Abstract
We construct a new opinion formation of the Deffuant-Weisbuch model with the interference of the outer noise, where there are finite agents and the evolution is discrete-time. The opinion interaction occurs by one randomly chosen pair at each time step. The difference to the original Deffuant-Weisbuch model is that communications of any selected pairs will be affected by noises. The aim of this paper is to study the robust consensus of this noisy Deffuant-Weisbuch model. We first define the noise strength as the maximum noise absolute value. We will then show that when the noise strength is less than a certain threshold, this noisy model will achieve robust consensus when is sufficiently large; next we prove that the noisy model achieves robust consensus with a positive probability; finally, we demonstrate these results and provide numerical relations among the noise strength and some model parameters.
1. Introduction
For grasping a deeper understanding of human social cognition and behaviours, sociologists and psychologists had been interested in understanding opinion formations and dynamics phenomena. This study had become of great interest in physics in the last decades [1, 2]. In a social system, the character of their mutual opinion interactions is one of the main research points. Sorts of interaction methods, the voter method [3], the network method [4–8], the influence method [9], the multilevel method [10, 11], and the self-organization method [12], are proposed. In the field of studying the interactions among agents, one general opinion update rule is realized by averaging neighbors opinions. Moreover, an individual often interacts with only those individuals whose opinions are close enough to his or her own. This update rule comes from a kind of social psychological phenomenon, selective exposure, a psychological concept broadly defined as “behaviors that bring the communication content within reach of ones sensory apparatus.” These opinion models are called bounded confidence opinion models [13].
The most famous bounded confidence models are Hegselmann-Krause (HK) model [14] and Deffuant-Weisbuch (DW) model [15]. The fundamental difference between both models is materialized in the different definitions. In the DW model, two randomly chosen individuals meet and a pairwise averaging is implemented, while in the HK model, the communication takes place in more large groups and individuals move their own opinions to the average opinion of all individuals which lies in the area of confidence. Although opinion update rules are much different, it has been well established that they always approach to a limited state in which either perfect consensus is reached or the population splits into a set of opinion clusters with each of them holding the same opinion [16–18].
However, in real social phenomena, opinion agreement among individuals in a society is quite rare, whereas opinion disagreement is quite normal [19]. Opinion disagreement phenomena can be roughly classified as fragmentation and fluctuation. In fact, fluctuation had been studied in some physical areas widely such as electroacoustics [20], through simulations and calculations of the random variable variance [21].
For one special case of opinion fluctuation phenomena, some scholars [22–24] considered the existence of random noise among people communications. In detailed, [22, 23] studied the DW model for continuous-opinion dynamics with the influence of noises, where individuals are given the opportunity to change their opinion, with a given probability, to a randomly selected opinion inside the whole opinion space. Reference [24] studied the effects of noise in the continuous-time HK model and obtains approximate analytical results for critical conditions of opinion cluster formation. Reference [25] studied the noisy HK model and obtain that opinions reach robust consensus when the confidence bound satisfies a certain condition. Reference [26] demonstrated the HK model with the influence of opinion leaders and the noises.
Different from theirs, especially different from the probability’s change in the noisy environment in [22, 23, 25, 26], we consider the opinion values for the DW model. Opinions will vibrate because of the existence of noises. Our main results are as follows:(i)We propose a noisy DW model, where both selected agents are affected by the external noise if their distance is not larger than the confidence bound. Specially, the noises are confined in a certain interval;(ii)We prove the robust consensus of this noisy DW model holds with a positive probability, when the noisy strength is small sufficiently. Besides, we provide an upper bound of the noise strength less which agent opinions will reach robust consensus;(iii)The threshold of the noise strength and some relations among model parameters are demonstrated in finally.
This paper is organized as follows. Section 2 presents the original DW model, and then Section 3 gives a class of noisy DW models, and shows some basic facts. Following that, Section 4 shows that the noisy DW model achieves robust consensus when the noisy strength is sufficiently small and with a positive probability. Section 5 demonstrates how the opinion ranges change and compares the numerical relation among the threshold value of noise strength and some model parameters. Finally, Section 6 gives the conclusions.
