Boltzmann and Fokker–Planck equations modelling opinion formation in the presence of strong leaders

Abstract

We propose a mathematical model for opinion formation in a society that is built of two groups, one group of ‘ordinary’ people and one group of ‘strong opinion leaders’. Our approach is based on an opinion formation model introduced in Toscani (Toscani 2006 Commun. Math. Sci. 4 , 481–496) and borrows ideas from the kinetic theory of mixtures of rarefied gases. Starting from microscopic interactions among individuals, we arrive at a macroscopic description of the opinion formation process that is characterized by a system of Fokker–Planck-type equations. We discuss the steady states of this system, extend it to incorporate emergence and decline of opinion leaders and present numerical results.

Boltzmann-type control of opinion consensus through leaders

“…Indeed since |w| ≤ 1 it follows that |v − w| ≤ 1 and, as 0 ≤ P(·, ·) ≤ 1, it is easily seen that w * , v * ∈ I. Finally, as shown in [10], the post-interaction opinion of followers w * * , in the leader-follower interaction (3.5), takes values in the reference interval I if the hypothesis of the following proposition is satisfied. Proposition 3.2.…”

“…[16,20,33,34]). Similarly to [10], we are interested in the opinion formation process of a followers' population steered by the action of a leaders' group. The major novelty here is that the leaders' behaviour is driven by a suitable control strategy based on the interplay between the desire to force followers towards a given state and the necessity to keep a position close to the mean opinion of the followers in order to influence them.…”

“…In particular, we consider models where the control strategy is based on hierarchical leadership. This hierarchical leadership concept is discussed in [10], where a population of leaders is considered giving rise to aggregate opinions and convergence towards specific patterns. Opinion dynamics in the presence of different populations has been previously been introduced in [24][25][26].…”

“…The general setting consists in a control problem involving a very large number of agents where both the evolution of the state and the objective functional of each agent are influenced by the collective behaviour of all other agents. Typical examples in socio-economical sciences, biology and engineering are represented by the problems of persuading voters to vote for a specific candidate, influencing buyers towards a given good or asset, forcing human crowds or a group of animals to follow a specific path or to reach a desired zone, or optimizing the traffic flow in road networks and supply chains [2,[6][7][8][9][10][11][12][13][14]. In this paper, we focus on control problems where the collective behaviour corresponds to the formation process of opinion consensus [15][16][17][18][19][20][21][22][23].…”

Partial differential equation models in the socio-economic sciences

“…Besides the fact that now systems have to be considered instead of single kinetic equations, the interactions naturally become asymmetric, which complicates the mathematical analysis (e.g. [10]). In this issue, Albi et al [11] deal with such a problem, considering Boltzmann-type interactions with leaders also at the microscopic level, and explaining how they can control opinion from the arising macroscopic models.…”