In order to study the formation process of group opinion in real life, we put forward a new opinion interactive model based on Deffuant model and its improved models in this paper because current models of opinion dynamics lack considering individual persuasiveness. Our model has following advantages: firstly persuasiveness is added to individual’s attributes reflecting the importance of persuasiveness, which means that all the individuals are different from others; secondly probability is introduced in the course of interaction which simulates the uncertainty of interaction. In Monte Carlo simulation experiments, sensitivity analysis including the influence of randomness, initial persuasiveness distribution, and number of individuals is studied at first; what comes next is that the range of common opinion based on the initial persuasiveness distribution can be predicted. Simulation experiment results show that when the initial values of agents are fixed, no matter how many times independently replicated experiments, the common opinion will converge at a certain point; however the number of iterations will not always be the same; the range of common opinion can be predicted when initial distribution of opinion and persuasiveness are given. As a result, this model can reflect and interpret some phenomena of opinion interaction in realistic society.
In the last decades, there are fruitful research achievements in learning behaviors of individual and complex group phenomena. There is little doubt that the behavior of individual and group phenomenon is inextricably linked. Group consists of lots of individuals, while common behaviors of plenty of individuals constitute many macroscopic emergences. Hence, it is an effective way to study the group phenomenon based on individual’s behavior. Since the human society can also be considered as a complex multicomponent system consisting of individuals interacting with themselves and with their material environment, it is a challenge to develop a strategy allowing of a general quantitative modeling procedure for collective dynamic macro processes in the society [
In the research of cognitive learning behavior of individual, Brenner [
So far, models of opinion dynamics provided by scholars can depict the opinion interactive situation well in realistic society in some special situation. Based on the form of opinion, opinion dynamics can be divided into discrete model and continuous model. In discrete model, opinion is a discrete numerical variable which includes Voter Model [
In the early research, individual is homogeneous. With the deepening of researching, many scholars add the heterogeneous attribute to individuals, such as softhead individuals, amiable individuals, bigoted individuals, stubborn individuals, opinion leaders, and authoritative individuals [
This paper puts forward a new opinion interactive model based on Deffuant models and its improved models. This model emphasizes the importance of individual’s persuasiveness and simulates the variation trend of group opinion. With this aim, the remainder of this paper is organized as follows. Previous work about Deffuant model is introduced in Section
The main idea of Deffuant model [
Honestly speaking, there are plenty of models after proposing of Deffuant model, especially some heterogeneous models. Lorenz [
However, all the improved Deffuant models can only deal with one aspect problem. Most of them focus on the threshold
Suppose a scene that a group of people discuss a topic. Everyone has its own attitude and interest to a certain topic. However people cannot only insist on their own opinion, because as an individual of society, it should take other people’s opinion into account. After discussion, the group should reach an agreement. This scene often appears in reality society, such as the discussion of some topics in Congress and the discussion of entertainment place where to go in the weekend among office members.
This paper simulates the scene through modeling and experiments. Any two individuals chosen randomly in group can exchange their opinions. Opinions are published according to the round, and opinions of current round are only affected by opinions of last round. In the process of interaction, the sequence of speech of individual is ignored which means that the speech of the whole group is parallel. Individual acquires the last round opinion of itself and the opposite individual’s opinion and then calculates the current round opinion of itself in line with some behavior rules and after that publishes its new opinion in the next round. With the evolution of group opinion, it forms a series of polymerization of macroopinion cluster eventually.
In this paper, the model is introduced in the form of agent. Individuals are abstracted as agents while each group consists of a lot of agents. Every agent has the same attributes and behavior rules, while they may be different from concrete attribute values or concrete behaviors. Through the interaction of microscopic interactive process, some macroscopic emergence phenomena of group can be found.
Attributes of agent are abstracted as index, opinion, and persuasiveness. Explanation to these attributes is as follows.
To distinguish from other agents, every agent has a unique identity.
To a certain topic in the discussion, every agent has its own opinion. Opinion is the position or attitude that one observes things. In the mathematic model, opinion is abstracted as a discrete variable or continuous variable. Although it is maybe too simple to model the complex human society, it has some advantages to some certain problems, such as “turning left or turning right,” “approve or disapprove,” and “going to classroom or going to library.” Due to the variety of opinions to a certain topic and for the convenience of analysis, the opinion is modeled by continuous interval between 0 to 1 referring to Deffuant model. Among it, 0 and 1 represent the opposite opinion and 0.5 represents neutrality that opinion values have different meanings in different cases.
