https://hal-polytechnique.archives-ouvertes.fr/hal-00494262Gómez-Serrano, JavierJavierGómez-SerranoICMAT - Instituto de Ciencias Matematicas - UAM - Universidad Autónoma de Madrid - CSIC - Consejo Superior de Investigaciones Científicas [Madrid]Graham, CarlCarlGrahamCMAP - Centre de Mathématiques Appliquées - UVSQ - Université de Versailles Saint-Quentin-en-Yvelines - X - École polytechnique - CNRS - Centre National de la Recherche ScientifiqueBoudec, Jean-Yves LeJean-Yves LeBoudecEPFL - Ecole Polytechnique Fédérale de LausanneThe Bounded Confidence Model Of Opinion DynamicsHAL CCSD2012Social networksreputationopinionmean-field limitpropagation of chaosnonlinear integro-differential equationkinetic equationnumerical experiments.numerical experiments[MATH.MATH-PR] Mathematics [math]/Probability [math.PR][NLIN.NLIN-AO] Nonlinear Sciences [physics]/Adaptation and Self-Organizing Systems [nlin.AO]Graham, Carl2010-06-22 15:28:142023-03-24 14:52:532010-06-22 15:28:14enJournal articles10.1142/S02182025115000721The bounded confidence model of opinion dynamics, introduced by Deffuant et al., is a stochastic model for the evolution of [0,1]-valued opinions within a finite group of peers. We show that as time goes to infinity, the opinions evolve into a random non-interacting set of clusters, and subsequently the opinions in each cluster converge to their barycenter; the limit empirical distribution is called a partial consensus. Then, we prove a mean-field limit result: for i.i.d. initial opinions, as the number of peers increases and time is rescaled accordingly, the peers asymptotically behave as i.i.d. peers, each influenced by opinions drawn independently from the unique solution of a nonlinear integro-differential equation. As a consequence, the (random) empirical distribution process converges to this (deterministic) solution. We also show that as time goes to infinity, this solution converges to a partial consensus, and identify sufficient conditions for the limit not to depend on the initial condition, and for formation of total consensus. Finally, we show that if the equation has an initial condition with a density, then its solution has a density at all times, develop a numerical scheme to solve the corresponding functional equation of the Kac type, and show, using numerical examples, that bifurcations may occur.