# Competition of Dynamic Self-Confidence and Inhomogeneous Individual Influence in Voter Models

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## Abstract

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## 1. Introduction

## 2. The Model

_{i}represents agent i’s confidence level, meaning agents have the tendency to maintain their original opinions [36]. The parameter μ

_{ij}denotes neighbor’s influence with values of +1 or –1 for a given probability. The parameter C

_{i}as well as μ

_{ij}hasthe same distribution for each agent. The parameters can be either annealed or quenched. The annealed parameters are assigned at each Monte Carlo step, but the quenched parameters are fixed during the evolution.

_{i}, which is a continuous variable taking value from [0, 1]. With the probability C

_{i}, agent i keeps its original opinion; otherwise, it will update its opinion following its neighbors. With the probability 1 − C

_{i}, agent i selects a neighbor j according to the influence strength of its neighbors and adopts the neighbor’s opinion. The probability that agent i will choose one of its neighbors j is directly proportional to neighbor j’s influence. After N such update events, the time step is increased by 1. There are many methods to measure agents’ influence. For the sake of simplicity and without loss of generality, considering the network structure, we use node degree and betweenness as their individual influence.

_{i}does not always remain constant; instead, its conviction changes when its opinion is confirmed by neighbors. If most of neighbors support the agent’s opinion, its confidence about its opinion is increased; however, if many neighbors hold an opposite opinion, the agent may doubt its own opinion, so that its confidence declines. This phenomenon agrees with social reinforcement, and multiple sources of support are required to convince people about a given behavior [37]. We assume that all agents have the same initial conviction C

_{0}. When agent i decides to follow its neighbor’s opinion, its confidence C

_{i}may change during the interaction. In each update event, if the neighbor j that is selected in terms of its influence has the same opinion as agent i, agent i’s conviction C

_{i}increases linearly by h, otherwise, its conviction decreases by h. The variation h satisfies 0 ≤ h < 1, and cannot be too large in order to avoid stopping the dynamics quickly. The variation of agents’ conviction may have many different forms. In Reference [31], individual conviction of keeping its current state changes linearly with the interacting time elapsed. Similarly, here we use the linear variation h of conviction for the sake of simplicity.

_{i}is fixed at C

_{0}. The global density of opinion +1 for agents with degree k at time t is defined as f (k,t). Considering the degree-based influence, for each agent with degree k, a neighbor with degree k

_{u}is selected with the probability ${k}_{u}P\left({k}_{u}|k\right)/{\displaystyle \sum _{v}{k}_{v}P\left({k}_{v}|k\right)}$, where $P\left({k}_{u}|k\right)$ is the degree-degree correlation function.

_{u}is identical, and then we get $f\left(k,t+1\right)=f\left(k,t\right)$. Therefore, without the evolution of agents’ conviction, the average magnetization of the system is conserved in any network, in accordance with the standard voter model.

## 3. Simulation Results

**Figure 1.**Distribution of betweenness in a scale-free network and random network. The average node degree is 10, and N = 1,000.

**Figure 2.**Convergence time as a function of system size N. Agents’ conviction is not considered in this figure. In the beginning, opinions are assigned uniformly at random. In the left plot, the underlying topology is scale-free networks, and in the right plot, random networks mediate the interaction. Every plot is an average of 200 different simulations.

_{0}does not have any distinct impact on the average magnetization.

**Figure 3.**Final density of opinion +1 versus initial density of opinion +1 f (0). The underlying topology is a scale-free network, N = 1,000 and h = 0.1. The results are averaged over 200 different simulations. The dotted curve refers to the standard voter model.

_{0}almost eliminates the fluctuation of clusters due to the instant occurrence of some extremists. Although the initial individual conviction does not affect the final average opinion, it changes the formation of opinion clusters greatly. Nontrivially, the system with large initial conviction in scale-free networks has the most clusters finally, but the number of clusters with the same condition in random networks is the smallest. The reason is that, even if agents’ conviction can become so strong within a quite short time, these agents in random networks still have some neighbors with the same opinion as a result of shortcuts of topology, and therefore, large-scale clusters are not split into fragments.

**Figure 4.**The number of opinion clusters as a function of time. In the beginning opinions are assigned uniformly at random, h = 0.1 and N = 1,000. The blue solid curve indicates that the degree-based individual influence is used and C

_{0}= 0.5. The green dotted curve describes the situation that agents have degree-based influence and C

_{0}= 0.8. The red dash-dotted curve refers to the system with betweenness-based individual influence and C

_{0}= 0.5 for both plots. The left plot uses a scale-free network as interacting topology, while the right plot is obtained from a random network. Every plot is an average of 100 different simulations.

**Figure 5.**The size of largest (square line) and second largest (circle line) opinion clusters as a function of h. In the beginning opinions are assigned uniformly at random, C

_{0}= 0.5 and N = 1,000. A scale-free network is used for both plots. In the left plot, the degree-based influence is used, while agents have betweenness-based influence in the right plot. Every plot is an average of 100 different simulations.

**Figure 6.**The proportion of update events adopting neighbor’s opinion versus h in a scale-free network, and initial opinions are assigned uniformly at random. C

_{0}= 0.5 and N = 1,000. Every plot is an average of 100 different simulations.

**Figure 7.**The distribution of number of times persuading neighbors to change opinions for each agent in a scale-free network, C

_{0}= 0.5 , h = 0.1 and N = 1,000. The exponents of power-law fitting for degree-based or betweenness-based influence are γ = −1.147, and γ = −0.937, respectively.

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Xiong, F.; Liu, Y.; Zhu, J. Competition of Dynamic Self-Confidence and Inhomogeneous Individual Influence in Voter Models. *Entropy* **2013**, *15*, 5292-5304.
https://doi.org/10.3390/e15125292

**AMA Style**

Xiong F, Liu Y, Zhu J. Competition of Dynamic Self-Confidence and Inhomogeneous Individual Influence in Voter Models. *Entropy*. 2013; 15(12):5292-5304.
https://doi.org/10.3390/e15125292

**Chicago/Turabian Style**

Xiong, Fei, Yun Liu, and Jiang Zhu. 2013. "Competition of Dynamic Self-Confidence and Inhomogeneous Individual Influence in Voter Models" *Entropy* 15, no. 12: 5292-5304.
https://doi.org/10.3390/e15125292