2. DW Models
2.1. Review of the Original DW Model
The original DW model was introduced as a nonlinear extension of previous models of social influence [15]. We consider the discrete-time continuous opinion DW model with individuals. The opinion on a given topic is a real variable in the interval . We also assume that for are randomly distributed in this interval. Dynamics is introduced to reflect that individuals interact, discuss, and modify their opinions.(1)At time , a pair of agents are equal randomly chosen;(2)If their opinions satisfy , then both keep unchanged;(3)If their opinions satisfy , then the averaged opinion of both is their opinion at next time ;(4)All others keep unchanged.
This rule can be rewritten aswhere is the indicator function: if is true and otherwise ([27]).
The theoretical results on this original DW model shows that agent opinions will aggregate into several limits a.s. An opinion evolution example based on the original DW model is shown in Figure 1.
2.2. The Noisy DW Model
In this section we propose a noisy DW model. Compared to the original DW model, any communicated agents are affected by noises, and the model iswhere is independent bounded noise, satisfying (i), a.s., is the noisy strength;(ii);(iii), ().
When , the model (2) is degenerated into the original DW model (1).
By the constraint of that locating in , we adjust the model (2) intowhere the selection pair is at time () and
An opinion evolution example based on the noisy DW model is shown in Figure 2. Compared with Figure 1, it seems that noises promote opinion aggregations and the aggregated group will be not divided in a certain condition.
3. Fundamental Definitions and Lemmas
In this section, we will give our definitions and some basic lemmas. For the original DW model (1), we first provide the following basic result [18].
Lemma 1. For the original DW model (1), one of the following two results holds almost surely for any :(i),(ii).
Remark 2. If , then at last opinions will be divided into several clusters. Generally, the cluster number varies as model parameters, agent number, confidence bound, selection sequences, etc. change. It is found that when the confidence bound increases, the cluster number decreases a.s. until opinions reach consensus a.s., which can be defined as , a.s.
However, for the noisy DW model, opinion consensus becomes extravagant hopes. Figure 2 shows that agent opinions always fluctuate before any large termination time. We provide the following definition, where the robust consensus is the main role in this work.
Definition 3. The opinion state is said to be fluctuating a.s. if , whereis the opinion range at time .
Note that also play the role of “aggregation degree” in social opinion system for one topic. Similarly, we can define , where is a subset of . Next, to study opinion fluctuation strength of the noisy DW model (3), we provide the definition of robust consensus as follows [28].
Definition 4. Given the initial opinions and the confidence bound , if there exists and and holds a.s., where , then we call the noisy DW model robust consensus. Generally, for any , if holds a.s., then we call the noisy DW model achieves robust consensus. Here , , and approaches 0 when .
4. Robust Consensus of the Noisy DW Model
In this section, we will prove the robust consensus of the noisy DW model (3).
We first demonstrate how the fluctuation strength changes as increases (Figure 3). This example shows that when , always holds until the termination time.
Secondly, we will provide some lemmas for understanding how the opinions change for the noisy DW model (3).
Lemma 5. If and , then the noisy DW model (3) achieves T
Proof. We can prove it by the mathematical induction method.
For , if and , then holds if . Similarly, holds for any if , . By the random of noise, exists a.s. Defining , then the conclusion holds obviously.
For , we assume and . Without loss of generality, we set . If agent and agent are selected at time , thenwhile and . Thus, if agent and agent always select each other and for , then there exists a finite time a.s., such that when , . By the Borel-Cantelli (B-C) lemma [29], we can get that . Further, because agent always keeps unchanged,Thus, there exists a finite time a.s., such that and when , .
Similarly, by the B-C lemma, there exists a finite time a.s., such that and when , . In a sum, there exist a stopping time sequence a.s., such that , , . Furthermore, among the interval sequence , by the B-C lemma and the ergodicity of or , there always exists a time interval subsequence, denoted by , at which or , , . Because , Denote . Thus, the system is robust consensus for the event that the selection pair is always for any time .
