Usually, opinion formation models assume that individuals have an opinion about a given topic which can change due to interactions with others. However, individuals can have different opinions on different topics and therefore n-dimensional models are best suited to deal with these cases. While there have been many efforts to develop analytical models for one dimensional opinion models, less attention has been paid to multidimensional ones. In this work, we develop an analytical approach for multidimensional models of continuous opinions. We show that for any generic reciprocal interactions between agents, the mean value of initial opinion distribution is conserved. Moreover, for positive social influence interaction mechanisms, the variance of opinion distributions decreases with time and the system converges to a delta distributed function. In particular, we calculate the convergence time when agents get closer in a discrete quantity after interacting, showing a clear difference between cases where the approach is through Manhattan or Euclidean distance.

通常，意见形成模型假设个人对给定主题有意见，该意见可能会因与他人的互动而改变。但是，个人可能对不同的主题有不同的看法，因此 n 维模型最适合处理这些情况。虽然已经为一维意见模型开发分析模型做了很多努力，但对多维模型的关注较少。在这项工作中，我们为连续意见的多维模型开发了一种分析方法。我们表明，对于代理之间的任何通用互惠交互，初始意见分布的平均值是守恒的。此外，对于积极的社会影响交互机制，意见分布的方差随着时间的推移而减小，系统收敛到一个 delta 分布函数。