# Applications of the Investor Sentiment Polarization Model in Sudden Financial Events

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. J–A Model

#### 2.2. B–A Scale-Free Network Model

#### 2.3. Noise Trading Theory

#### 2.4. Summary and Innovation

## 3. Research Methods

#### 3.1. B–A Scale-Free Network

#### 3.2. J–A Model

#### 3.3. Multi-Agent System

#### 3.4. DSSW Model

## 4. Model Building

#### 4.1. Modeling Ideas

#### 4.2. Construct a Polarization Model of Investor Sentiment

#### 4.2.1. Definition of Parameters and Variables

- (1)
- Radiation range ${R}_{i}$: the number of edges formed by node $i$ and adjacent nodes.The size most intuitively reflects the number of neighboring nodes that the node $i$ connects, and the larger the ${R}_{i}$, the greater the importance of the node $i$ in the network. In the financial market, nodes with a large range of influence, such as financial media and professional institutions, tend to have larger ${R}_{i}$.
- (2)
- Degree of interaction participation ${T}_{i}$: the ratio of the actual number of edges ${R}_{i}$ of node $i$ to the most polygonal number $Max$ that may exist.The formula expression for the most polygonal number that the node $i$ may have is:$${T}_{i}=1/{C}_{Max}^{2}$$The size of ${T}_{i}$ intuitively reflects the degree of aggregation of nodes, and in general, individuals are more inclined to create groups with high concentrations. In financial markets, ${T}_{i}$ measures the extent to which an investor participates in emotional interactions, and the higher the ${T}_{i}$, the greater the impact of the interaction on investors. The formula expression for Degree of interaction participation ${T}_{i}$ is:$${C}_{i}={R}_{i}/{C}_{Max}^{2}$$
- (3)
- Junction coefficient ${C}_{ij}$: the affinity relationship between node $i$ and node $j$.The size most intuitively reflects the degree to which two nodes interact with each other. The range is between 0 and 1, and as the value becomes larger, the relationship between nodes goes from sparse to intimate. In real-world financial markets, investors tend to be influenced by investors they trust and relatively less influenced by other investors.
- (4)
- Investor sentiment ${X}_{i}\left(t\right)$: an indicator of the attitude of node $i$ at the t-moment.${X}_{i}^{+}\left(t\right)$ represents the sentiment value of node i that holds a positive view; ${X}_{i}^{-}\left(t\right)$ represents the sentiment value of node i that holds a negative view.Combined Ambient sentiment value $\overline{X}\left(t\right)$: the average sentiment value of the node connected to the node $i$ at t-time. Its formula expression is:$$\begin{array}{c}\overline{X}\left(t\right)=\frac{{{\displaystyle \sum}}_{j=1}^{n}{C}_{ij}{X}_{j}\left(t\right)}{{{\displaystyle \sum}}_{j=1}^{n}{k}_{ij}}\end{array}$$
- (5)
- Investment expectations quantification point ${Z}_{i}$: the percentage quantile of the node $i$’s investment expectation in the investment expectations of the surrounding nodes.The value measures the order of an investor’s investment expectations among the investors around it, and being at an excessively high or too low index means that the investor’s investment expectations deviate greatly from the surrounding nodes, and there is often a tendency to center. The formula for the investment expectations quantification point ${Z}_{i}$ is:$${Z}_{i}=P({X}_{j}>{X}_{i})\times 100\%$$
- (6)
- Coefficient of destabilization $\epsilon $: how easy it is for nodes to change their emotional values.$\epsilon $ is a constant in the short term, measuring how easy it is for investors to change their original emotions in the face of the differences in the emotions around them and instead follow the emotions of the group. The larger the $\epsilon $, the more conformist the investor, and it is easier to change their expectations when faced with the difference between the surrounding emotions and their own emotions.
- (7)
- Comprehensive impact ${S}_{i}$: an indicator to determine whether ${X}_{i}\left(t\right)$ will undergo further changes.${S}_{i}$ is a comprehensive reflection of the environmental information in which the node $i$ is located, and its numerical value determines whether the sentiment value of the node $i$ has undergone further changes. This paper argues that an investor’s decision to change expectations depends on the individual’s interaction participation and the expected deviation from the surrounding nodes. ${X}_{i}\left(t\right)$ will undergo further changes when ${S}_{i}\ge 1$; ${X}_{i}\left(t\right)$ does not change when ${S}_{i}<1$. The formula expression for ${S}_{i}$ is:$${S}_{i}=\{\begin{array}{c}{T}_{i}+\left|Zi-0.5\right|+\epsilon \frac{{{\displaystyle \sum}}_{j=1}^{n}\left|{C}_{ij}{X}_{j}^{+}\left(t\right)-Xi\left(t\right)\right|}{{{\displaystyle \sum}}_{j=1}^{n}\left|{C}_{ij}{X}_{j}\left(t\right)-Xi\left(t\right)\right|}{{\displaystyle \sum}}_{j=1}^{n}{C}_{ij}{X}_{j}\left(t\right)>0\\ {T}_{i}+\left|Zi-0.5\right|+\epsilon \frac{{{\displaystyle \sum}}_{j=1}^{n}\left|{C}_{ij}{X}_{j}^{-}\left(t\right)-Xi\left(t\right)\right|}{{{\displaystyle \sum}}_{j=1}^{n}\left|{C}_{ij}{X}_{j}\left(t\right)-Xi\left(t\right)\right|}{{\displaystyle \sum}}_{j=1}^{n}{C}_{ij}{X}_{j}\left(t\right)<0\end{array}$$
- (8)
- Weighted variance of environmental sentiment $\sigma $: the degree of difference in the sentiment values of surrounding investors.The smaller the value, the more convergent the opinions of surrounding investors, and vice versa, the more divergent the opinions of the surrounding investors. Its formula expression is:$$\sigma =\frac{{{\displaystyle \sum}}_{j=1}^{n}{C}_{ij}{\left(Xj-\overline{X}\left(t\right)\right)}^{2}}{{{\displaystyle \sum}}_{j=1}^{n}{C}_{ij}}$$
- (9)
- Effect interval threshold ${D}_{1},{D}_{2}$: the threshold for the change of node sentiment value.When the $\mathsf{\sigma}$ is less than ${D}_{1}$, it means that the surrounding investors are unanimous and follow the rules of assimilation; when the $\sigma $ is greater than ${D}_{2}$, it means that the opinions of the surrounding investors are more diverse, and the exclusion rule is followed. Simultaneous requirements ${D}_{1}$ and ${D}_{2}$ meet: ${D}_{1}\le {D}_{2}$.
- (10)
- Assimilation and rejection parameters $\alpha ,\beta $: the degree of change of control.$\alpha $ represents the degree of change parameter when following the assimilation rule, and $\beta $ represents the degree of change parameter when the exclusion rule is followed. Both $\alpha $ and $\beta $ range in value (0,1].

