# Multi-Agent Simulation of Product Diffusion in Online Social Networks from the Perspective of Overconfidence and Network Effects

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Product Diffusion in OSNs

#### 2.2. Overconfidence Theory

#### 2.3. Network Effects

#### 2.4. Multi-Agent Simulation and Game Theory

## 3. Methodology

#### 3.1. Assumptions

- Products are abstractions of common features and attributes of the product that are diffused in OSNs. The unique characteristics of some internet products are not considered.
- We only consider the diffusion of one product on the OSNs.
- Consumers are assumed to have bounded rationality. They only know the strategies of their neighbors and their ultimate goal in decision-making is to maximize their interests.
- Individual overconfidence leads to bias in expected utility estimation.
- Some consumers have free-riding behavior and they are punished for it.

#### 3.2. Conceptual Model of Product Diffusion in OSNs

#### 3.2.1. Conceptual Model of Product Diffusion in OSNs Based on Game Theory

#### 3.2.2. Conceptual Model of Product Diffusion in OSNs Considering Network Effects

#### 3.2.3. Conceptual Model of Product Diffusion in OSNs Considering Overconfidence

- Overestimation. We defined that $k$ is the degree of overestimation and $k=({B}_{1}-b)/b$, where ${B}_{1}$ and $b$ represent the perceived benefit and actual benefits of individuals adopting the product. The value utility of adoption and rejection under the global and local network effects at the timestamp $t$ can be expressed as follows, respectively:$${U}_{a-global}\left(t\right)={B}_{1}-c+h\times p=\left(1+k\right)b-c+h\times p,$$$${U}_{a-local}\left(t\right)={B}_{1}-c+h\times \frac{{G}_{i}\left(t-1\right)}{{d}_{i}}=\left(1+k\right)b-c+h\times \frac{{G}_{i}\left(t-1\right)}{{d}_{i}},\text{}$$$${U}_{r-global}\left(t\right)=p({B}_{1}-d)=p\left[\left(1+k\right)b-d\right],$$$${U}_{r-local}\left(t\right)=p({B}_{1}-d)=p\left[\left(1+k\right)b-d\right],$$
- Overprecision. We defined $\beta $ as the degree of overprecision. Additionally, we assume the linear function of each individual’s expected benefit can be expressed by ${B}_{2}=b+\overline{X}=b+\frac{\mathsf{\sigma}}{\beta}\times X$ where ${B}_{2}~N(b,{\frac{\sigma}{\beta}}^{2})$. ${B}_{2}$ represents the perceived benefit, $b$ represents the actual benefits, $\overline{X}$ is a random disturbance, $\sigma $ is the actual variance, $X$ is the stabilized random disturbance. The value utility of adoption and rejection under the global and local network effects at the timestamp $t$ can be expressed as follows, respectively:$${U}_{a-global}\left(t\right)={B}_{2}-c+h\times p=b+\frac{\sigma}{\beta}\times X-c+h\times p,$$$${U}_{a-local}\left(t\right)={B}_{2}-c+h\times \frac{{G}_{i}\left(t-1\right)}{{d}_{i}}=b+\frac{\sigma}{\beta}\times X-c+h\times \frac{{G}_{i}\left(t-1\right)}{{d}_{i}},$$$${U}_{r-global}\left(t\right)=p({B}_{2}-d)=p(b+\frac{\sigma}{\beta}\times X-d),$$$${U}_{r-local}\left(t\right)=p({B}_{2}-d)=p(b+\frac{\sigma}{\beta}\times X-d).\text{}$$

#### 3.2.4. The Game Learning Algorithm for Users in the Product Diffusion

#### 3.3. Simulation Model with Multi-Agents

#### 3.3.1. Basic Multi-Agent Simulation Model

- $\Theta $ is the set of agents. $\Theta =\left\{Agen{t}_{1},Agen{t}_{2},\dots ,Agen{t}_{n}\right\}$, where $\mathit{n}$ is the number of consumers in the network. Each agent is a consumer in the online social network.
- $Z$ is the decision-making state of the consumer at time $t$. $Z=\left\{{Z}_{1},{Z}_{2}\right\}$, where ${Z}_{1}$ means purchase and ${Z}_{2}$ means refusal.
- $N$ is the neighbor set of the Agent. $N=\left\{{N}_{1},{N}_{2},\dots ,{N}_{n}\right\}$, where ${N}_{i}=\left\{Agen{t}_{i}\to Agen{t}_{j}\right\}$. That is, $N$ consists of agents connected to the Agent.
- $P$ is the set of overconfidence parameters, and $P=\left\{k,\beta \right\}$. $k$ is the degree of overestimation, and $\beta $ is the degree of overprecision.
- $F$ is the consumer’s decision transfer function, that is, the current state of an individual is related to the state of itself and its neighbors, overconfidence, and network effects at the last moment.
- $t$ is the system clock, and $t=\left\{{t}_{1},{t}_{2},\dots ,{t}_{n}\right\}$ is the basis of the simulation system.

#### 3.3.2. State Transition Rules of Multi-Agent Simulation Model

#### 3.3.3. Model Validation of Multi-Agent Simulation Model

## 4. Simulation Results

#### 4.1. Experimental System and Default Parameters

- The number of agents is at as 400.
- For the random network, the node layout is set as a ring type and the number of agents in the neighborhood of each node (i.e., average degree distribution) is set at 6.
- For the small-world network, the node layout is set as an arranged type. The average degree distribution is set at 6, and the reconnection probability of each edge (i.e., rewiring probability) is 0.6.
- For the scale-free network, the node layout is set at random and the number of large nodes is 5.

