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

Frank P. Pijpers; Benedikt V. Meylahn; Michel R. H. Mandjes

physics.soc-phMar 5, 2026

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

Authors: Frank P. Pijpers, Benedikt V. Meylahn, Michel R. H. Mandjes

Organizations: Statistics Netherlands The Hague · Korteweg-de Vries Institute for Mathematics University of Amsterdam Amsterdam · Institute of Physics Dutch Institute for Emergent Phenomena University of Amsterdam Amsterdam · Mathematical Institute Leiden University Leiden

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 & tables

Panel of Figure 1 — 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 ( 1 ). (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.

Panel of Figure 3 — 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’.

Panel of Figure 4 — 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.

Panel of Figure 5 — 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.

Panel of Figure 6 — Figure 6: The estimated probability of (a) consensus and (b) opinion bi-polarization under the topical weights extension to the bounded confidence model. The results are plotted in stars and a dotted line for the toroidal opinion space and with down chevrons and a solid line for the cubic opinion space. (c) Distribution of the time to absorption (survival probability). Notice that with this extension, for the first time there are runs which do not absorb in the simulated period.

$b$ :	11	12	13	14	15
Cubic	1	0	0	1	2
Toroidal	1	12	9	28	276

Table 1: Number of simulation runs (out of 1000) that did not converge to a steady state within the simulated 250 000 time steps.

Panel of Figure 7 — Figure 7: The mean number of groups over time for the simulations with clip sizes in $\{11,13,15\}$ with $\alpha=2$ on weighted cubic and weighted toroidal opinion spaces (marked C. and T. respectively).

Figure 8: Histogram of the change caused by the uniform weighting of opinions in both cubic and toroidal opinion space. Change is measured: $d_{\text{weighted}}-d_{\text{unweighted}}$ , thus a negative change implies that the weighting brought the opinions closer together.

Panel of Figure 9 — Figure 9: The distribution of distances under (a) cubic and (b) toroidal opinion space with and without topical weights.

Panel of Figure 10 — Figure 10: The estimated probability of consensus under network rewiring at differing levels of bounded confidence.

Panel of Figure 11 — Figure 11: The effect of the average length of the shortest path between vertices in the network on the number of groups at termination of the simulation. That is the per simulation network configuration the average is taken over the shortest paths between vertices.

Panel of Figure 12 — Figure 12: The effect of the spectral gap of the network termination time $\tau$ of the simulation.

Panel of Figure 13 — Figure 13: The mean number of groups over time for the simulations with clip sizes in $\{10,13,15\}$ and rewiring probability $p\in\{0.0,0.2,0.4\}$ with $\alpha=2$ on weighted cubic and weighted toroidal opinion spaces (marked C. and T. respectively).

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

	Cube				Torus
$\alpha$	$b=12$		$b=15$		$b=12$		$b=15$
	BC	TW	BC	TW	BC	TW	BC	TW
0.5	0.904	0.939	0.005	0.0901	0.04	0.3418	0.0	0.0
1.0	0.92	0.9489	0.008	0.1298	0.038	0.4039	0.00	0.00
2.0	0.928	0.948	0.006	0.1022	0.067	0.4534	0.0	0.00

Appendix

Table 2: Probability of consensus under the bounded confidence (BC) and weighted topics (TW) for

\alpha\in\{0.5,1,2\}

	Cube				Torus
$\alpha$	$b=12$		$b=15$		$b=12$		$b=15$
	BC	TW	BC	TW	BC	TW	BC	TW
0.5	0.09	0.0590	0.031	0.1972	0.098	0.3531	0.00	0.00
1.0	0.077	0.0480	0.03	0.2354	0.144	0.3215	0.00	0.00
2.0	0.065	0.051	0.0270	0.2345	0.139	0.3300	0.00	0.0028

Appendix

Table 3: Probability of bi-polarization under the bounded confidence (BC) and weighted topics (TW) for for

\alpha\in\{0.5,1,2\}

Panel of Figure 14 — Figure 14: The mean number of groups over time for the simulations with $b=12$ at different values of $\alpha=0.5,1.0,2.0$ under bounded confidence (BC) and topical weights (TW).

Panel of Figure 15 — Figure 15: The mean number of groups over time for the simulations with $b=15$ at different values of $\alpha=0.5,1.0,2.0$ under bounded confidence (BC) and topical weights (TW).

Panel of Figure 16 — Figure 16: (a): the transition probabilities as a function of opinionspace distance $d$ , scaled by $M(q-1)$ , using either the standard shape option (dotted and solid) or the alternative eq. ( 16 ) (dashed) for the function $f$ . Right panel: the equivalent Hamiltonian, which, using Glauber dynamics, would induce those probabilities. The parameters chosen for this example are either $\alpha=1/2$ (dotted) or $\alpha=2$ (solid) for function $f$ , and $b=1/\sqrt{6}\approx 0.408$ . For shape ( 16 ) the parameters $\beta=7.2$ and $\sigma_{b}=0.2$ for the function $f_{b}$ . In setting these particular choices it is deliberate that the values of the functions match for $d\downarrow 0$ and that the probabilities drop away near the same value for $d$ .

Panel of Figure 17 — Figure 17: The estimated probability of consensus at differing levels of bounded confidence.

Panel of Figure 18 — Figure 18: The estimated mean number of groups at the termination of the simulation at different levels of bounded confidence.

Figure 19: The frequency of $|S(\tau)|$ in the cubic opinion space under bounded confidence $b=15$ and rewiring from circulant networks. Note that empty squares imply no simulation runs with that outcome, and the annotations are suppressed for values $<0.05$ for legibility.

Panel of Figure 20 — Figure 20: The number of groups at termination under rewiring from the (a) circulant network and (b) the grid network in the toroidal and cubic opinion spaces for $b=15$ .

Panel of Figure 21 — Figure 21: Boxplots showing how rewiring the grid and the circulant graphs changes the spectral gap which is defined as the second eigenvalue of the Laplacian of the graph.

Panel of Figure 22 — Figure 22: The effect of the spectral gap of the network on the termination time $\tau$ of the simulation and the effect of the shortest path on the number of groups at termination for $b=13$ .

Panel of Figure 23 — Figure 23: The mean number of groups over time for the simulations for $b=13$ and rewiring probability $p\in\{0.0,0.2,0.4\}$ with $\alpha=2$ on weighted cubic and weighted toroidal opinion spaces (marked C. and T. respectively).

Opinion Dynamics

Opinion dynamics studies how people and agents influence one another’s beliefs, producing consensus, disagreement, or polarization.

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Multi-Agent LLM Systems Opinion Dynamics Agent-Based Modeling

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Large language models (LLMs) increasingly interact with one another in multi-agent systems, from simulations of human discourse to influence operations and fully LLM-driven social platforms. These interactions give rise to new regimes of opinion propagation that are not yet well understood. We investigate whether classical opinion dynamics models, which have long been used to explain how interactions shape collective beliefs in human societies, can capture the behavior of LLM networks. Specifically, we quantify the relative contributions of classical opinion-dynamics mechanisms, including neighbor influence, stubbornness, homophily, and bias, to the evolution of LLM opinions. Among the mechanisms considered, a bias term capturing regression toward a model- and topic-dependent opinion emerges as a significant explanatory component, with its inclusion reducing cumulative estimated mean opinion error by up to

88\%

. We additionally observe similar trends across the evaluated model families, discussion topics, and network structures.

Abstract

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