When minor issues matter: symmetries, pluralism, and polarization in similarity-based opinion dynamics

TL;DR

A stochastic agent-based model where individuals hold binary opinions on multiple issues of heterogeneous weights and interact through both attraction and repulsion reveals how issue salience and social tolerance jointly shape collective opinion evolution.

Abstract

Polarization is a problem in modern society. Understanding how opinions evolve through social interactions is crucial for addressing conditions that lead to polarization, consensus, or opinion diversity. Classical opinion dynamics models have explored bounded confidence and homophily, but most assume equal issue importance and purely attractive forces. We extend these frameworks by developing a stochastic agent-based model where individuals hold binary opinions on multiple issues of heterogeneous weights and interact through both attraction (with similar others) and repulsion (from dissimilar others). Our model reveals that the similarity threshold determining friend-or-foe interactions fundamentally shapes outcomes, which in this model can be of three types: consensus, polarization, and persistent pluralism, where each opinion combination occurs in the population. Low thresholds promote consensus, while high thresholds lead to polarization or persistent pluralism. Surprisingly, introducing even a single issue of arbitrarily small weight can destabilize stable states, thus changing the solution type and increasing convergence times by orders of magnitude. To explain these phenomena, we derive a deterministic system of ordinary differential equations and analyze equilibrium symmetries. For up to five-issue systems, we provide a complete characterization: all weight configurations fall into a number of cases, each exhibiting distinct symmetry cascades as the threshold varies. Our analysis shows polarization risk increases when importance concentrates on few issues. This suggests mitigation strategies: fostering cross-cutting social ties, broadening discourse beyond core issues, and introducing new topics to disrupt polarization. The symmetry-based framework reveals how issue salience and social tolerance jointly shape collective opinion evolution.

Figures (5)

• Figure 1: Stochastic and deterministic dynamics of the opinion model. (a) The rules of opinion change are illustrated using two interactions in an $L=6$ system, where in the first case, the similarity happens to be higher than the threshold, $\alpha$, and the focal individual (the upper string) updates one of the issues to be more like the interlocutor (the string lower string); in the second case, the similarity is lower than the threshold and the focal individual updates one of the issues to be further away from the interlocutor. (b) Two types of absorbing states of the stochastic model with $r=0$ are illustrated with an $L=4$ example. (c) Examples of an $L=4$ system behavior for two different threshold values. The intervals of $\alpha$ are shown on the left for weights $\frac{1}{205}[85,50,40,30]$ (case 8 of Table \ref{['tab:symmetries']}). For intervals 2 (panels (c4,c5,c6)) and 7 (panels (c1,c2,c3)), a typical stochastic trajectory (left, (c1) and (c4)) is shown, together with a zoomed-in grayed fragment on a log-scale (middle, (c2) and (c5)), and a typical solution of the corresponding ODE (right, (c3) and (c6)). Different lines correspond to relative abundance of each of the 16 possible opinion strings. $N=3,000$ individuals, $r=0$.
• Figure 2: Convergence behavior of the stochastic model varies significantly with $\alpha$. (a) The effect of threshold, $\alpha$, on the probability of the stochastic model ending in a consensus/polarization/persistent pluralism state (green/yellow/purple, respectively), when initialized uniformly at random, where the length of the bars represents the probability of different outcomes for different $\alpha$ intervals. Parameters are $L=4$, $N=100$, $r=0$, and weights $\frac{1}{205}[85, 50, 40, 30]$ (case 8 of Table \ref{['tab:symmetries']}). (b) The effect of $\alpha$ on the absorption times with the same weights as in (a), and initial conditions chosen uniformly at random. The inset explains the color code for panels (a) and (c). (c,d) The same as (a,b), except the weights are $[0.346494 , 0.29631 , 0.215498 , 0.141697]$ (case 13 of Table \ref{['tab:symmetries']}). The number of simulations in (a-c) is $2,000$ per $\alpha$, except for the largest intervals with persistent pluralism, where we used $100$ simulations. Other parameters are $L=4$, $N=100$, and $r=0$. (e) The intervals of $\alpha$ for the weights in (c,d) are shown on the left. For intervals 5 and 11-15, a typical stochastic trajectory and the ODE solution are shown. Different lines correspond to relative abundance of each of the 16 possible opinion strings.
• Figure 3: Introducing issues of small weight can destabilize a population, causing every opinion string being held equally and significantly delaying convergence. The proportions of each opinion string in a run of the stochastic model with weights $\frac{1}{312}[108, 104, 100]\approx[0.35,0.33,0.32]$, $\alpha=\frac{101}{312}\approx0.324$, $N=100$, and $r=0.001$. Once the population converges, in this case to a polarization state a bit before ten thousand iterations, a fourth issue is added (i.e. the weights are changed to $\frac{1}{314}[108, 104, 100, 2]\approx[0.34, 0.33, 0.32, 0.01]$) and individuals with opinion string $s$ are split uniformly at random into the opinion strings $s0$ and $s1$. For clearer comparison of the dynamics, opinion strings with the same opinions on the first three issues are combined for plotting. We observe the predicted symmetry, first where polar opposites have equal proportions (corresponding to the third row and fifth column of Fig. S3), and then to a state of persistent pluralism.
• Figure 4: Classification of symmetries on the deterministic model with $L=4$. (a) Visual depiction of symmetries. The diagrams represent the possible symmetries of the hypercube $\{0,1\}^4$ seen among frequencies of opinion strings at equilibria in the deterministic model. Solid lines connect opinion strings with equal proportions at equilibrium, along with matching colors. (b) The 14 cases of weights. A complete characterization of when each symmetry occurs for each of these cases is given in Table \ref{['tab:symmetries']}.
• Figure 5: The equilibrium structure determined by symmetries explains the effects we observe. For $L=2$, the population composition is given by the proportions $(x_{00}, x_{01}, x_{10}, x_{11})$. The constraint $x_{00}+x_{01}+x_{10}+x_{11} = 1$ means one coordinate can be omitted, resulting in a tetrahedron (a 3D simplex) of points. Each vertex represents a population in a consensus state with a given opinion string, shown with blue dots. Lines between these represent populations with a combination of the two opinion strings at their endpoints. In particular, populations with only the opinion strings 00 and 11, or 01 and 10, are polarized and shown in red. A black trajectory indicates one possible path through this space the population composition could take under the opinion dynamics. For a large enough population size, this path will approach the manifold of equilibria shown in green. In the case on the left, this manifold connects to a polarized state, which will be reached relatively quickly. In contrast, the right panel features a case with only one attracting equilibrium. The dynamics will keep pulling the population back to this point and away from polarization or consensus, resulting in longer absorption times. Further, this case is significantly more likely to reach polarization than consensus, as the latter requires more opinion strings to vanish from the population than the former. This same principle applies to the larger $L$ cases.