Opinion dynamics modelling: distinct attraction and repulsion topologies highlight quantitative effects of trolling

TL;DR

The paper models opinion dynamics on networks by coupling a linear attraction term with a repulsive tanh interaction controlled by controversialness, alpha. By placing attraction and repulsion on separate networks and analyzing multiple topologies (BA, caveman variants), it reveals a critical threshold at alpha = 1 where consensus can break down into polarisation, clustering, or dissensus, with the outcome highly dependent on network structure. It provides analytical insights (potential function, Laplacian-based linear regime) and a comprehensive statistical framework, including a continuous clustering-based paradigm classification, to quantify how trolls or provocative content can structurally drive division. The approach gives a structured way to predict and diagnose how network topology and controversy interact to shape collective opinion, with implications for mitigating mis/disinformation and designing robust online communities.

Abstract

We introduce a model of opinion dynamics based on networked non-linear differential equations. The model combines a linear attraction with a repulsive hyperbolic tangent interaction, labeled controversialness. For low controversialness the model displays universal consensus, which is typical of opinion models. As controversialness increases, opinion behaviours such as polarisation, clustering and dissensus emerge, dependent on the network topology. By placing attractive and repulsive interactions on distinct networks, this model is able to simulate the manipulative effects of trolls by introducing controversy, which may be associated with mis/disinformation, toxic messaging, and encouraging provocative questioning and/or emotional posting. This work offers an analytical and statistical analysis of model results, under a wide variety of topologies and initial conditions, whilst also generalising cluster detection algorithms typically applied to discrete models.

Figures (20)

• Figure 1: For six types of network, $\{BA_1, BA_2$, $BA_3$, $BA_5, C_C$, $C_R$}, we show the average degree values $\langle d_i \rangle$, of each node $i$, sorted from highest to lowest, sampled from 25 graph instances. Each panel additionally displays a specific instance of the graph under consideration.
• Figure 2: Left panel displays one instance of initial conditions $x_i(t=0)$, $i \in \{1,100\}$, applied in this work, randomly generated from a normal distribution, with standard deviation equal to $\frac{1}{3}$. Middle panel displays the same initial conditions as on the left, now ordered. Right panel displays the ordered average of 25 of the initial conditions considered in this work.
• Figure 3: Model output examples for same attraction and repulsion networks (${\cal A} = {\cal R}$) with top and bottom rows showing $BA_1$ and $BA_2$ networks, respectively. Columns 1, 2, and 3 show opinion trajectories for $\alpha$ values of $0.75$, $1.01$, and $5$, respectively. Insets on each panel list the order parameter value of Eq.(\ref{['order']}) for each model output. All outputs have the same initial conditions.
• Figure 4: Examples of the potential function in Eq.(\ref{['N_2']}) for $N=2$ with different values of $\alpha$.
• Figure 5: Average results of the order parameter for increasing $\alpha$ for different classes of $BA$ networks, plotted with the theoretical limit of the all-to-all network ($r=2/3$) given in Eq.(\ref{['theor']}).