Network dynamics of social influence in the wisdom of crowds

Abstract

A longstanding problem in the social, biological, and computational sciences is to determine how groups of distributed individuals can form intelligent collective judgments. Since Galton's discovery of the "wisdom of crowds" [Galton F (1907) Nature 75:450-451], theories of collective intelligence have suggested that the accuracy of group judgments requires individuals to be either independent, with uncorrelated beliefs, or diverse, with negatively correlated beliefs [Page S (2008) The Difference: How the Power of Diversity Creates Better Groups, Firms, Schools, and Societies]. Previous experimental studies have supported this view by arguing that social influence undermines the wisdom of crowds. These results showed that individuals' estimates became more similar when subjects observed each other's beliefs, thereby reducing diversity without a corresponding increase in group accuracy [Lorenz J, Rauhut H, Schweitzer F, Helbing D (2011) Proc Natl Acad Sci USA 108:9020-9025]. By contrast, we show general network conditions under which social influence improves the accuracy of group estimates, even as individual beliefs become more similar. We present theoretical predictions and experimental results showing that, in decentralized communication networks, group estimates become reliably more accurate as a result of information exchange. We further show that the dynamics of group accuracy change with network structure. In centralized networks, where the influence of central individuals dominates the collective estimation process, group estimates become more likely to increase in error.

Keywords: collective intelligence; experimental social science; social learning; social networks; wisdom of crowds.

Fig. 1.

Effect of social influence on group accuracy in centralized and decentralized networks. Average error and SD in 13 experimental trials for each network condition are shown. (A) In decentralized networks, both the mean and the median became more accurate over two rounds of social influence. (B) In centralized networks, the effect of social influence on the accuracy of the group mean and group median was determined by the initial estimate of the central node. Results were conditioned on whether the central node was in the direction of truth relative to the group estimate. (C) Total change from round 1 to round 3 with bootstrapped 95% error bars, indicating that changes shown in A and B are significant. Both the mean and median of estimates in decentralized networks became more accurate (n = 13, P < 0.01 for mean, P < 0.001 for median). For centralized networks, the mean and median became less accurate when the central node provided an estimate in the opposite direction of truth (n = 13, P < 0.01 for both mean and median). Both the mean and median became more accurate when the central node provided an estimate in the direction of truth (n = 12, P < 0.01 for the mean and median). (D) In both network conditions, the SD in the distribution of estimates (i.e., diversity of opinions) decreased significantly after each round of revision (n = 13, P < 0.001 for both conditions).

Fig. 2.

Correlation between revision magnitude and individual error. Each point in the main graph shows the average size of individuals’ revisions from round 1 to 3 for individuals located in each decile of the distribution of individual error (i.e., average distance from zero error), measured for n = 4,340 estimates provided by 1,040 individuals assigned to one of 13 decentralized networks or 13 centralized networks. The graph shows a positive “revision coefficient,” such that individuals with greater error in their initial estimates made significantly larger revisions. Controlling for correlation between estimates by the same individual (SI Appendix), we find a positive correlation between individual error and individual revision magnitude (n = 4,340, $\rho$ = 0.41, 95% CI [0.39, 0.43], P < 0.001). (Inset) On the y axis, positive values indicate larger revisions than would be expected based on the distance between an individual’s estimate and their neighborhood estimate. On the x axis, positive values indicate greater initial error than would be expected given the distance between an individual’s estimate and their neighborhood estimate. After controlling for the distance between each individual’s initial estimate and the average estimate of their neighborhood, there is still a significant correlation between individual error and individual revision magnitude (n = 4,340, $\rho$ = 0.25, 95% CI [0.22, 0.28], P < 0.001).

Fig. 3.

Correlations with changes in group mean. Shown are all 59 estimation tasks completed over 13 experimental trials. In centralized networks, two estimation tasks are omitted, where the central node did not provide any response. Decentralized networks show all 59 estimation tasks. (A) In decentralized networks, the revision coefficient for each group estimate—i.e., the partial correlation for all members of a network between individuals’ accuracy and their revision magnitudes on a given estimation task—is highly correlated with the change in the error of the group mean (n = 59, $\rho$ = −0.71, 95% CI [−0.84, −0.51]). On estimation tasks in which groups exhibited larger revision coefficients, they showed significantly greater improvements in the accuracy of the group mean. (B) By contrast, in centralized networks, there was no significant correlation between the revision coefficient and the change in group mean (n = 57, $\rho$ = −0.16, 95% CI [−0.33, 0.10]). (C) In centralized networks, the change in the group mean is strongly correlated with the behavior of the central node. The difference between the initial group estimate and the initial estimate of the central node is highly correlated with the change in the group’s estimate (n = 57, $\rho$ = 0.92, 95% CI [0.88, 0.95]). When the central node had an estimate larger than group mean, the group mean typically increased; when the central node was below the group mean, the group mean typically decreased.