Persuasiveness is the ability to persuade the other individual. In order to protect its own interest, everyone wants to persuade others and make them accept its opinion in the discussion. However, the status is often unequal. Some people are of higher qualification, elder age, or an opinion leader in some fields. So their opinions have a guiding role to others and their persuasiveness is higher than others. Most people have similar knowledge to a certain topic, and they do not have a deep research in it. So their opinions are just reference to others and their persuasiveness is lower. Drawing lessons from the mathematic model of opinion, the persuasiveness is modeled by continuous interval ranging from 0 to 1. Among it, 0 represents the lowest persuasion, 1 represents the highest persuasion, and 0.5 represents that individual has their own opinions to a topic but does not have an insight into it.
The paper does not take topology of network into consideration in this model. What we focus on is the influence of individual’s persuasiveness and interactive probability on the process of interaction. In real life, if the difference between two individuals is low, the probability of thorough interaction between them may be high. However, it cannot be guaranteed that two sides will interact, because there are many other factors influencing interacting such as personal character. If the difference between two sides is large, the probability of thorough interaction is small. Similar to the former, it cannot guarantee that two sides will not interact of which the probability of interaction is just low. As to persuasiveness, an individual with high persuasiveness can change other’s opinion easier than that with low persuasiveness.
Suppose that there are
each time Agent
if
Among formula (
This model draws the lessons from Deffuant model and some other homogeneous Deffaunt models. Honestly speaking, if every individual possess the same and fixed parameters
Classical opinion evolution processes (initial opinion distributions of group obey the uniform distribution ranging from 0 to 1); number of individuals is 1000. The horizontal ordinate represents the iteration number, and vertical ordinate represents individual opinion. (a)
From Figure
This serial of experiments set a certain scale of agents, which does not take the topology of network into consideration. When the absolute difference of two opinions is less than 0.001, we assume that their opinions are the same. The simulation will come to an end when all the agents have the same opinion. The experiment flow is as follows:
assign initial values to all the agents;
in each time step, every agent can randomly choose one agent to exchange opinion. If the total number of individuals is odd, an agent will miss a turn in this round of interaction that it will not exchange its opinion with others;
all the agents change their opinions based on the opinion updating rule in section three except the one who misses a turn;
repeating step
In addition, for the convenience of analysis of the evolution character in macroscopic aspect, two indexes are defined here. Number of iterations (NOI) is the total time steps when all the agents have the same opinion, which indicates the convergent speed of group opinion. Common opinion (CO) is the opinion of group when all the agents have the same opinion, which determines variation trend of group opinion.
In the opinion updating model, there are two factors influencing CO and NOI. One factor is initial distribution of persuasiveness and opinion and the corresponding relationship between them. The other factor is the probability in the interaction. It is liable that the group may obey different kinds of distributions. However, for the convenience of analysis, we assume three persuasiveness distributions, namely, normal distribution, exponent distribution, and uniform distribution. Three experiments have been done in this paper. Experiment 4.1 and Experiment 4.2 are based on the same initial condition. The difference between them is that Experiment 4.1 focuses on the analysis of factors (influence of randomness, initial distribution, and number of individuals) influencing CO, while Experiment 4.2 stresses factors (influence of randomness, initial distribution, and number of individuals) influencing the NOI. Experiment 4.3 takes two initial distributions (normal distribution and exponential distribution) into account to predict the range of CO.
In this experiment, we discuss three group experiments and each group experiment is also divided into three experiments. These three group experiments have the same initial opinion distribution obeying uniform distribution. The difference among them is that initial persuasiveness distribution of the first group experiment follows normal distribution, which can be denoted as “Nor,” initial persuasiveness distribution of the second group experiment follows uniform distribution which is denoted as “Unif,” and the remained group experiment follows the exponent distribution which can be represented as “Exp.” Each group experiment is also divided into three experiments, which are distinguished by the number of individuals. Each group experiment, respectively, has 100 agents, 1000 agents, and 10000 agents. So there are nine experiments in total; three experiments of “Nor” can be denoted as “Nor100,” “Nor1000,” and “Nor10000,” respectively; three experiments of “Unif” are represented as “Unif100,” “Unif1000,” and “Unif10000,” respectively; three experiments of “Exp” are denoted as “Exp100,” “Exp1000,” and “Exp10000,” respectively. Each experiment is independently repeatedly calculated for 100 times.
Random number generator separately generates initial opinion distribution and initial persuasiveness distribution of group based on above requirements. Suppose that the initial opinion distribution of all the experiments obeys the uniform distribution ranging from 0 to 1. As for “Nor100,” “Nor1000,” and “Nor10000,” the initial persuasiveness of group follows normal distribution of which the mean value
Mean values and standard deviations of COs of nine experiments are shown in Table
COs of nine experiments.
Group name  Nor100  Nor1000  Nor10000 
Mean value  0.5818  0.5978  0.5838 
Deviation 





