# The Robust Consensus of a Noisy Deffuant-Weisbuch Model

Authors: Jiangbo Zhang; Yiyi Zhao
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

We construct a new opinion formation of the Deffuant-Weisbuch model with the interference of the outer noise, where there are finiten 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 T-robust consensus when t 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.


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## Body

## 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. Section2 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 n individuals. The opinion xi(t) on a given topic is a real variable in the interval [0,1]. We also assume that xi(0) for i∈V={1,2,…,n} are randomly distributed in this interval. Dynamics is introduced to reflect that individuals interact, discuss, and modify their opinions.(1)
At timet, a pair of agents (i,j) are equal randomly chosen;(2)
If their opinions satisfyxit-xit≥η, then both keep unchanged;(3)
If their opinions satisfyxit-xit<η, then the averaged opinion of both is their opinion at next time t+1;(4)
All others keep unchanged.This rule can be rewritten as(1)xit+1=xit+121xjt-xit<ηxjt-xit,xjt+1=xjt+121xjt-xit<ηxit-xjt,xkt+1=xkt,k≠i,j.where 1S is the indicator function: 1S=1 if S is true and 1S=0 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 Figure1.Figure 1
Opinion evolution of 20 agents for (1), where n=20 and η=0.2.


### 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 is(2)xit+1=xit+1xjt-xit<η12xjt-xit+ξit+1,xjt+1=xjt+1xjt-xit<η12xit-xjt+ξjt+1,xkt+1=xkt,k≠i,j,where {ξi(t)} is independent bounded noise, satisfying (i)
ξit≤δ, a.s., δ>0 is the noisy strength;(ii)
E[ξi(t)]=0;(iii)
E[ξi2(t)]≥cδ2, (c∈(0,1]).Whenδ=0, the model (2) is degenerated into the original DW model (1).By the constraint of thatxi(t) locating in [0,1], we adjust the model (2) into(3)xit+1=1x¯it>1+x¯it·1x¯it∈0,1+0·1x¯it<0,xjt+1=1x¯jt>1+x¯jt·1x¯jt∈0,1+0·1x¯jt<0,xkt+1=xkt,k≠i,j,where the selection pair St is (i,j) at time t (St=(i,j)) and(4)x¯it=xit+1xjt-xit<η12xjt-xit+ξit+1.An opinion evolution example based on the noisy DW model is shown in Figure2. Compared with Figure 1, it seems that noises promote opinion aggregations and the aggregated group will be not divided in a certain condition.Figure 2
Opinion evolution of 20 agents for (3), where n=20, η=0.2, and δ=0.01.


## 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 n individuals. The opinion xi(t) on a given topic is a real variable in the interval [0,1]. We also assume that xi(0) for i∈V={1,2,…,n} are randomly distributed in this interval. Dynamics is introduced to reflect that individuals interact, discuss, and modify their opinions.(1)
At timet, a pair of agents (i,j) are equal randomly chosen;(2)
If their opinions satisfyxit-xit≥η, then both keep unchanged;(3)
If their opinions satisfyxit-xit<η, then the averaged opinion of both is their opinion at next time t+1;(4)
All others keep unchanged.This rule can be rewritten as(1)xit+1=xit+121xjt-xit<ηxjt-xit,xjt+1=xjt+121xjt-xit<ηxit-xjt,xkt+1=xkt,k≠i,j.where 1S is the indicator function: 1S=1 if S is true and 1S=0 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 Figure1.Figure 1
Opinion evolution of 20 agents for (1), where n=20 and η=0.2.


## 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 is(2)xit+1=xit+1xjt-xit<η12xjt-xit+ξit+1,xjt+1=xjt+1xjt-xit<η12xit-xjt+ξjt+1,xkt+1=xkt,k≠i,j,where {ξi(t)} is independent bounded noise, satisfying (i)
ξit≤δ, a.s., δ>0 is the noisy strength;(ii)
E[ξi(t)]=0;(iii)
E[ξi2(t)]≥cδ2, (c∈(0,1]).Whenδ=0, the model (2) is degenerated into the original DW model (1).By the constraint of thatxi(t) locating in [0,1], we adjust the model (2) into(3)xit+1=1x¯it>1+x¯it·1x¯it∈0,1+0·1x¯it<0,xjt+1=1x¯jt>1+x¯jt·1x¯jt∈0,1+0·1x¯jt<0,xkt+1=xkt,k≠i,j,where the selection pair St is (i,j) at time t (St=(i,j)) and(4)x¯it=xit+1xjt-xit<η12xjt-xit+ξit+1.An opinion evolution example based on the noisy DW model is shown in Figure2. Compared with Figure 1, it seems that noises promote opinion aggregations and the aggregated group will be not divided in a certain condition.Figure 2
Opinion evolution of 20 agents for (3), where n=20, η=0.2, and δ=0.01.


