# Opinion Dynamics with Higher-Order Bounded Confidence

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Numerical Simulation

#### 3.1. Experiment Design

- ${P}_{C}$, the frequency of consensus in multiple runs ($0\le {P}_{C}\le 1$). In a run, if there is only one opinion cluster left in the system (e.g., Figure 2c), we say the system achieves consensus;
- ${r}_{1}$, the lower bounded confidence above which the system may consistently achieve consensus (i.e., ${P}_{C}<1$, $\forall r<{r}_{1}$, and ${P}_{C}=1$, $\exists r\ge {r}_{1}$). Similarly, ${r}_{0}$, the upper bounded confidence below which the system cannot achieve consensus (i.e., ${P}_{C}=0$, $\forall r<{r}_{0}$, and ${P}_{C}>0$, $\exists r\ge {r}_{0}$);
- ${N}_{C}$, the number of opinion clusters. For example, in Figure 2a–c, we have ${N}_{C}=7$, ${N}_{C}=2$ and ${N}_{C}=1$, respectively. ${N}_{C}=1$ means the system achieves consensus;
- ${C}_{\mathrm{max}}$, the relative size of the largest opinion cluster. We find the opinion cluster with the highest number of agents and divide it by N. Obviously, this quantity yields $1/N\le {C}_{\mathrm{max}}\le 1$;
- $\rho \left[{x}_{i}\left({T}^{*}\right)\right]$, the distribution of stability opinions. We divide the range between 0 and 1 into 100 equal parts, and denote $\Delta x=1/100=0.01$. If $n\Delta x\le {x}_{i}\left({T}^{*}\right)<(n+1)\Delta x$, we add 1 to the distribution function at the nth part ($n=1,2,\cdots ,100$). After going through $i=1,2,\cdots ,N$, we divide the result in each part by N, and acquire the normalized opinion distribution;
- ${T}^{*}$, the convergence time. If $|{x}_{i}\left(t\right)-{x}_{i}(t-1)|<0.0001$, $i=1,2,\cdots ,N$, then, we denote ${T}^{*}=t$.

#### 3.2. Results

#### 3.3. Discussion

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the analogy, from games with higher-order interactions (

**left**), to opinion dynamics with higher-order bounded confidence (

**right**). (

**left**) five agents on a regular square lattice. The purple agent organizes a multiplayer game among the five agents (its nearest neighbors and itself), while also participates in the games organized by the other four agents. (

**right**) five agents on a continuous one-dimensional opinion space. The purple agent organizes a group opinion discussion among the blue, purple, and red agents within its bounded confidence, while also participates in the discussions organized by the blue and red agents.

**Figure 2.**Each agent’s opinion ${x}_{i}\left(t\right)$, $i=1,2,\cdots ,N$, as a function of time t at $\alpha =1$ and different r. (

**a**) $r=0.05$. (

**b**) $r=0.15$. (

**c**) $r=0.25$. There are N curves in each panel, where one curve represents the opinion evolution of one agent.

**Figure 3.**(

**a**) The frequency of consensus ${P}_{C}$ as a function of the bounded confidence r at different $\alpha $; (

**b**) the frequency of consensus ${P}_{C}$ as a function of the fraction of decentralized agents $\alpha $ at different r.

**Figure 4.**The lower bounded confidence ${r}_{1}$, above which the system may consistently achieve consensus (i.e., ${P}_{C}<1$, $\forall r<{r}_{1}$, and ${P}_{C}=1$, $\exists r\ge {r}_{1}$), as a function of $\alpha $. The upper bounded confidence ${r}_{0}$, below which the system cannot achieve consensus (i.e., ${P}_{C}=0$, $\forall r<{r}_{0}$, and ${P}_{C}>0$, $\exists r\ge {r}_{0}$), as a function of $\alpha $. The “data” derive from simulation, while the “fitting” derives from fitting a linear function to “data” using the least squares method.

**Figure 5.**(

**a**) The number of opinion clusters ${N}_{C}$ as a function of the bounded confidence r at different $\alpha $; (

**b**) the number of opinion clusters ${N}_{C}$ as a function of the fraction of decentralized agents $\alpha $ at different r.

**Figure 6.**(

**a**) The relative size of the largest opinion cluster ${C}_{\mathrm{max}}$ as a function of the bounded confidence r at different $\alpha $; (

**b**) the relative size of the largest opinion cluster ${C}_{\mathrm{max}}$ as a function of the fraction of decentralized agents $\alpha $ at different r.

**Figure 7.**The distribution of stability opinions $\rho \left[{x}_{i}\left({T}^{*}\right)\right]$ at $r=0.2$ and different $\alpha $. (

**a**) $\alpha =0$; (

**b**) $\alpha =1$. The results are the average of ${10}^{5}$ independent runs.

**Figure 8.**(

**a**) The distribution of stability opinions $\rho \left[{x}_{i}\left({T}^{*}\right)\right]$ as a function of the bounded confidence r at $\alpha =1$; (

**b**) the distribution of stability opinions $\rho \left[{x}_{i}\left({T}^{*}\right)\right]$ as a function of the fraction of decentralized agents $\alpha $ at $r=0.2$.

**Figure 9.**The convergence time ${T}^{*}$ as a binary function of the bounded confidence r and the fraction of decentralized agents $\alpha $.

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**MDPI and ACS Style**

Wang, C.
Opinion Dynamics with Higher-Order Bounded Confidence. *Entropy* **2022**, *24*, 1300.
https://doi.org/10.3390/e24091300

**AMA Style**

Wang C.
Opinion Dynamics with Higher-Order Bounded Confidence. *Entropy*. 2022; 24(9):1300.
https://doi.org/10.3390/e24091300

**Chicago/Turabian Style**

Wang, Chaoqian.
2022. "Opinion Dynamics with Higher-Order Bounded Confidence" *Entropy* 24, no. 9: 1300.
https://doi.org/10.3390/e24091300