#### 4.2.2. Construction of the Network

#### 4.2.3. Setting of Interaction Rules

## 5. Simulation Experiments

- ${H}_{0}$: the mean of the sentiment value $\mu \ne 0$;
- ${H}_{1}$, the alternative hypothesis: the mean of the sentiment value $\mu =0$.

#### 5.1. Effect of the Shaking Coefficient ε on the Polarization Result

#### 5.2. Effect of Assimilation and Repulsion Interval Thresholds ${D}_{1}$, ${D}_{2}$ on Polarization Results

#### 5.3. Effects of Assimilation and Exclusion Parameters $\alpha ,\beta $ on Polarization Results

#### 5.4. Analysis and Discussion

## 6. A Real Case Study

## 7. Conclusions

#### 7.1. Findings of the Study

#### 7.2. Policy Recommendations

#### 7.2.1. Guide the Emotional Expression of Opinion Leaders

#### 7.2.2. Strengthen Information Disclosure Supervision

#### 7.2.3. Strengthen the Quality of Education of Investors

#### 7.2.4. Deepen the Reform of the Registration System

#### 7.3. Deficiencies and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Distribution of emotion values under different times of interaction. (

**a**) time = 0. (

**b**) time = 100. (

**c**) time = 200. (

**d**) time = 400.

**Figure 3.**Changes in the proportion of extreme emotions with the number of interactions. (

**a**) amount = 1000. (

**b**) amount = 2000. (

**c**) amount = 5000. (

**d**) amount = 10000.

**Figure 5.**Distribution of different $\epsilon $. (

**a**) $\epsilon $ = 0.2. (

**b**) $\epsilon $ = 0.4. (

**c**) $\epsilon $ = 0.6. (

**d**) $\epsilon $ = 0.8.

**Figure 6.**Distribution map under different interval thresholds. (

**a**) ${D}_{1}$ = 0.1, ${D}_{2}$ = 0.7. (

**b**)${D}_{1}$ = 0.5, ${D}_{2}$ = 0.7. (

**c**) ${D}_{1}$ = 0.3, ${D}_{2}$ = 0.5. (

**d**) ${D}_{1}$ = 0.3, ${D}_{2}$ = 0.9.

**Figure 7.**Distribution diagram under different effect parameters. (

**a**) $\alpha $ = 0.001, $\beta $ = 0.001. (

**b**) $\alpha $ = 0.002, $\beta $ = 0.002. (

**c**) $\alpha $ = 0.004, $\beta $ = 0.004. (

**d**) $\alpha $ = 0.01, $\beta $ = 0.01.

Parameter | Definition |
---|---|

${R}_{i}$ | Radiation range |

${T}_{i}$ | Degree of interaction participation |

${C}_{ij}$ | Junction coefficient |

$\epsilon $ | Coefficient of destabilization |

${D}_{1}$ | Assimilation interval threshold |

${D}_{2}$ | Exclusion interval threshold |

$\alpha $ | Coefficient of a degree of assimilation |

$\beta $ | Coefficient of a degree of rejection |

Parameter | Definition |
---|---|

${X}_{i}\left(t\right)$ | Investor sentiment |

${Z}_{i}$ | Investment expectations quantification point |

${S}_{i}$ | Comprehensive impact |

$\sigma $ | Weighted variance of environmental sentiment |

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**MDPI and ACS Style**

Yu, Y.; Wei, H.; Chen, T.
Applications of the Investor Sentiment Polarization Model in Sudden Financial Events. *Systems* **2022**, *10*, 75.
https://doi.org/10.3390/systems10030075

**AMA Style**

Yu Y, Wei H, Chen T.
Applications of the Investor Sentiment Polarization Model in Sudden Financial Events. *Systems*. 2022; 10(3):75.
https://doi.org/10.3390/systems10030075

**Chicago/Turabian Style**

Yu, Yuanyuan, Hongjia Wei, and Tinggui Chen.
2022. "Applications of the Investor Sentiment Polarization Model in Sudden Financial Events" *Systems* 10, no. 3: 75.
https://doi.org/10.3390/systems10030075