence model, we recover the results of the discrete model on on the boss and approval sets for the command games equivalent to some influence functions.

DOI : https://doi.org/10.1051/ro/2012009
Classification : C7,  C6,  D7
Mots clés : social network, inclination, importance weight, decision, influence function, differential equations
@article{RO_2012__46_1_83_0,
     author = {Maruani, Emmanuel and Grabisch, Michel and Rusinowska, Agnieszka},
     title = {A study of the dynamic of influence through differential equations},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {83--106},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {1},
     year = {2012},
     doi = {10.1051/ro/2012009},
     zbl = {1248.91086},
     mrnumber = {2934894},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro/2012009/}
}
Maruani, Emmanuel; Grabisch, Michel; Rusinowska, Agnieszka. A study of the dynamic of influence through differential equations. RAIRO - Operations Research - Recherche Opérationnelle, Tome 46 (2012) no. 1, pp. 83-106. doi : 10.1051/ro/2012009. http://www.numdam.org/articles/10.1051/ro/2012009/

[1] C. Asavathiratham, Influence model : a tractable representation of networked Markov chains. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA (2000). | MR 2716979

[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. | MR 624343 | 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. | 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. Networks 19 (1997) 209-222.

[8] B. Golub and M.O. Jackson, Naïve learning in social networks and the wisdom of crowds. American Economic Journal : Microeconomics 2 (2010) 112-149.

[9] M. Grabisch and A. Rusinowska, Measuring influence in command games. Soc. Choice Welfare 33 (2009) 177-209. | MR 2520240 | Zbl 1190.91017

[10] M. Grabisch and A. Rusinowska, A model of influence in a social network. Theor. Decis. 69 (2010) 69-96. | MR 2657905 | 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. | MR 2721692 | 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.

[13] M. Grabisch and A. Rusinowska, Influence functions, followers and command games. Games Econ Behav. 72 (2011) 123-138. | MR 2829682 | Zbl 1236.91021

[14] M. Grabisch and A. Rusinowska, A model of influence with a continuum of actions. GATE Working Paper, 2010-04 (2010). | Zbl 1236.91061

[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).

[16] C. Hoede and R. Bakker, A theory of decisional power. J. Math. Sociol. 8 (1982) 309-322. | MR 655913 | Zbl 0485.92019

[17] X. Hu and L.S. Shapley, On authority distributions in organizations : equilibrium. Games Econ. Behav. 45 (2003) 132-152. | MR 2022864 | Zbl 1054.91011

[18] X. Hu and L.S. Shapley, On authority distributions in organizations : controls. Games Econ. Behav. 45 (2003) 153-170. | MR 2022865 | Zbl 1071.91006

[19] M.O. Jackson, Social and Economic Networks. Princeton University Press (2008). | MR 2435744 | Zbl 1203.91001

[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). | MR 1792007 | Zbl 0988.39004

[22] J. Lorenz, A stabilization theorem for dynamics of continuous opinions. Physica A 355 (2005) 217-223. | MR 2143285

[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 http://ccc.cs.uni-duesseldorf.de/COMSOC-2010/slides/invited-rusinowska.pdf (2010).