The existing opinion dynamics models mainly concentrate on the impact of opinions on other opinions and ignore the effect of the social similarity between individuals. Social similarity between an individual and their neighbors will also affect their opinions in real life. Therefore, an opinion evolution model considering social similarity (social-similarity-based HK model, SSHK model for short) is introduced in this paper. Social similarity is calculated using individual properties and is used to measure the social relationship between individuals. By considering the joint effect of confidence bounds and social similarity in this model, the role of neighbors’ selection is changed significantly in the process of the evolution of opinions. Numerical results demonstrate that the new model can not only obtain the salient features of the opinion result, namely, fragmentation, polarization, and consensus, but also achieve consensus more easily under the appropriate similarity threshold. In addition, the improved model with heterogeneous and homogeneous confidence bounds and similarity thresholds are also discussed. We found that the improved heterogeneous SSHK model could acquire opinion consensus results more easily than the homogeneous SSHK model and the classical models when the confidence bound was related to the similarity threshold. This finding provides a new way of thinking and a theoretical basis for the guidance of public opinion in real life.
Over the past decades, opinion dynamics as a special type of complex human behavior has attracted a great deal of interest from researchers in different scientific fields [
In the existing study of opinion dynamics, models can be divided into discrete opinion models and continuous models [
However, the classical HK model and these improved HK models only consider the influence of neighbors’ opinions on an individual’s opinion in the process of communication. Many studies have proved that the change in an individual’s opinion is also influenced by the social attributes of other agents [
This paper combines the joint influence of neighbors’ opinions and social attributes on an individual’s opinion. Likewise, the opinion of some agents outside the confidence bound, which can also communicate, is considered. In Section
The intimacy of a social relationship between individuals can be described by their social similarity. “Birds of a feather flock together” is a saying which means that people of the same kind show stronger interpersonal attraction. Interpersonal attraction is a state of mutual dependence and is a positive form of relationship. One of the most important principles of interpersonal attraction is the similarity principle [
The possible values of each social attribute are limited, and the range of values of each social attribute is summarized according to empirical analysis. To facilitate the modeling calculation, each attribute value is simply quantified. There are two kinds of quantitative methods—one is a certain number of discrete values corresponding to the value of the property, and the other quantifies social attributes as a range of continuous values, and the social attributes themselves are represented by a range of values. It is obvious that the latter method has better representation effects, so that continuous values are used to represent individual attributes. Each agent will be assigned attributes forming an attribute set, including age, gender, education level, economic status, and geographic location according to the fifth population census data of China. For example, age is a nonnegative integer represented by a number between 0 and 100, and education level can be expressed as a float number between 0 and 1, where 0 stands for the lowest education level (illiteracy) and 1 stands for the highest education level. Gender can be specified as male or female, and geographic location can be represented by the latitude and longitude of an individual’s residence.
Specifically, agent
The intimacy of the social relationship between individuals can be described by their social similarity
By combining
In the classical HK model, each agent selects another agent whose opinion is limited in the confidence bound as communication neighbors [
Based on the MHK model, this paper proposes the SSHK model, which introduces social similarity to represent the social relationship between an agent and his neighbors. An agent’s neighbors can be selected by their social similarity and confidence bound. The SSHK model has
The classical bounded confidence model shows that when communicating with each other, the agents may be convinced by their neighbors who share a similar opinion with him, and every one may change his opinion toward his neighbors. However, the agent also changes his opinion toward his close or reliable neighbors regardless of his own opinion, indicating that people’s dependence on social relations and the objective judgment of opinion in communication matter in the process of opinion exchange. To consider the above two situations simultaneously, social similarity between agents is introduced to extend the confidence bound. A social similarity threshold
In the process of neighbor selection, two constraints are set up simultaneously:
This paper ignores the influence of network topology and assumes that the network is fully connected. Therefore, each agent can communicate with all other agents in the system. However, every agent can only communicate with the neighbors calculated by formula (
The SSHK model is studied using agent-based modeling and simulation. The simulation result is averaged 100 times. The system assumes a fully connected network of
Steady state is defined when the opinion of all agents no longer changes; that is, the system is stable [
Homogeneous and heterogeneous models are discussed in the following sections. A homogeneous model means that the confidence bound and similarity threshold are equal. A heterogeneous model means that the confidence bound and similarity threshold are unequal.
The homogeneous model is studied in this section. The confidence bound is neglected to analyze the impact of social similarity on opinion dynamics. The influences of the confidence bound and social similarity threshold are compared when they work simultaneously. The SSHK model is verified by comparing it with the classical HK and MHK model.
First, we assume that
Opinion evolution results of the SSHK model by changing the social similarity with different
Figure
The reasonable value of
The influence of
Based on the assumption that most individuals are generally objective, we assume that the weight of the confidence bound is greater than the weight of the similarity (
Opinion evolution result under the mutual influence of social similarity and the confidence bound of different
The similarity threshold is small (
The similarity threshold is large (
It can be seen that when the weight of similarity is at the intermediate value (
It is noteworthy that when
The influence of the similarity threshold in different confidence bounds is also studied. The results are shown in Figure
Influence of the similarity threshold on opinion evolution results under different confidence bounds.
