Control strategies and trends to equilibrium for kinetic models of opinion dynamics driven by social activity

TL;DR

The paper develops a kinetic framework for opinion dynamics where each agent has an opinion $w\in[-1,1]$ and a time-evolving activity level $A$ that modulates interaction propensity. Through a Boltzmann description and a quasi-invariant scaling, it derives a coupled Fokker-Planck model with binary interactions inside the population and with opinion leaders, revealing that inactive agents tend to polarize while active ones converge to consensus; a control strategy is proposed to increase activity and reduce inactivity, and its mean-field effects are analyzed. Analytical results show convergence to equilibrium for the controlled system, with entropy-based methods proving relaxation toward Beta-type equilibria and demonstrating the influence of leaders in steering the mean opinion toward leaders’ averages. The work connects micro-level interactions to macroscopic trends, provides conditions under which the control is effective, and lays groundwork for numerical and graph-structured extensions to study clustering and targeted interventions in large populations.

Abstract

We introduce new kinetic equations modeling opinion dynamics inside a population of individuals, whose propensity to interact with each other is described by their level of social activity. We show that opinion polarization can arise among agents with a low activity level, while active ones develop a consensus, highlighting the importance of social interactions to prevent the formation of extreme opinions. Moreover, we present a realistic control strategy aimed at reducing the number of inactive agents and increasing the number of socially active ones. At last, we prove several (weak and strong) convergence to equilibrium results for such controlled model. In particular, by considering additional interactions between individuals and opinion leaders capable of steering the average opinion of the population, we use entropy method-like techniques to estimate the relaxation toward equilibrium of solutions to a Fokker--Planck equation with degenerate time-dependent coefficients.

Figures (3)

• Figure 1: Graph of the function $\bar{A}$ defined by (\ref{['eq2']}). The three different parts of the population are highlighted.
• Figure 2: Partial equilibrium distributions for a fixed time and different values of the activity level. Inactive agents are characterized by opinion polarization, while active ones by consensus formation. In (A) $m_{\bar{A}}=0$ while in (B) $m_{\bar{A}}=-0.2$, the other values are $\rho_{\bar{A}}=0.65$, $\omega_p=0.8$, $\varepsilon=0.05$, and $\nu_p=0.04$. The function $c$ was so chosen so that the area subtended by the graphs was equal to 1, but other choices would have been acceptable (and they would have led analogous results).
• Figure :

Theorems & Definitions (32)

• Proposition 1
• proof
• Conjecture 1
• Remark 1
• Remark 2
• Proposition 2
• proof
• Proposition 3
• proof
• Remark 3