# 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

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topMaruani, 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- C. Asavathiratham, Influence model : a tractable representation of networked Markov chains. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA (2000).
- C. Asavathiratham, S. Roy, B. Lesieutre and G. Verghese, The influence model. IEEE Control Syst. Mag.21 (2001) 52–64.
- R.L. Berger, A necessary and sufficient condition for reaching a consensus using DeGroot’s method. J. Amer. Statist. Assoc.76 (1981) 415–419.
- M.H. DeGroot, Reaching a consensus. J. Amer. Statist. Assoc.69 (1974) 118–121.
- P. DeMarzo, D. Vayanos and J. Zwiebel, Persuasion bias, social influence, and unidimensional opinions. Quart. J. Econ.118 (2003) 909–968.
- N.E. Friedkin and E.C. Johnsen, Social influence and opinions. J. Math. Sociol.15 (1990) 193–206.
- N.E. Friedkin and E.C. Johnsen, Social positions in influence networks. Soc. Networks19 (1997) 209–222.
- B. Golub and M.O. Jackson, Naïve learning in social networks and the wisdom of crowds. American Economic Journal : Microeconomics2 (2010) 112–149.
- M. Grabisch and A. Rusinowska, Measuring influence in command games. Soc. Choice Welfare33 (2009) 177–209.
- M. Grabisch and A. Rusinowska, A model of influence in a social network. Theor. Decis.69 (2010) 69–96.
- M. Grabisch and A. Rusinowska, A model of influence with an ordered set of possible actions. Theor. Decis.69 (2010) 635–656.
- 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.
- M. Grabisch and A. Rusinowska, Influence functions, followers and command games. Games Econ Behav.72 (2011) 123–138.
- M. Grabisch and A. Rusinowska, A model of influence with a continuum of actions. GATE Working Paper, 2010-04 (2010).
- 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).
- C. Hoede and R. Bakker, A theory of decisional power. J. Math. Sociol.8 (1982) 309–322.
- X. Hu and L.S. Shapley, On authority distributions in organizations : equilibrium. Games Econ. Behav.45 (2003) 132–152.
- X. Hu and L.S. Shapley, On authority distributions in organizations : controls. Games Econ. Behav.45 (2003) 153–170.
- M.O. Jackson, Social and Economic Networks. Princeton University Press (2008).
- M. Koster, I. Lindner and S. Napel, Voting power and social interaction, in SING7 Conference. Palermo (2010).
- 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).
- J. Lorenz, A stabilization theorem for dynamics of continuous opinions. Physica A355 (2005) 217–223.
- E. Maruani, Jeux d’influence dans un réseau social. Mémoire de recherche, Centre d’Economie de la Sorbonne, Université Paris 1 (2010).
- 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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