The effect of a toroidal opinion space on opinion bi-polarisation

Abstract

Many models of opinion dynamics include measures of distance between opinions. Such models are susceptible to boundary effects where the choice of the topology of the opinion space may influence the dynamics. In this paper we study an opinion dynamics model following the seminal model by Axelrod, with the goal of understanding the effect of a toroidal opinion space. To do this we systematically compare two versions of the model: one with toroidal opinion space and one with cubic opinion space. In their most basic form the two versions of our model result in similar dynamics (consensus is attained eventually). However, as we include bounded confidence and eventually per agent weighting of opinion elements the dynamics become quite contrasting. The toroidal opinion space consistently allows for a greater number of groups in steady state than the cubic opinion space model. Furthermore, the outcome of the dynamics in the toroidal opinion space model are more sensitive to the inclusion of extensions than in the cubic opinion space model.

Figures (23)

• Figure 1: (a) Survival probability over time for both the toroidal (T.) and the cubic (C.) opinion space for different values of $\alpha$ from the interaction function (\ref{['eq:fa']}). (b) The mean distance between neighbours against time measured every 1000 time steps for $\alpha=0.5,2$. 1000 iterations were run for each parameter setting up to a maximum of 250 000 simulated time steps.
• Figure 2: The number of groups over time for the toroidal and cubic opinion space with $\alpha=0.2$. The bars shown indicate the sample mean plus/minus 2 standard deviations. We see that the differences in the dynamics disappear after roughly 15 000 time steps. For legibility we only plot until time step 30 000.
• Figure 3: The 'pull' felt by individuals in a fully connected population with one individual on each possible belief in (a) Cubic opinion space and (b) in Toroidal opinion space. Note here we set $M=2$ for clarity. (c) Illustration of how the bounded confidence is applied to the interaction function $f(d)$, and the interpretation of the 'clip'.
• Figure 4: The effect of bounded confidence has on the estimated probability of reaching a consensus and the estimated probability of opinion bi-polarization in both the cubic and toroidal opinion model, as well as the time it takes for dynamics to settle down.
• Figure 5: The mean number of groups over time for the simulations with $b$ in $\{5, 10, 12,13,14, 15\}$ with $\alpha=2$ on cubic and toroidal opinion spaces (marked C. and T. respectively). We see that the difference between dynamics on a cubic vs a toroidal opinion space grow as the bounded confidence allows for less interaction.