Opinion dynamics in bounded confidence models with manipulative agents: Moving the Overton window

TL;DR

This study analyzes how manipulative agents shape opinion dynamics within bounded-confidence frameworks, connecting their influence to the Overton window concept. It develops a general manipulative-augmented model and analyzes HK, DW, and Weighted HK variants, deriving conditions under which manipulators can steer normal agents and shift the central opinion interval. The results show that manipulative groups can induce consensus around their trajectory in HK, while DW yields similar trends but with greater fragmentation; Weighted HK requires larger manipulator groups or slower changes to achieve comparable influence. The findings have implications for understanding how targeted campaigns or bots could move acceptable public opinion, and point to future work on competitive manipulation and window-centering strategies.

Abstract

This paper focuses on the opinion dynamics under the influence of manipulative agents. This type of agents is characterized by the fact that their opinions follow a trajectory that does not respond to the dynamics of the model, although it does influence the rest of the normal agents. Simulation has been implemented to study how one manipulative group modifies the natural dynamics of some opinion models of bounded confidence. It is studied what strategies based on the number of manipulative agents and their common opinion trajectory can be carried out by a manipulative group to influence normal agents and attract them to their opinions. In certain weighted models, some effects are observed in which normal agents move in the opposite direction to the manipulator group. Moreover, the conditions which ensure the influence of a manipulative group on a group of normal agents over time are also established for the Hegselmann-Krause model.

Figures (29)

• Figure 1: Simultaneous HK--model: opinion dynamics for an initial distribution of 100 equispaced points $x_i(0)=-1+\frac{2i}{101}$ for $i=1,\ldots,100$. The constant values of $\varepsilon$ are specified above every graph.
• Figure 2: Simultaneous HK--model: opinion dynamics for the same initial distribution as figure \ref{['figure-scBC1rev1']} with one group of 20 stubborn agents each with opinion at $x=0.75$ (marked with a dashed line).
• Figure 3: Simultaneous HK--model: dynamics of an equispaced initial opinion distribution of 100 normal agents in $[-0.6,0.6]$ with a confidence threshold $\varepsilon=0.1$. The pictures represent the opinion evolution without manipulative agents (left), with a group of $K=10$ manipulators (center) and a group of $K=15$ manipulator (right). The manipulative group changes its opinion linearly from $f(0)=-0.6$ to $f(80)=1$. The yellow area represents the initial Overton window and the dashed line the trajectory of the opinion of the manipulative group.
• Figure 4: Simultaneous HK--model: initial opinion distribution of 100 normal agents in $[-1,1]$ with a confidence threshold $\varepsilon=0.1$ and a group of 8 manipulators changing their opinion linearly from $f(0)=-1$ to $f(t_{\Delta})=0.75$ for $t_{\Delta}\in\{70,140,210\}$ iterations keeping it constant for $t\geq t_{\Delta}$. The dashed line is the trajectory of the opinion of the manipulative group.
• Figure 5: Simultaneous HK--model: size of the cluster in the final distribution of the same simulation of figure \ref{['figure-scBCMan2rev1A']}. The manipulators groups the opinions of neighbouring normal agents along its way until the cluster is big enough not to satisfy the condition (\ref{['eq-condition-one-group']}).

Theorems & Definitions (6)

• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Corollary 3.3
• proof