A long-standing debate is whether social influence improves the collective wisdom of a crowd or undermines it. This paper addresses this question based on a naive learning setting in influence systems theory: in our models individuals evolve their estimates of an unknown truth according to the weighted-average opinion dynamics. A formal mathematization is provided with rigorous theoretical analysis. We obtain various conditions for improving, optimizing and undermining the crowd accuracy, respectively. We prove that if the wisdom of finite-size group is improved, then the collective estimate converges to the truth as group size increases, provided individuals' variances are finite. We show that whether social influence improves or undermines the wisdom is determined by the social power allocation of the influence system: if the influence system allocates relatively larger social power to relatively more accurate individuals, it improves the wisdom; on the contrary, if the influence system assigns less social power to more accurate individuals, it undermines the wisdom. At a population level, individuals' susceptibilities to interpersonal influence and network centralities are both crucial. To improve the wisdom, more accurate individuals should be less susceptible and have larger network centralities. Particularly, in democratic influence networks, if relatively more accurate individuals are relatively less susceptible, the wisdom is improved; if more accurate individuals are more susceptible, the wisdom is undermined, which is consistent with the reported empirical evidence. Our investigation provides a theoretical framework for understanding the role social influence plays in the emergence of collective wisdom.

翻译：一个长期存在的争议是，社会影响力究竟是提升还是削弱了群体的集体智慧。本文基于影响力系统理论中的朴素学习设定来探讨这一问题：在我们的模型中，个体根据加权平均意见动态更新其对未知真相的估计。我们提供了形式化的数学表达及严谨的理论分析，分别获得了提升、优化和削弱群体准确性的多种条件。我们证明，若有限规模群体的智慧得到提升，则随着群体规模增大，集体估计将收敛于真相，前提是个体方差有限。研究表明，社会影响力是提升还是削弱群体智慧，取决于影响力系统中的社会权力分配：若影响力系统将相对更大的社会权力分配给相对更准确的个体，则提升群体智慧；反之，若系统将更少的社会权力分配给更准确的个体，则削弱群体智慧。在群体层面，个体对人际影响的易感性和网络中心性均至关重要。为提升群体智慧，更准确的个体应具有更低的易感性和更高的网络中心性。特别地，在民主影响力网络中，若相对更准确的个体相对不易受影响，则群体智慧提升；若更准确的个体更易受影响，则群体智慧被削弱，这与已有实证证据一致。本研究为理解社会影响力在集体智慧涌现中的作用提供了理论框架。