## 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,j∈V:(i)
limt→∞xit-xjt=0,(ii)
limt→∞xit-xjt>η.Remark 2.
Iflimt→∞xit-xjt>η, 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 limt→∞xi(t)=x∗, ∀i∈V a.s.However, for the noisy DW model, opinion consensus becomes extravagant hopes. Figure2 shows that agent opinions always fluctuate before any large termination time. We provide the following definition, where therobust consensus is the main role in this work.Definition 3.
The opinion state{xi(t),i∈V} is said to be fluctuating a.s. if P(lim¯t→∞d(t)>0)=1, where(5)dt=maxi,j∈Vxit-xjtis theopinion range at time t.Note thatd(t) also play the role of “aggregation degree” in social opinion system for one topic. Similarly, we can define dK(t)=maxi,j∈Kxit-xjt, where K is a subset of V. Next, to study opinion fluctuation strength of the noisy DW model (3), we provide the definition ofrobust consensus as follows [28].Definition 4.
Given the initial opinionsxi(0)∈[0,1] and the confidence bound η∈(0,1), if there exists T∗>0 and T>0 and (6)dt≤fδholds a.s., where T∗≤t≤T∗+T, then we call the noisy DW model T-robust consensus. Generally, for any t≥T∗, if(7)dt≤fδ holds a.s., then we call the noisy DW model achievesrobust consensus. Here f:R+→R+, f(0)=0, and f(δ) approaches 0 when δ→0.


