Volume 46, Number 1, January-March
|Page(s)||83 - 106|
|Published online||15 May 2012|
A study of the dynamic of influence through differential equations∗
Royal Bank of Canada, New York, USA
2 Paris School of Economics, Université Paris I Panthé on-Sorbonne 106-112 Bd. de l’Hôpital, 75647 Paris Cedex 13, France
3 Paris School of Economics – CNRS, Centre d’Economie de la Sorbonne, France
Accepted: 2 April 2012
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.
Mathematics Subject Classification: C7 / C6 / D7
Key words: Social network / inclination / importance weight / decision / influence function / differential equations
Emmanuel Maruani was a student at the Université de Paris 1 and École Nationale des Ponts et Chaussées when this research has been conducted. Michel Grabisch and Agnieszka Rusinowska acknowledge support by the National Agency for Research (Agence Nationale de la Recherche), Reference : ANR-09-BLAN-0321-01.
© EDP Sciences, ROADEF, SMAI, 2012
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.