舆论动态的有界置信模型
The Bounded Confidence Model Of Opinion Dynamics
摘要：
由Deffuant等人提出的意见动态的有界置信模型是一个描述有限群内连续值意见演化的随机模型。我们证明，随着时间的推移到无穷大，意见演变成一个随机集的集群太远，相互作用，此后所有的意见，在每个集群收敛到他们的重心。然后，我们证明了一个平均场极限的结果，混沌的传播：随着对等体的数量在充分启动的系统中趋于无穷大，时间相应地重新调整，意见过程收敛到独立同分布。非线性马尔可夫（或McKean-Vlasov）过程;极限意见过程的演变就好像是在从其自身的瞬时定律得出的意见的影响下，这是Kac型的非线性积分微分方程的唯一解。这意味着（随机）经验分布过程收敛于这个（确定性）解。然后，我们证明，随着时间的推移到无穷大，这个解决方案收敛到一个法律集中在孤立的意见太远，相互作用，并确定充分条件的限制，不依赖于初始条件，并集中在一个单一的意见。最后，我们证明了，如果方程有一个初始条件的密度，那么它的解决方案有一个密度在任何时候，开发相应的函数方程的数值方案，并显示数值可能会发生分叉。
英文摘要：
The bounded confidence model of opinion dynamics, introduced by Deffuant et al, is a stochastic model for the evolution of continuous-valued opinions within a finite group of peers. We prove that, as time goes to infinity, the opinions evolve globally into a random set of clusters too far apart to interact, and thereafter all opinions in every cluster converge to their barycenter. We then prove a mean-field limit result, propagation of chaos: as the number of peers goes to infinity in adequately started systems and time is rescaled accordingly, the opinion processes converge to i.i.d. nonlinear Markov (or McKean-Vlasov) processes; the limit opinion processes evolves as if under the influence of opinions drawn from its own instantaneous law, which are the unique solution of a nonlinear integro-differential equation of Kac type. This implies that the (random) empirical distribution processes converges to this (deterministic) solution. We then prove that, as time goes to infinity, this solution converges to a law concentrated on isolated opinions too far apart to interact, and identify sufficient conditions for the limit not to depend on the initial condition, and to be concentrated at a single opinion. Finally, we prove that if the equation has an initial condition with a density, then its solution has a density at all times, develop a numerical scheme for the corresponding functional equation, and show numerically that bifurcations may occur.
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