Multi-Dimensional Opinion Formation

TL;DR

The paper develops a kinetic framework for multi-dimensional opinion formation where individuals carry opinions in $[-1,1]^d$ and topic weights $α$, coupled through a distance $p_a(x,y,α)$. It derives a binary-interaction model and its mean-field Vlasov-type PDE, and analyzes global existence, mass conservation, and moment evolution, highlighting how weight heterogeneity can drive mean shifts and even increase variance. Stationary states include consensus, separated clusters, and interacting clusters, with detailed results on when each form arises and how the geometry of interactions limits cluster numbers in 2D. Simulations illustrate how distance choices and weight distributions affect interaction symmetry, clustering, and ideology shifts, underlining the crucial role of individual topic weights in shaping long-term opinion structure. Overall, the work provides analytical and computational insights into how multi-topic opinions and topic importance interact to produce complex, weight-dependent stationary patterns.

Abstract

In this paper we propose and investigate a multi-dimensional opinion dynamics model where people are characterised by both opinions and importance weights across these opinions. Opinion changes occur through binary interactions, with a novel coupling mechanism: the change in one topic depends on the weighted similarity across the full opinion vector. We state the kinetic equation for this process and derive its mean-field partial differential equation to describe the overall dynamics. Analytical computations and numerical simulations confirm that this model generates complex stationary states, and we demonstrate that the final opinion structures are critically determined by the peoples' opinion weights.

Figures (6)

• Figure 1: Inferaction radius defined by the $p_\alpha$-distance \ref{['eq:p_distance']} for an interaction function $\phi$ with compact support on $[0,R]$
• Figure 2: Example of an interacting cluster discussed in Example \ref{['ex:statsol']}
• Figure 3: Interacting cluster \ref{['eq:statsol_epsilon']}, in which the Dirac measures are arbitrary close (see Example \ref{['ex:closeness']})
• Figure 4: Initial distribution given by \ref{['eq:ex1_new']} and the corresponding stationary states illustrating the impact of different distances discussed in Subsection \ref{['sec:sim_ex1']}
• Figure 5: Initial and final particle density illustrating the swing from 'left' to 'right' discussed in Section \ref{['sec:ltr']}

Theorems & Definitions (19)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Example 3.1
• Proposition 3.3
• proof