Modeling confirmation bias and polarization

Abstract

Online users tend to select claims that adhere to their system of beliefs and to ignore dissenting information. Confirmation bias, indeed, plays a pivotal role in viral phenomena. Furthermore, the wide availability of content on the web fosters the aggregation of likeminded people where debates tend to enforce group polarization. Such a configuration might alter the public debate and thus the formation of the public opinion. In this paper we provide a mathematical model to study online social debates and the related polarization dynamics. We assume the basic updating rule of the Bounded Confidence Model (BCM) and we develop two variations a) the Rewire with Bounded Confidence Model (RBCM), in which discordant links are broken until convergence is reached; and b) the Unbounded Confidence Model, under which the interaction among discordant pairs of users is allowed even with a negative feedback, either with the rewiring step (RUCM) or without it (UCM). From numerical simulations we find that the new models (UCM and RUCM), unlike the BCM, are able to explain the coexistence of two stable final opinions, often observed in reality. Lastly, we present a mean field approximation of the newly introduced models.

Figure 1. Probability density functions (PDFs) of final opinion, after a maximum of 10⁵ time steps or until convergence is reached, for four different combinations of the parameters (ε, μ).

In the upper left figure we have (ε, μ) = (0, 0.05), in the upper right (ε, μ) = (0, 0.1), in the lower left (ε, μ) = (0.2, 0.05), and in the lower right (ε, μ) = (0.2, 0.1). In all figures the blue solid curve is for RUCM, the green dot-dashed one for UCM, the violet dotted one for BCM, and the pale orange dashed one for RBCM. We observe a bimodal distribution for RUCM and UCM, representing the coexistence of two polarized stable opinions.

Figure 2. Summary statistics (mean, standard deviation, 1st quantile, and 3rd quantile) of the final opinion distributions for varying ε and three different values of μ: violet denotes μ = 0.05, blue denotes μ = 0.25, and orange denotes μ = 0.5.

The left column is for BCM, the central one for UCM, and the right one for RUCM.

Figure 3. Final distribution of peaks for the BCM, with varying (ε, μ) ∈ [0, 0.5] × [0, 0.5].

The Monte Carlo simulations are carried on a Scale-Free network with 2000 nodes for T = 10⁵ times steps, that are sufficient to reach the final state of the system under the different parameters combinations (all results are averaged over 5 repetitions).

Figure 4. Final distribution of peaks for the UCM (left) and RUCM (right), with varying (ε, μ) ∈ [0, 0.5] × [0, 0.5].

The Monte Carlo simulations are carried on a Scale-Free network with 2000 nodes for T = 10⁵ times steps, that are sufficient to reach the final state of the system under the different parameters combinations (all results are averaged over 5 repetitions).