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Modelling heterogeneity and an open-mindedness social norm in opinion dynamics

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  • We study heterogeneous interactions in a time-continuous bounded confidence model for opinion formation. The key new modelling aspects are to distinguish between open-minded and closed-minded behaviour and to include an open-mindedness social norm. The investigations focus on the equilibria supported by the proposed new model; particular attention is given to a novel class of equilibria consisting of multiple connected opinion clusters, which does not occur in the absence of heterogeneity. Various rigorous stability results concerning these equilibria are established. We also incorporate the effect of media in the model and study its implications for opinion formation.

    Mathematics Subject Classification: Primary: 91D30, 37N99; Secondary: 70F99.


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  • Figure 1.  Schematic diagram of opinion interactions along the opinion continuum $[-1,1]$ in the presence of a social norm of open-mindedness (note that opinion space is shown on the vertical axis in all other figures). Green circles represent warm and open-minded individuals (in $O$), while diamonds correspond to individuals in $C$, those having colder and less open communication styles. (i) The interactions on the left show the influences acting on the open-minded agent at $x_1$, which experiences attractive social forces (green arrows) from both open-and closed-minded individuals with opinions within the bound of confidence $\epsilon$; the influence function $\phi(|x - x_1|) = 1_{[x_1 - \epsilon, x_1 + \epsilon]}$ is shown in green. (ii) The social forces acting on the closed-minded individual with non-moderate opinion $x_2$ (large red diamond; note $|x_2| > X_c$), in the presence of a social norm of open-mindedness, are shown on the right. Interactions with other members of $C$, both moderate (blue diamond at $x_3$) and extremist (red diamond), reinforce the opinion $x_2$ and drive it closer to the nearest extreme at 1 (blue and red arrows). However, given an open-mindedness social norm (model (6)), the interaction with an agent in $O$ within its bound of confidence (green circle near $x_2$) induces an open-minded response and attractive interaction (green arrow) according to the influence function shown in red. In the absence of a social norm of open-mindedness (model (5)), this interaction would instead also drive the individual at $x_2$ to the extreme at 1.

    Figure 2.  Evolution to equilibria of solutions to model (4) with (6) with $N=80$ for some choices of the parameters $m$ (number of open-minded agents), $X_c$ (critical threshold for extreme-seeking dynamics) and $\epsilon$ (bound of confidence), and with both type NS (non-symmetric) and type S (symmetric) initial conditions. The plots illustrate various qualitatively different outcomes as discussed in the text: (a) $m=80$ ($X_c$ irrelevant), $\epsilon=0.2$, type S; (b) $m=54$, $X_c=1/3$, $\epsilon=1.5010$, type NS; (c) $m=54$, $X_c=1/3$, $\epsilon=0.7909$, type NS; (d) $m=0$, $X_c=0$ $\epsilon=2$, type S; (e) $m=8$, $X_c=2/3$, $\epsilon=0.5222$, type NS; (f) $m=8$, $X_c=2/3$, $\epsilon=1.0020$, type S.

    Figure 3.  Numerical illustration of local stability: At time $t=50$ a perturbation of size $\eta $ is applied to the 5-cluster solution in Figure 2(f); the perturbed system is then evolved to $t=100$. (a) For $\eta =0.02$, the solution returns to the original 5-cluster configuration; (b) with $\eta =0.03$, the system approaches a 4-cluster state.

    Figure 8.  Bifurcation with respect to the location of the media source. The filled circles represent the locations of equilibrium clusters, while the dashed line indicates the media bias; parameter values are $X_c = 0$ and $\epsilon= 1.2$, with (a) $m=0$, (b) $m=26$, and (c) $m=54$. In the absence of open-minded individuals (a), the media has no effect, as all closed-minded individuals approach extremist views. Plots (b) and (c) correspond to non-symmetric (type NS) initial data, with inserts that show the results obtained from symmetric (type S) initializations. Asymmetry seems to enhance the effect of the media, in particular when the number of open-minded individuals is relatively low ($m=26$, plot (b)).

    Figure 4.  Dependence of the number and stability of cluster equilibria on $X_c$ (threshold for extreme-seeking dynamics) and $\epsilon$ (bound of confidence): (a) $m = 80$ (all open-minded), type S initial data; (b) $m = 0$ (none open-minded), $\epsilon = 0.2$, type S data; (c) $m = 8$, $X_c = 2/3$, type NS data (type S data in insert). Cluster locations for linearly stable equilibria are represented by blue filled circles, while neutrally stable equilibria are denoted by red stars.

    Figure 5.  Extremist consensus arising as bifurcations in parameter space. Within each plot, the same non-symmetric (type NS) initial conditions are used to generate equilibria for all parameter values. The inserts show equilibria obtained from symmetric initial data (type S), where no extremist consensus can arise. (a) Dependence on $X_c$ with $m=26$, $\epsilon=1.2$; (b) dependence on $m$ with $X_c=1/3$, $\epsilon=2$; (c) dependence on $\epsilon$ with $m=54$, $X_c=1/3$. Symbols are as in Figure 4.

    Figure 6.  Effect of a social norm of open-mindedness on opinion convergence. The initial data is of type NS (non-symmetric); parameters are set at $N = 80$, $\epsilon=2$ and $X_c=0$, and symbols are as in Figure 4. Increasing the proportion $m/N$ of open-minded individuals has a negligible effect on the opinion distribution when there is no social norm of open-mindedness (plot (a)), but increases agreement significantly when a social norm of open-mindedness is present (plot (b)).

    Figure 7.  Evolution to equilibria of the opinion model with media (19)-(20) with $(6)$ with Type NS (non-symmetric) initial data, and parameter values $m=26$ (number of open-minded agents), $X_c=0$ (threshold for extreme-seeking dynamics) and $\epsilon=1.2$ (bound of confidence). The circles indicate the media location $\mu$. (a) $\mu=1$: all individuals approach the extremist media bias. (b) $\mu=0.1$: open-minded individuals converge to the media bias, while the opinions of non-open-minded individuals form two clusters balanced between attraction to open-minded agents and to the extremes.

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