Source-linked AI summary
Modeling confirmation bias and polarization
Michela Del Vicario, Antonio Scala, Guido Caldarelli, H Eugene Stanley, Walter Quattrociocchi
TL;DR
The paper addresses why consensus-oriented opinion models do not capture polarized online exchanges. It develops BCM-based models with rewiring and negative feedback, finding that UCM and RUCM reproduce coexistence of two stable final opinions, while deriving mean field approximations.
Problem
Classical opinion models often reach consensus, although face-to-face and online exchanges commonly exhibit polarization and non-consensus outcomes.
Method
The paper extends BCM with network rewiring and negative feedback, introducing RBCM, RUCM, and UCM for continuous-opinion social interactions.
Results
UCM and RUCM, unlike BCM, explain the coexistence of two stable final opinions; the paper also derives mean field approximations for the three new models.
Takeaways & Limitations
The convergence parameter µ tunes the number of final opinion peaks, so convergence or divergence speed changes the final opinion landscape.
Takeaways & Limitations
The analysis assumes a continuous interval of opinions and restricts simulations to ε ∈ [0,0.5] under periodic boundary conditions.
Abstract
from arXiv · showhide
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.
Introduction
Classical opinion-dynamics models commonly reach consensus, whereas online and face-to-face exchanges often produce polarization and homogeneous communities. The paper models these dynamics using continuous opinions, peer-to-peer interactions, confirmation bias, social influence, and network evolution.
- Classical opinion-dynamics models, including BCM, generally reach consensus under suitable conditions.
- Online opinion exchanges often produce polarized communities or echo chambers rather than consensus.
- Confirmation bias and social influence are presented as forces associated with polarization and homogeneous links.
- The study uses a continuous interval of opinions and peer-to-peer interactions to represent multiple stable opinions and social exchanges.
- The proposed framework extends the BCM to include online-network interconnection, complexity, and competition among information sources.
- It introduces RBCM, RUCM, and UCM variations and analyzes their dynamics through simulations and mean field approximation.
Methods
The models represent agents with continuous opinions on a periodic domain and update connected pairs according to bounded confidence. BCM dynamics conserve total mass and mean opinion, producing consensus for sufficiently large tolerance and separated clusters for smaller tolerance.
- Agents hold initial opinions x_i uniformly distributed in [0,1], with periodic boundary conditions addressing boundary effects.The alternative opinion distance ranges from 0 to 0.5.
- In BCM, connected agents interact only when their opinion distance is below tolerance ε.Otherwise, they do not interact.
- Interacting opinions update symmetrically using parameter µ, which lies in the interval (0,0.5).Each agent moves toward the other agent’s opinion according to rule (2).
- The opinion density evolves toward a final distribution P(x,∞), while total mass and mean opinion remain conserved.The paper analyzes the time evolution of P(x,t) and its conserved moments.
- For ε ≥ 1, the BCM second moment decays exponentially and all agents approach the center opinion, reaching consensus.With periodic boundary conditions, the corresponding simulation threshold is ε ≥ 1/2.
- Simulations use scale-free networks after the paper reports that network structure does not influence the simulation results.The considered networks include Erdős–Rényi, scale-free, and small-world networks.
- For smaller ε, consensus is not reached and opinions form isolated clusters separated by distances larger than ε.Each cluster evolves toward a Dirac delta function; cluster masses and the mean opinion satisfy conservation laws.
Results and Discussion
The paper compares three BCM-based variants through simulations and mean-field analysis, finding that UCM and RUCM support stable two-opinion coexistence unlike BCM. Rewiring in RUCM changes convergence and peak transitions relative to UCM.
- Models: RBCM rewires discordant links until every connected pair is concordant, then applies BCM updating on the rewired network.The rewiring phase ends when all linked opinion distances fall below ε.
- Models: UCM allows every randomly selected neighbor pair to interact, using BCM updating for concordant opinions and negative updating for discordant opinions.The negative rule models repulsion between sufficiently distant opinions.
- Models: RUCM combines interaction among all randomly selected pairs with rewiring after discordant interactions.Discordant pairs receive the negative update, their link is broken, and the selected agent is connected to a new user.
- Final opinion peaks: UCM and RUCM show a broad parameter region with two coexisting final opinions, whereas BCM follows its established cluster pattern.RUCM also converges faster than UCM, with direct transitions from many opinions to two and from two to consensus.
- Final opinion peaks: Consensus occurs for ε ∈ (0.45,0.5) in UCM and ε ∈ (0.3,0.5) in RUCM, except when µ is near zero.For smaller ε outside the two-opinion region, many opinions separated by more than ε coexist.
Conclusions
Classical opinion-dynamics models generally reach consensus, whereas real face-to-face and online exchanges often do not. The paper proposes BCM-based variations that reproduce coexistence of two stable opinions through rewiring and negative feedback.
- Classical opinion-dynamics models reach consensus under suitable conditions, but consensus is not commonly achieved in face-to-face and online exchanges.
- The paper proposes a model capable of reproducing the empirically observed coexistence of two stable opinions.
- RBCM breaks discordant links until convergence, while UCM permits discordant interactions with negative updating, with RUCM adding rewiring.
- Numerical simulations show that UCM and RUCM, unlike BCM, explain coexistence of two stable final opinions often observed in reality.
Simulation Results for RBCM
RBCM first rewires discordant links until the network is fully concordant, after which users interact through the BCM dynamics. Simulations show that this rewiring lowers the tolerance needed for consensus relative to BCM.
- RBCM performs random rewiring until every connected pair has an opinion difference smaller than ε.
- The estimated mean number of rewiring steps decreases with ε and is fitted by ax−b, with a = 5.105 and b = 1.072.
- RBCM reaches consensus for ε ≥0.15, whereas BCM reaches consensus for ε ≥0.25.
- Figure 6 reports RBCM final peak distributions across varying ε and µ using Scale-Free simulations with 2000 nodes.