#### 4.2. Parameters Related to Overconfidence

#### 4.3. Parameters Related to Network Effects

#### 4.4. Parameters Related to Network Structure

## 5. Conclusions

- Enterprises should fully display the product information so that consumers have a proper overestimate of the benefits of buying the product. However, at the same time, enterprises should not exaggerate the utility of products too much, to prevent consumers from changing their purchase decisions when their expectations are inconsistent with reality.
- Companies should not invest too much capital in enhancing network effects. Moderate promotion can yield high returns, but over-enhancing social network effects often backfires, for example by making consumers loathe them. When the social network effects intensity is low, enterprises should try to reduce the degree of overconfidence of consumers. When it is high, they should try to improve the overprecision of consumers but control their overestimation.
- Enterprises should try to introduce influential KOLs (Key Opinion Leaders) into social networks to transform the small-world network into a scale-free network to achieve a higher level of product diffusion. This avoids fluctuations in consumer overconfidence and network effects on product proliferation.

- The psychological effects (overconfidence) and network effects perspectives opened up an entirely new way of looking at product diffusion in OSNs. This study adds to the theoretical foundation for the new product diffusion model by including additional behavioral theories and complex network theories.
- This research uncovered several different modeling strategies for complicated group behavior resulting from individual interaction. The micro-basis of the multi-agent simulation model was overconfidence theory, which was utilized to discover probable individual-level processes. This multi-method provides a fundamental framework for testing dynamic group behavior using a mix of a multi-agent method, network effects theory, and a psychological theory.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

**1. The code of Main model:**

**//Set the consumer’s initial strategies:**

**2. The code of Person class:**

**//Calculate the global network effect**

**//Calculate the local network effect**

**//Calculate the utility of agent**

**3. The code of interaction rules between agents:**

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**Figure 2.**Internal validation indicates simulation model conformance with the conceptual model. (

**a**) the adoption rate increases with c decreasing; (

**b**) consumers will choose to buy the product and the adoption rate will approach 1 over time if d is very large.

**Figure 3.**The influence of overestimation on product diffusion in OSNs. (

**a**) k = 0; (

**b**) k = 0.1; (

**c**) k = 0.2; (

**d**) k = 0.3; (

**e**) k = 0.4; (

**f**) k = 0.5; (

**g**) k = 0.6; (

**h**) k = 0.7; (

**i**) k = 0.8.

**Figure 4.**The influence of overprecision on product diffusion in OSNs: (

**a**) β = 0.1; (

**b**) β = 0.5; (

**c**) β = 1; (

**d**) β = 4; (

**e**) β = 7; (

**f**) β = 10.

**Figure 5.**The influence of the network effects intensity on product diffusion in OSNs: (

**a**) h = 20 under local network effect; (

**b**) h = 40 under local network effect; (

**c**) h = 60 under local network effect; (

**d**) h = 20 under global network effect; (

**e**) h = 40 under global network effect; (

**f**) h = 60 under global network effect.

**Figure 6.**The influence of network effects on product diffusion in OSNs under different overconfidence scenarios: (

**a**) global network effect; (

**b**) local network effect.

**Figure 7.**The influence of network structure on product diffusion in OSNs: (

**a**) overestimation scenario under global network effect; (

**b**) overestimation scenario under local network effect; (

**c**) overprecision scenario under global network effect; (

**d**) overprecision scenario under local network effect.

Player 1 | Player 2 | |
---|---|---|

Adoption | Rejection | |

Adoption | $b-c,b-c$ | $b-c,b-d$ |

Rejection | $b-d,b-c$ | $0,\text{}0$ |

No. | Parameter | Description | Default |
---|---|---|---|

1 | Network-type | Type of network | Small-world |

2 | $\mathit{b}$ | Value of $\mathit{b}$ in Table 1 | 55 |

3 | $\mathit{c}$ | Value of $\mathit{c}$ in Table 1 | 35 |

4 | $\mathit{d}$ | Value of $\mathit{d}$ in Table 1 | 15 |

5 | $\mathit{p}$ | Percentage of users holding an adoption strategy | 0.5 |

6 | $\mathit{n}$ | Information noise | 0.1 |

7 | $k$ | The overestimation parameter | 0 |

8 | $\beta $ | The overprecision parameter | 1 |

9 | $\mathit{h}$ | The network effects intensity | 100 |

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## Share and Cite

**MDPI and ACS Style**

Wei, X.; Zhang, Y.; Liao, Q.; Nie, G.
Multi-Agent Simulation of Product Diffusion in Online Social Networks from the Perspective of Overconfidence and Network Effects. *Sustainability* **2022**, *14*, 6589.
https://doi.org/10.3390/su14116589

**AMA Style**

Wei X, Zhang Y, Liao Q, Nie G.
Multi-Agent Simulation of Product Diffusion in Online Social Networks from the Perspective of Overconfidence and Network Effects. *Sustainability*. 2022; 14(11):6589.
https://doi.org/10.3390/su14116589

**Chicago/Turabian Style**

Wei, Xiaochao, Yanfei Zhang, Qi Liao, and Guihua Nie.
2022. "Multi-Agent Simulation of Product Diffusion in Online Social Networks from the Perspective of Overconfidence and Network Effects" *Sustainability* 14, no. 11: 6589.
https://doi.org/10.3390/su14116589