Group name  Unif100  Unif1000  Unif10000 
Mean value  0.6353  0.6643  0.6656 
Deviation 





Group name  Exp100  Exp1000  Exp10000 
Mean value  0.7272  0.7482  0.7500 
Deviation 



Based on Table
Without formula derivation, some conclusions can be drawn through these experiment results.
The initial condition of this experiment is the same with Experiment 4.1. In this experiment, the influence of randomness, initial distribution of agent’s attributes, and the number of individuals on NOI are discussed. Figure
NOIs of nine experiments.
Many phenomena can be found from Figure
Without formula derivation, three conclusions can be acquired based on simulation experiment results.
In this experiment, a situation is considered that initial opinion distribution and initial persuasiveness distribution of group are known but we do not know the corresponding relationship between them. The CO cannot be acquired because the value of opinion and persuasiveness of every agent cannot be acquired in advance. Calculating the boundary condition is a proper solution which can decrease the degree of error of predicting CO. There are two situations of discussion in real life: one scene is that to a topic most people have their own opinion while only a few people have indepth knowledge or have no idea of it; the other scene is that most people do not have idea of it while only a few people have an insight into the topic. Normal distribution and exponent distribution can substitute for these two scenes. Through simulation experiment, the range of CO in the light of initial persuasiveness distribution and initial opinion distribution can be predicted.
The range of CO is based on two boundaries: one boundary is corresponding of positive sequence between initial persuasiveness distribution and initial opinion distribution, and the other boundary is corresponding of negative sequence between them. Opinion distribution and persuasiveness distribution are onetoone correspondence from small to large in positive sequence corresponding while they are onetoone correspondence that one distribution is from small to large and the other distribution is from large to small in negative sequence corresponding.
Positive sequence corresponding is taken into account at first. Suppose that the initial persuasiveness of group obeys normal distribution and initial opinion of group obeys the uniform distribution ranging from 0 to 1. The number of individuals is 10000. Because the range of persuasiveness is from 0 to 1, in order to confine the persuasiveness within the boundary it will generate persuasiveness again if persuasiveness is bigger than 0.99 or smaller than 0.01. The normal distribution has two parameters (deviation and mean value). A series of discrete values of standard deviation which are 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, and 0.14 separately and discrete values of mean value which are 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, and 0.7 separately are picked up. Figure
The influence of
From Figure
Among it, parameters
Table
Indexes of fitted curve.
Index  Value 

SSE  0.0005162 
RMSE  0.003009 

0.9896 
Adjusted 
0.9887 
Figure
The fitted curve based on formula (
In addition, the negative sequence corresponding is considered. All the conditions are the same with the above except the relationship of sequence corresponding between initial opinion distribution and initial persuasiveness distribution. However, it is not necessary to calculate again because the initial opinion distribution obeys uniform distribution ranging from 0 to 1 which has a symmetrical characteristic. CO of negative sequence corresponding is symmetrical with that of positive sequence corresponding and the axis of symmetry is
Suppose that the initial persuasiveness distribution obeys normal distribution which mean that value is
Positive sequence corresponding is taken into account at first. Suppose that the initial persuasiveness of group obeys exponent distribution and initial opinion of group obeys the uniform distribution ranging from 0 to 1. The number of individuals is 10000. Because the range of persuasiveness is from 0 to 1, in order to confirm persuasiveness within boundary, it will generate persuasiveness again if persuasiveness is larger than 0.99 or smaller than 0.01. The parameter in exponent distribution is denoted as
The influence of parameter
Figure
In addition, the negative sequence corresponding is considered. All the conditions are the same with the above except the relationship of sequence corresponding. However, it is not necessary to calculate again because the initial opinion distribution obeys uniform distribution ranging from 0 to 1 which has a symmetrical characteristic. The calculating process is the same with Experiment
Suppose that the initial persuasiveness distribution obeys exponent distribution of which parameter is
Through simulation experiments, some meaningful conclusions and predicted methods are summarized as follows.
No matter how many independently replicate experiments we do, if initial values of all the individuals are fixed, the CO of group is fixed.
Randomness, initial distribution, and number of agents have a great influence on the NOI of experiment. The more the initial persuasiveness distribution concentrates on 0.5, the less the NOI is. In the realistic society, the persuasiveness value of 0.5 represents that individual has their own opinions to a topic but does not have an insight into it.
A method of predicting interval of CO is put forward in the condition that the distribution of initial opinion and initial persuasiveness are known in advance. When initial opinion distribution obeys uniform distribution ranging from 0 to 1 and initial persuasive distribution obeys normal distribution which mean that value is
The paper focuses on the influence of individual persuasiveness on opinion interaction and builds a new opinion interactive model based on Deffuant models and its improved models. The most different thing between our model and Deffuant model is that all individuals in our model are different from others, namely, owning different persuasiveness. If such a mechanism of taking a decision by a community is correct, our model leads to the following conclusions.
In a closed (isolated) community there are a variety of possible opinions ranging from person to person in original state. After a short and long time our model tend to be one of the “ordered” states, there is a possibility of reflecting the case that the group takes a common decision.
As for a certain discussion topic of closed community, when the initial conditions of the whole group are known, the result may be predicted ahead of time. What may not be predicted easily is the convergent time because of many uncertain factors, like number of individuals, individual persuasiveness, and so forth.
One of the most important factors is the distribution of persuasiveness. Taking Section
To sum up, the proposed very simple rules trigger a rather complicated dynamics. However, one can doubt if these rules properly describe real mechanisms of taking a decision. There are of course other possibilities within the improved model. Because humans are exactly the opposite of such simple entities, such as atoms and molecules, indeed the detailed behavior of each of them is already the complex outcome of one’s interest, benefits, and especially many physiological and psychological processes. However in the model, we just consider a certain case mentioned in Section
The authors declare that there is no conflict of interests regarding the publication of this paper.
In this paper, the research was supported by National Nature and Science Foundation of China (nos. 91024030, 71303252, 61403402, 41201544, and 71373282).