Leveraging Uncertainty in Collective Opinion Dynamics with Heterogeneity

TL;DR

It is demonstrated that uncertainty-driven adaptive weighting leads to increased accuracy and speed of consensus, especially with increasing heterogeneity, and the detrimental effect of overconfident central agents on consensus accuracy is shown.

Abstract

Natural and artificial collectives exhibit heterogeneities across different dimensions, contributing to the complexity of their behavior. We investigate the effect of two such heterogeneities on collective opinion dynamics: heterogeneity of the quality of agents' prior information and of centrality in the network, i.e., the number of immediate neighbors. To study these heterogeneities, we not only consider them in our model, proposing a novel network generator with heterogeneous centrality, but also introduce uncertainty as an additional dimension. By quantifying the uncertainty of each agent, we provide a mechanism for agents to adaptively weigh their individual against social information. As uncertainties develop according to the interactions between agents, they capture information on heterogeneities. Therefore, uncertainty is a relevant additional observable in the study of complex collective opinion dynamics that we use to show the bidirectional relationship of heterogeneous centrality and information. Furthermore, we demonstrate that uncertainty-driven adaptive weighting leads to increased accuracy and speed of consensus, especially under heterogeneity, and provide guidelines for avoiding performance-decreasing errors in uncertainty modeling. These opportunities for improved performance and observability suggest the importance of uncertainty both for the study of natural and the design of artificial heterogeneous systems.

Figures (7)

• Figure 1: Centrality changes with uncertainty. By quantifying uncertainty in opinion dynamics, we can observe the complex development of centrality driven by relative differences in uncertainty. We show this for a $k$-regular homogeneous graph of $26$ agents with $k=6$. We distribute initial certainty heterogeneously by assigning higher certainty to only one agent. Following uncertainty-driven updating (BI-AI), agents with lower uncertainty relative to their neighbors gain higher in-degree centrality.
• Figure 2: Centrality determines the rate of uncertainty reduction of nodes (a-c), while uncertainty provides an adaptive weighting mechanism (as in d-f). This adaptivity alleviates the negative effects observed in heterogeneous networks, such as central nodes pulling the collective opinion. This pulling effect can be seen in g, j, and m, with the collective consensus leaning toward the central node, as opposed to the neutral consensus in d. The first row (a-c) shows how the uncertainty of the most central node compares to that of the other nodes in a Bayesian approach (BI-AI) with a homogeneous initial distribution of uncertainty. The following rows show the time-development of opinions of all nodes, of the central node, and the average opinion of the collective, for different opinion updating mechanisms. All settings share the same initial opinions, each column corresponds to a different network centrality, as shown in the first row.
• Figure 3: Quantifying uncertainty enables agents to outperform the naive averaging methods, especially in heterogeneous settings (b). The mean performance of naive averaging for different self-weights (NA) has low precision, having locally equal weights (NA-LEW) enhances this precision, similar to choosing an optimal universal weight (NA-OUW) for all agents. By assuming independence (BI-AI) agents capture more information about the heterogeneity in the network, leading to overall lower errors under heterogeneity, but higher bias (trueness error) in homogeneous conditions (c).
• Figure 4: Heterogeneity speeds up the collective consensus. While Bayesian inference methods (c, d) leverage the heterogeneity of information and centrality, naive averaging methods (a, b) lack a mechanism to take advantage of information heterogeneity. The independence assumption accelerates the consensus of the Bayesian approach even further (BI-AI, d). We show this by evaluating the precision error for different methods under two different dimensions of heterogeneity. The final precision error normalized with respect to the initial one indicates the speed of convergence, given a fixed amount of time.
• Figure 5: Uncertainty-driven methods are robust against modeling errors unless agents are highly over-confident, in particular if they are central. We show this by evaluating performance under varying uncorrelated systematic errors (illustrated in a) and varying centrality-correlated errors (illustrated in d) for different Bayesian mechanisms. The performance, measured as accuracy error, only decreases for uncorrelated errors when the information heterogeneity is high and agents are over-confident about their uncertainty (top-left corner in b and c). Whereas for centrality-correlated errors, the combination of heterogeneous networks and over-confident central nodes (top-right corner in e and f) is detrimental to the collective performance. Initial opinions are drawn from a Gaussian distribution, but uncertainties are assigned according to the modeling error.