According to the rotational symmetry of agent selection, if agent and agent always select each other, then the conclusion also follows. Generally, note thatif and . Thus, if , , and , then . Note that this event happens infinite often by the B-C lemma. Therefore, there exists a time subsequence a.s. in which . In a sum, the system is robust consensus for the general condition.
Inductively, for any , we assume and .
Firstly, if agent and agent select each other at time 0, then Secondly, if agent and agent keep unchanged at time 0, thenFinally, if agent selects another agent , , but agent keeps unchanged at time 0, then Note that agents and satisfy , and then by the similar method, the system is robust consensus for the event that any two agents are selected periodically. By the B-C lemma, we get that the no-periodical selection event is a zero event; thus the system is robust consensus for the general case.
Figure 4 shows how the opinion range changes as increases. Note that is infinite often, .
Besides, the ergodicity of any agent’s opinion values can be shown in Figure 5. This is deduced by the randomness of the external noise , , .
Remark 6. In fact, Lemma 5 shows that when is sufficiently small, the noisy DW model will be robust consensus. However, it is still a problem how opinions evolve if ?
In the following, we study one special condition such that there are two agent subsets, where both satisfy the condition in Lemma 5 but their opinions’ distances would be larger than . We will show that the system (3) can also achieve robust consensus finally.
Lemma 7. Suppose there exists a partition of , satisfying ; then the system (3) achieves T if .
Proof. For , we assume and .
If , then we get that the system is robust consensus by Lemma 5.
If , for simplicity, , then we can get that if the selection pair is at time . With positive probability, happens to be infinite often and is ergodic among a.s. Similarly, among the time interval sequence , there is a finite time at which a.s., and then we get that the system is robust consensus by Lemma 5.
For , we fixed a partition of the agent set . Denote , suppose the time subsequence at which agent and agent select each other and . In fact, with the same method used in Lemma 5, the time subsequence exists with the probability 1.
Note that is a random walk among , . Therefore, the agent interval can also change randomly among . Select and , then the event happens i.o. by the B-C lemma. Similarly, the event that agent and agent select each other also happens i.o.. There also exists the time subsequence a.s. such that agent and agent select each other and . For the finiteness of the selection pair set , we get that there exists a time subsequence at which i.o.. Consequently, we get that the system is robust consensus by Lemma 5. Thus, the conclusion follows.
Finally, based on these lemmas, it is obviously to get the following theorem. Hence, we neglect its proof in the following.
Theorem 8. For any initial states and , the noisy DW model (3) achieves T a.s..
Besides, we further get the following theorem by weakening the probability condition.
Theorem 9. For any initial states and , the noisy DW model (3) achieves robust consensus with a positive probability.
Proof. If , then if the selection pair is at time . With the similar method used in Lemma 7, we only need to consider the initial condition .
By Lemma 7, suppose there exists a finite partition of , satisfying for , then the system (3) achieves robust consensus if .
In the following we will prove that the system is also robust consensus. Denote and as the repeated times of the selection pair in the following proof. Furthermore, the time interval sequence of the repeated selection pair of is denoted by .
For , all selection pairs are , and . In fact, as approaches to 0, Lemmas 5 and 7 also hold. Consider the eventBecause , in the event , note that holds for any . We get that if the selection pair is at time and for other conditions. Thus, denote , thus, the system achieves robust consensus in the event .
Similarly, define Because and , there exists at least one moment such that ’s symbol is different from the ones in some other moments. Thus, agent would not get out of the confidence range of another agent , , , or . Thus, with the similar method, the system achieves robust consensus in the event .
With a similar method, we can define an event sequence , where Based on the randomness of , we get that Thus, the conclusion holds when .
For , we can generate the analysis for into . Similarly, we analyse the event sequence where By Lemma 7, we get that the event happens i.o.; thus, happens with the probability larger than , . Along with the same method used before, we can get that the conclusion holds for general cases.
