A study of the dynamic of influence through differential equations∗

Emmanuel Maruani; Michel Grabisch; Agnieszka Rusinowska

RAIRO - Operations Research (2012)

  • Volume: 46, Issue: 1, page 83-106
  • ISSN: 0399-0559

Abstract

top
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.

How to cite

top

Maruani, Emmanuel, Grabisch, Michel, and Rusinowska, Agnieszka. "A study of the dynamic of influence through differential equations∗." RAIRO - Operations Research 46.1 (2012): 83-106. <http://eudml.org/doc/276397>.

@article{Maruani2012,
abstract = {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.},
author = {Maruani, Emmanuel, Grabisch, Michel, Rusinowska, Agnieszka},
journal = {RAIRO - Operations Research},
keywords = {Social network; inclination; importance weight; decision; influence function; differential equations; social network},
language = {eng},
month = {5},
number = {1},
pages = {83-106},
publisher = {EDP Sciences},
title = {A study of the dynamic of influence through differential equations∗},
url = {http://eudml.org/doc/276397},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Maruani, Emmanuel
AU - Grabisch, Michel
AU - Rusinowska, Agnieszka
TI - A study of the dynamic of influence through differential equations∗
JO - RAIRO - Operations Research
DA - 2012/5//
PB - EDP Sciences
VL - 46
IS - 1
SP - 83
EP - 106
AB - 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.
LA - eng
KW - Social network; inclination; importance weight; decision; influence function; differential equations; social network
UR - http://eudml.org/doc/276397
ER -

References

top
  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.  Zbl0455.60004
  4. M.H. DeGroot, Reaching a consensus. J. Amer. Statist. Assoc.69 (1974) 118–121.  Zbl0282.92011
  5. P. DeMarzo, D. Vayanos and J. Zwiebel, Persuasion bias, social influence, and unidimensional opinions. Quart. J. Econ.118 (2003) 909–968.  Zbl1069.91093
  6. N.E. Friedkin and E.C. Johnsen, Social influence and opinions. J. Math. Sociol.15 (1990) 193–206.  Zbl0712.92025
  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
  10. M. Grabisch and A. Rusinowska, A model of influence in a social network. Theor. Decis.69 (2010) 69–96.  Zbl1232.91579
  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
  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.  Zbl1331.91148
  13. M. Grabisch and A. Rusinowska, Influence functions, followers and command games. Games Econ Behav.72 (2011) 123–138.  Zbl1236.91021
  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).  Zbl1232.91579
  16. C. Hoede and R. Bakker, A theory of decisional power. J. Math. Sociol.8 (1982) 309–322.  Zbl0485.92019
  17. X. Hu and L.S. Shapley, On authority distributions in organizations : equilibrium. Games Econ. Behav.45 (2003) 132–152.  Zbl1054.91011
  18. X. Hu and L.S. Shapley, On authority distributions in organizations : controls. Games Econ. Behav.45 (2003) 153–170.  Zbl1071.91006
  19. M.O. Jackson, Social and Economic Networks. Princeton University Press (2008).  Zbl1149.91051
  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).  Zbl0988.39004
  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).  Zbl1331.91148URIhttp://ccc.cs.uni-duesseldorf.de/COMSOC-2010/slides/invited-rusinowska.pdf

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.