A data-driven kinetic model for opinion dynamics with social network contacts (2307.00906v2)

Abstract: Opinion dynamics is an important and very active area of research that delves into the complex processes through which individuals form and modify their opinions within a social context. The ability to comprehend and unravel the mechanisms that drive opinion formation is of great significance for predicting a wide range of social phenomena such as political polarization, the diffusion of misinformation, the formation of public consensus, and the emergence of collective behaviors. In this paper, we aim to contribute to that field by introducing a novel mathematical model that specifically accounts for the influence of social media networks on opinion dynamics. With the rise of platforms such as Twitter, Facebook, and Instagram and many others, social networks have become significant arenas where opinions are shared, discussed, and potentially altered. To this aim after an analytical construction of our new model and through incorporation of real-life data from Twitter, we calibrate the model parameters to accurately reflect the dynamics that unfold in social media, showing in particular the role played by the so-called influencers in driving individual opinions towards predetermined directions.

• The paper introduces a kinetic model that leverages Boltzmann equations to capture opinion dynamics driven by social network interactions.
• It calibrates the model with Twitter data, revealing a log-normal distribution of followers that validates the framework against empirical trends.
• Numerical experiments demonstrate consensus and polarization scenarios by simulating heterogeneous interaction kernels and bounded confidence models.

A Data-Driven Kinetic Model for Opinion Dynamics with Social Network Contacts

Introduction

In this paper, a new kinetic model is proposed to paper opinion dynamics within social networks. The work extends traditional models by incorporating the influence of social media platforms, like Twitter, on the formation and evolution of opinions. The research highlights the significance of opinion leaders or "influencers" and their widespread reach on social networks in shaping and driving public opinion.

Mathematical Framework

The paper uses a kinetic theory based on Boltzmann equations to describe the distribution and evolution of opinions over time. The model begins by capturing the microscopic level interactions between individuals on a social network and upscales to a mesoscopic framework. Each agent is characterized by a real-valued opinion and the number of followers or connections they possess on the platform.

The model's dynamics are described through several phases:

1. Network Formation: The model begins with the formation of a network predicated on individuals' attempts to increase their connections, driven by a satisficing number of followers that gives a user satisfaction.

   Figure 1: Representation of the social network using a sample of $N_s = 400$ accounts from the data extracted from Twitter.

2. Opinion Dynamics: These dynamics are governed by a set of interactions where agents update their opinions based on interactions with others. The degree of influence an agent can exert is proportional to their number of connections, effectively modeling the role of influencers.
3. Fokker-Planck Reformulation: Asymptotic expansions are employed to derive a Fokker-Planck equation, which models the large-scale distribution of opinions. This reflects a quasi-invariant limit where frequent but small interactions drive the dynamics.

Model Calibration and Validation

The model was calibrated using real-world data from Twitter. Specifically, the distribution of followers among a sample of accounts was characterized to help fit model parameters.

Figure 2 illustrates the comparison between the tails of the data distribution from Twitter and the equilibrium distributions of the proposed models. This fit was achieved using a non-linear least squares method, revealing that a log-normal distribution of followers aligns most closely with observed data.

Figure 2: Comparison between the tails of the data distribution and the different possible equilibrium distributions of the Fokker-Planck models.

Numerical Experiments

To validate and explore the model behavior, several numerical experiments were conducted:

1. Bounded Confidence Models: Simulations demonstrated the emergence of opinion clusters and eventual consensus by introducing and varying parameters governing confidence bounds and follower impact.
2. Heterogeneous Interaction Kernels: The paper explored scenarios where the influence of an opinion was weighted by the confidence bound between interacting agents, delineating scenarios of consensus and polarization.
3. Sznajd Model Dynamics: This extension considered situations where agents' willingness to align opinions depended on the variance in their confidence levels, highlighting the model’s ability to simulate realistic social dynamics.

Application to Real-World Data

The practical relevance of the model was underscored by its application to actual Twitter data, notably reactions to political events such as Donald Trump's re-admission to Twitter and discussions on climate change. Sentiment analysis using VADER provided a means to quantify opinions into a range suitable for model calibration. The fitting processes showed the model could capture real-world opinion trends with acceptable accuracy.

Conclusion

This research introduces an advanced model that integrates social network structures and opinion dynamics effectively. The incorporation of real-world data and kinetic theory provides a robust framework for understanding how opinions spread and evolve in the presence of influential network dynamics. Future research may focus on extending the model to incorporate additional social metrics and contextual factors, refining the real-world applicability even further.