Speaking of Opinions: Comparing Approaches to Modelling Opinion Manipulation

TL;DR

This paper surveys how opinion formation and manipulation are modelled across continuous and binary frameworks, contrasting control-based interventions with protocol-based manipulation. It highlights three continuous manipulation paradigms—direct opinion control, leaders-and-followers, and network-control—each yielding mean-field or Boltzmann descriptions and consensus objectives toward a target opinion $x_d$. It then surveys discrete-state models, including voter and majority dynamics, and algorithmic manipulation tasks such as influence maximisation, exact majority, and network inference, exposing fundamental trade-offs in controllability, computational complexity, and robustness. The work underlines the societal relevance of these models for understanding and mitigating manipulation via social media, while stressing ethical considerations and the potential for designing defenses against such manipulation.

Abstract

This review outlines the major approaches to modelling opinion formation and manipulation in mathematics and computer science. Key tools such as ordinary and partial differential equations, stochastic models, control theory, and interaction protocols are introduced and compared as methods for describing manipulation. The review is separated into those models using a continuous opinion space and those using discrete or binary opinions, with the advantages and disadvantages of each discussed. Finally, the authors provide an interdisciplinary perspective on the field of opinion dynamics and its social significance.

Figures (8)

• Figure 1: Examples of microscopic dynamics \ref{['eqn: HK-model']} for different interaction functions $\phi$. Each interaction function is symmetric and takes values in $[0,1]$. We note different numbers of clusters appearing for each interaction function.
• Figure 2: Example of an uncontrolled (left) and controlled (right) microscopic system with a bounded confidence interaction function with $R=0.2$. The target opinion, $x_d = 0$ is shown in the dashed black line. Here we consider $N = 100$ agents and the control is calculated according to cost function \ref{['Eqn: directly affect cost functional']} with regularisation parameter $\nu = 1$.
• Figure 3: Uncontrolled (left) and controlled (right) solutions to the Fokker-Planck equation \ref{['Eqn: Fokker-Planck Liars']} in the case where the liar's goal opinion $x_d = 0$. In the left-most plot, we have $\rho=0$ and the population interacts without any external influence and we observe clustering. In the right-most plot, we take $\rho=0.1$ and $\kappa = 0.01$ and observe that the population is brought to consensus at the liar's desired opinion. In both plots, the interaction function is equal to 1 for $|x-x'|<r_1 = 0.1$ and equal to 0 for $|x-x'|>r_2=0.2$ with smooth interpolation between these two values. The right-most plot is reproduced with permission from glendinning2025what.
• Figure 4: Reproduced with permission from nugent2024steering. Example optimal control of \ref{['eqn: ODE system']} with cost function \ref{['Eqn: network control cost functional']}. At each timepoint a blue star, grey circle or red triangle is placed at $(x_i,x_j)$ is the control $u_{ij}$ is equal to $+1,0$ and $-1$ respectively. The target opinion $x_d=0.5$ is shown with dashed lines. The control is successful as all points are converging towards this target.
• Figure 5: Realisation of the voter model with $N=1000$ individuals, showing the formation of spatial clusters. Agents are placed on a 2D lattice with connections to their four neighbours. Agents coloured yellow have opinion $0$ and agents coloured purple have opinion $1$.

Theorems & Definitions (6)

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