Remark 10. The main result, Theorem 8, shows that when is sufficiently small, the noisy DW model will achieve robust consensus. Furthermore, Theorem 9 tells us that the noisy DW model achieves robust consensus with a positive probability. However, it is nearly impossible to estimate the probability ; thus we could not provide it in Theorem 9. Besides, it is still unknown on the relation of and ? We denote the threshold values of as . By the previous result, we get that . For further study of the parameter , we will provide the experimental demonstrations in the next section.
5. Demonstrations for System Parameters
In this section, we exhibit the simulation results for the noisy DW model. The main aim is to study how the threshold of changes as the confidence bound changes. In fact, both Lemma 7 and Theorem 8 imply that separated opinion subsets will merge together. Firstly, we demonstrate how the limited opinion range changes as parameters increase. Secondly, the cluster number is studied as the noise strength changes. Finally, the parameter is simulated for further understanding the relation of the noise and opinion robust consensus. Specially, all simulations must be in finite steps and the termination time should be large enough. We will demonstrate them in the following subsections. All simulations are the averages of 1000 times experiments, respectively.
5.1. Opinion Range
First, we show how the opinion range changes as parameters change. In Figure 6, we show how the fluctuation strength for each cluster varies as the confidence bound and the noise strength change, respectively.
(a) How changes as increases
(b) How changes as increases
Secondly, we simulate the cluster number and average cluster width (Figure 7). It shows that when the noise strength is less than a certain value, the average cluster number is one. Specially, due to the finiteness of the termination time, the average cluster number is larger than one if the noise strength is small enough.
(a) How cluster number changes as increases
(b) How cluster width changes as increases
5.2. Noise Strength
Then, we demonstrate the threshold values less which opinions will achieve robust consensus when is sufficiently large. In Figure 8, we show how the threshold values of change when the confidence bound increases. It is worth noticing that the threshold values are nearly a linear relation with the confidence bound. Note that the average linear relation is nearly . However, because of the finiteness of the termination time , we get that the time complexity is very high when is very small. Thus, the estimated threshold values of may be no verifiable. Also, there exist some exceptional examples in Figure 7, and we will obtain the desirable results if we take a long but unrealistic termination time.
(a) How changes as increases
(b) How changes as increases
For Lemma 7 and Theorem 8, we consider the initial condition as follows: we assume the agent opinions are split into two parts, one in which opinions are located in , another in which opinions are located in .
6. Conclusions
In this paper, we proposed a noisy DW model in which any selected pairs will be affected by external noises. We mainly proved that when the strength of noises is small enough, opinions of this noisy DW model will reach robust consensus; that is to say, all opinions fluctuate but the fluctuation range’s value is less if the noise strength is less for finite time intervals. Beside, opinions of this noisy DW model will achieve robust consensus with a positive probability. The definition of the robust consensus was introduced to describe the relation of opinion fluctuation and noise strength . We proposed some lemmas. For one, we described that the aggregated opinions will not be divided into different parts with distance larger than the confidence bound , when the noisy strength is sufficiently small; For another, we described that the separated parts will merge together when is sufficiently small. Then we gave the theorem describing that the noisy DW model achieves robust consensus when is sufficiently small. Finally, we demonstrated threshold values of , where there exists a positive probability larger than which the aggregated agent opinions will be divided, and another positive probability larger than which separated parts will not merge together.
Data Availability
The simulation data are random values on the real interval .
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the NNSF of China under Grants 11601450 and 71503206 and the Social science project of Sichuan Education Department under Grant XQ15B10.
Supplementary Materials
Supplementary material contains five MATLAB programs on the noisy DW model. Program 1: The opinion evolution of the noisy DW model. Program 2: How the opinion fluctuation strength changes as the time increases. Program 3: How the opinion fluctuation strength changes as the model parameters change. Program 4: The clusters of opinion evolution. Program 5: How changes as the confidence bound increases. (Supplementary Materials)