In Figure
To verify the feasibility of the SSHK model, we compare it with that of the classical HK and MHK models. The opinion distributions of the three models with different confidence bounds are shown in Figure
Opinion evolution results in the classical HK model, MHK model, and SSHK model with different confidence bounds (
Figure
Comparison of stable time (
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Comparison of opinion evolution results in the MHK model and SSHK model (
In Figure
Table
Communication times of agents outside the confidence bound in the SSHK model with
In Figure
The communication times of agents outside the confidence bound have significantly decreased and are reduced to 0 after Step 50. This finding can be attributed to opinions that gradually converge to a few clusters with great differences in time and are constrained by the confidence bound to only allow agents with similar opinions to communicate.
Usually, the confidence bound and social similarity threshold of each agent are different. Therefore, the SSHK model of the heterogeneous confidence bound and social similarity threshold is studied in this section. Moreover, the heterogeneous model is then compared with the homogeneous model.
Given that the power-law distribution phenomenon widely exists, the heterogeneous confidence bound and social similarity threshold in this paper follow it. The power-law distribution
Each individual may be affected by his neighbors’ opinions and social relations. These two effects are different and related. The three relation types are as follows: The confidence bound and social similarity threshold have a positive correlation. When making a decision, the agent is easily affected by both his neighbors’ opinions and their social relationships. The confidence bound and social similarity threshold have a negative correlation. The psychology of agents is extreme. They change their opinions because of communication with their neighbors or their social relationships. The confidence bound and social similarity threshold are not correlated. The effects of these factors on agents are random.
In Figure
Comparison of
From Figure
In Figure
Comparison of
It can be seen from Figure
Combining the opinion cluster numbers and stable time, it can be concluded that the result obtained by opinion evolution is always the worst in any case when the similarity threshold and the confidence bound are irrelevant; that is, it achieves most opinion cluster numbers and takes the longest time to stabilize. However, as long as there is a correlation between the two factors, whether positive or negative correlation, the results are not significantly different. The results are the same when the confidence bound is not small.
The homogeneous and heterogeneous models are now compared. The heterogeneous model (
First, the homogeneous model is compared with the heterogeneous model with negative correlation between the confidence bound and social similarity threshold. The result is shown in Figure
Comparison of
Figure
The homogeneous model is now compared with the heterogeneous model of no correlation in Figure
Comparison of
In Figures
In conclusion, compared with the homogeneous model, the heterogeneous model can lead to decreasing opinion cluster numbers and promote the opinion to achieve consensus. This finding is related to the distribution of the confidence bound and social similarity threshold as well as their correlations.
This paper proposed an improved SSHK model based on a classical bounded confidence model by introducing social similarity between agents into the MHK model, simultaneously considering the influence of neighbors’ opinions and their social relationships as well. The SSHK model considers the special situation that a small number of agents outside the confidence bound have the chance to exchange opinions. The homogeneous and heterogeneous SSHK models are also studied and compared in this paper. The result shows that the new model also obtains the main features of the opinion result, namely, fragmentation, polarization, and consensus.
By studying the homogeneous SSHK model, it can be concluded that the new model can easily achieve consensus under an appropriate social similarity threshold compared to the classical models; that is, social similarity also influences opinion dynamics [
The heterogeneous model with the influence of correlation between the confidence bound and social similarity threshold is discussed and compared with the homogeneous model. Compared to the homogeneous model, the heterogeneous model promotes the opinion to reach consensus. When the confidence bound is related to social similarity threshold, the heterogeneous model enables the opinion to converge easily, especially when the confidence bound and social similarity threshold are positively correlated. This finding suggests that opinion consensus is easily reached when individuals treat the influence of neighbors’ opinions and social relationships as equal. Opinion is easier to converge when all individuals are in extreme conditions; that is, they are only affected by neighbors’ opinions, or they are only influenced by social relationships. However, opinion is difficult to predict and reaching consensus is difficult when there is no correlation. Thus, the correlation between the confidence bound and social similarity threshold influences the state of opinion stability. This finding may provide a theoretical basis and method for the control or guidance of public opinion. Some agents with a positive or negative correlation between similarity threshold and the confidence bound may join and lead the community to reach an agreement.
In summary, regardless of the nature of the results (i.e., reality or experimental), social similarity between individuals also influences the opinion evolution result. The impact of social relationships between individuals on opinion dynamics cannot be ignored. Introducing the social relationship and social attributes of individuals into the opinion evolution model makes the model more realistic. Moreover, the introduction of these concepts provides a theoretical framework for the opinion dynamics model, which considers complex individual social attributes and relationships. This study ignores the influence of network topology. The influence of complex social networks and heterogeneous individual social attributes on public opinion evolution needs further research.
The authors declare that there are no conflicts of interest regarding the publication of this article.
This work was supported by the National Natural Science Foundation of China (Grant no. 71571081, no. 61540032, and no. 91324203).