# Complex Transitions of the Bounded Confidence Model from an Odd Number of Clusters to the Next

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. The Models and Their Approximation with Distributions

#### 2.1. The Two Variants of the BC Model

Algorithm 1: runHK($[{a}_{1},\cdots ,{a}_{n}]$ (initial opinions), $\u03f5$ (confidence bound)) |

Algorithm 2: runDW($[{a}_{1},\cdots ,{a}_{n}]$ (initial opinions), $\u03f5$ (confidence bound)) |

#### 2.2. Distribution Version

Algorithm 3: distributionRunDW($\u03f5$ (confidence bound), ${n}_{\u03f5}$ ($\u03f5$ discretisation size)) |

Algorithm 4: distributionHK($\u03f5$ (confidence bound), ${n}_{\u03f5}$ ($\u03f5$ discretisation size)) |

#### 2.3. Dealing with Computational Instabilities: Forcing Symmetry and Adding Noise

## 3. Simulation Results

#### 3.1. DW Model

#### 3.2. HK Model with Odd or Even Discretisation Size

#### 3.3. HK Model with Noise on the Initial Conditions

- From $1/\u03f5\approx 2.53$ to $1/\u03f5\approx 2.62$, for an odd ${n}_{g}$, one central cluster is reached after a few hundred iterations, and for an even ${n}_{g}$, two clusters, at distance roughly $\pm \u03f5$ from the centre (slowly increasing with $1/\u03f5$), are reached after about 10 iterations.
- From $1/\u03f5\approx 2.62$ to $1/\u03f5\approx 2.85$, two clusters at a distance of roughly $\pm \u03f5$ from the centre (slowly increasing with $1/\u03f5$) with one minor cluster at the centre are reached after a dozen iterations,
- From $1/\u03f5\approx 2.85$ to $1/\u03f5\approx 3.35$, there are two clusters in asymmetric positions and of different masses. The cluster of smaller mass is located at $\pm \u03f5$ and the one of bigger mass is closer to the centre on the other side, and moves slowly closer to the centre when $1/\u03f5$ increases. The convergence is reached after a large number of iterations (several thousands), except at the beginning and in the second half the phase where some cases of fast convergence also appear.

## 4. Discussion

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Positions of the clusters ($a/\u03f5$) in the full (

**upper**) and zoomed (

**lower**) $a/\u03f5$ ranges for the opinion, a, range $[-1,1]$. The discretisation of $\u03f5$ is ${n}_{\u03f5}=101$. The colors and symbols represent the values of the total mass of the cluster divided by $2\u03f5$ as indicated. See text for details.

**Figure 2.**Positions of the clusters ($a/\u03f5$) for the opinion range $[-1,1]$) for the odd (

**upper**) and bottm (

**lower**) opinion range discretisation, ${n}_{g}$. $\u03f5$ discretisation size, ${n}_{\u03f5}=1001$. The colors and symbols represent the values of the total mass of the cluster divided by $2\u03f5$ as indicated.

**Figure 3.**A closer look at the first three transitions for odd (

**left**) and even (

**right**) opinion discretisation sizes shown in Figure 2.

**Figure 4.**Examples of runs of HK distribution model for $1/\u03f5\approx 3$ and ${n}_{\u03f5}=15$ when the total discretisation, ${n}_{g}$, is odd (

**upper**) or even (

**lower**). The two black lines are at $\pm \u03f5$. The colors and symbols represent the values of the total mass of the cluster divided by $2{n}_{\u03f5}$, as indicated.

**Figure 5.**Positions of the clusters ($a/\u03f5$) (opinion range $[-1,1]$) for odd (

**upper**) and even (

**lower**) opinion range discretisation, ${n}_{g}$. $\u03f5$ discretisation size ${n}_{\u03f5}=101$. The colors and symbols represent the values of the total mass of the cluster divided by $2{n}_{\u03f5}$, as indicated.

**Figure 6.**Positions of the clusters ($a/\u03f5$) (

**upper**) and convergence time (

**lower**) (opinion range $[-1,1]$). Discretisation ${n}_{g}$ is odd or even. $\u03f5$ discretisation size ${n}_{\u03f5}=500$. Upper: the colors and symbols represent the values of the total mass of the cluster divided by $2\u03f5$, as indicated.

**Figure 7.**Examples of runs with initial noise, for ${n}_{\u03f5}=500$ and ${n}_{g}=1000/\u03f5$ for the simulations in the first phase (

**top**and

**middle**) and in the third phase (

**bottom**) of the transition. The colors and symbols are as in Figure 1.

**Figure 8.**Positions of the clusters ($a/\u03f5$) (

**upper**) and convergence time (

**lower**) (opinion range $[-1,1]$). The vertical dotted lines delimit the phases of the transition. Upper: the colors and symbols represent the values of the total mass of the cluster divided by $2\u03f5$, as indicated.

**Figure 9.**Example of simulation in the fourth phase of the second transition. The colors and symbols are as in Figure 1.

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**MDPI and ACS Style**

Deffuant, G.
Complex Transitions of the Bounded Confidence Model from an Odd Number of Clusters to the Next. *Physics* **2024**, *6*, 742-759.
https://doi.org/10.3390/physics6020046

**AMA Style**

Deffuant G.
Complex Transitions of the Bounded Confidence Model from an Odd Number of Clusters to the Next. *Physics*. 2024; 6(2):742-759.
https://doi.org/10.3390/physics6020046

**Chicago/Turabian Style**

Deffuant, Guillaume.
2024. "Complex Transitions of the Bounded Confidence Model from an Odd Number of Clusters to the Next" *Physics* 6, no. 2: 742-759.
https://doi.org/10.3390/physics6020046