## 4. Robust Consensus of the Noisy DW Model

In this section, we will prove therobust consensus of the noisy DW model (3).We first demonstrate how the fluctuation strengthd(t) changes as t increases (Figure 3). This example shows that when t≥4×104, d(t)<0.2=η always holds until the termination time.Figure 3
Opinion evolution of 20 agents for the noisy DW model (left) and the corresponding opinion ranged(t) (right), where n=20, η=0.2, and δ=0.01.Secondly, we will provide some lemmas for understanding how the opinions change for the noisy DW model (3).Lemma 5.
Ifd(0)≤η and δ<(1/4)η, then the noisy DW model (3) achievesT-robustconsensus.Proof.
We can prove it by the mathematical induction method.
[ 1 ] For n=2, if d(0)<η and δ<(1/4)η, then (8)d1=x11-x21≤12x10+x20-x20-x10+ξ10-ξ20≤2δ<ηholds if S0=(1,2). Similarly, d(t)<2δ holds for any t∈N if Sk=(1,2), k≤t-1. By the random of noise, {tk:d(tk)≤2δ} exists a.s. Defining f(δ)=2δ, then the conclusion holds obviously.
[ 2 ] For n=3, we assume d(0)<η and δ<(1/4)η. Without loss of generality, we set x1(0)≤x2(0)≤x3(0). If agent 1 and agent 2 are selected at time 0, then(9)x11-x31=12x10+x20-x30+ξ10≤η+δwhile x11-x21≤2δ<η and x21-x31≤η+δ. Thus, if agent 1 and agent 2 always select each other and ξ1(t)+ξ2(t)<0 for t∈N, then there exists a finite time t0 a.s., such that when t<t0, d(t)≤η. By the Borel-Cantelli (B-C) lemma [29], we can get that P{⋂t=1inf{ξ1(t)+ξ2(t)<0}}=0. Further, because agent 3 always keeps unchanged,(10)Pmax⁡x1t-x3t,x2t-x3t>η,∀t≥1=0.
Thus, there exists a finite timet1 a.s., such that d(t1)≤η and when t0≤t<t1, d(t)>η.
Similarly, by the B-C lemma, there exists a finite timet2 a.s., such that d(t2)>η and when t1≤t<t2, d(t)≤η. In a sum, there exist a stopping time sequence {T(1,2)(k)} a.s., such that d(t)≤η, t∈[T(1,2)(2s-1),T(1,2)(2s)), s∈N+. Furthermore, among the interval sequence {[T(1,2)(2s-1),T(1,2)(2s))}, by the B-C lemma and the ergodicity of [x1(t),x2(t)] or [x2(t),x1(t)], there always exists a time interval subsequence, denoted by {[T(1,2)∗(k),T(1,2)∗(k+1))}, at which x1(t)≤x3(t)≤x2(t) or x2(t)≤x3(t)≤x1(t), t∈[T(1,2)∗(k),T(1,2)∗(k+1)), k∈N. Because x2t-x1t=ξ2t-1-ξ1t-1<2δ, (11)dt+1=maxi,j∈V⁡xit+1-xjt+1=max⁡x1t+1-x2t+1,x3t+1-x2t+1,x1t+1-x3t+1≤2δ.
Denotef(δ)=2δ. Thus, the system is T-robust consensus for the event that the selection pair is always (1,2) for any time t∈N.
According to the rotational symmetry of agent selection, if agent1 and agent 3 always select each other, then the conclusion also follows. Generally, note that(12)xit-xjt=12xit-1+xjt-1-xjt-1-xit-1+ξit-1-xjt-1≤2δif St-1=(i,j) and xjt-1-xit-1<η. Thus, if d(t)=x1t-x2t<η, (1/2)(x1(t)+x2(t))-δ<x3(t)<(1/2)(x1(t)+x2(t))+δ, and St=(1,2), then d(t+1)<2δ. Note that this event happens infinite often by the B-C lemma. Therefore, there exists a time subsequence {tk} a.s. in which d(tk)<2δ. In a sum, the system is T-robust consensus for the general condition.
[ 3 ] Inductively, for any n∈N, we assume d(0)<η and δ<(1/4)η.
Firstly, if agenti and agent j select each other at time 0, then (13)xi1-xj1=12xi0+xj0-xi0-xj0+ξi0-ξj0≤ξi0+ξj0≤2δ<η;
Secondly, if agenti and agent j keep unchanged at time 0, then(14)xi1-xj1=xi0-xj0<η;
Finally, if agenti selects another agent k, k≠j, but agent j keeps unchanged at time 0, then(15)xi1-xj1=12xi0+xk0-xj0+ξi0≤η+δ. Note that agents i and k satisfy xi1-xk1=ξi0-ξk0≤2δ<(1/2)η, and then by the similar method, the system is T-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 T-robust consensus for the general case.Figure4 shows how the opinion range d(t) changes as t increases. Note that d(t)<2δ=0.1 is infinite often, t∈N.Figure 4
Opinion evolution of 20 agents for the noisy DW model (left) and the correspondingd(t) (right), where n=20, η=0.2, and δ=0.05.Besides, the ergodicity of any agent’s opinion values can be shown in Figure5. This is deduced by the randomness of the external noise {ξi(t)}, i∈V, t∈N.Figure 5
Agent1’s opinion locations in the opinion evolution process for the noisy DW model with parameters n=20, T=105, η=0.2, and δ=0.05.Remark 6.
In fact, Lemma5 shows that when d(0) is sufficiently small, the noisy DW model will be T-robust consensus. However, it is still a problem how opinions evolve if d(0)>η?In the following, we study one special condition such that there are two agent subsets, where both satisfy the condition in Lemma5 but their opinions’ distances would be larger than η. We will show that the system (3) can also achieve T-robust consensus finally.Lemma 7.
Suppose there exists a partitionV1,V2 of V, satisfying dV1(0),dV2(0)<η; then the system (3) achievesT-robustconsensus if δ<(1/4)η.Proof.
[ 1 ] For n=3, we assume V1={1,2} and V2={3}.
IfdV1,V2(0)≜maxi∈V1,j∈V2xi0-xj0≤η, then we get that the system is T-robust consensus by Lemma 5.
IfdV1,V2(0)>η, for simplicity, x30-x10,x30-x10>η, then we can get that x3(1)=x3(0)+ξ3(0) if the selection pair is (3,3) at time 0. With positive probability, (3,3) happens to be infinite often and {x3(t)} is ergodic among [0,1] a.s. Similarly, among the time interval sequence {[T(1,2)∗(k),T(1,2)∗(k+1))}, there is a finite time t0 at which d(t0)<η a.s., and then we get that the system is T-robust consensus by Lemma 5.
[ 2 ] For n>3, we fixed a partition of the agent set V. Denote i,j∈V1, suppose the time subsequence {t(i,j)(k)} at which agent i and agent j select each other and xiti,jk-1-xjti,jk-1<η. In fact, with the same method used in Lemma 5, the time subsequence exists with the probability 1.
Note that{∑s=1t‍ξi(s)} is a random walk among [0,1], i=1,2. Therefore, the agent interval [mini∈V1xi(t),maxi∈V1xi(t)] can also change randomly among [0,1]. Select i∈V1 and w∈V2, then the event {xit-xwt<η} happens i.o. by the B-C lemma. Similarly, the event that agent i and agent w select each other also happens i.o.. There also exists the time subsequence {t(i,w)(k)} a.s. such that agent i and agent w select each other and xiti,wk-1-xwti,wk-1<η. For the finiteness of the selection pair set {(i,w),i∈V1,w∈V2}, we get that there exists a time subsequence {t∗(k)} at which d(t∗(k))≤η i.o.. Consequently, we get that the system is T-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 andδ<(1/4)η, the noisy DW model (3) achievesT-robustconsensus a.s..Besides, we further get the following theorem by weakening the probability condition.Theorem 9.
For any initial states andδ<(1/4)η, the noisy DW model (3) achievesrobust consensus with a positive probability.Proof.
Ifmin{i,j∈V}xi0-xj0>η, then xi(1)=xi(0)+ξi(0) if the selection pair is (i,i),i∈V at time 0. With the similar method used in Lemma 7, we only need to consider the initial condition mini,j∈Vxi0-xj0≤η.
By Lemma7, suppose there exists a finite partition V1,V2,…,VK of V, satisfying dVi(0)<η for i=1,2,…,K, then the system (3) achieves T-robust consensus if δ<(1/4)η.
In the following we will prove that the system is also robust consensus. DenoteM=max{⌊η/2δ⌋,1} and ♯{(i,j)} as the repeated times of the selection pair (i,j) in the following proof. Furthermore, the time interval sequence of the repeated selection pair of (i,j) is denoted by {T(i,j)(k)}.
[ 1 ] For n=2, all selection pairs are (1,1), (2,2) and (1,2). In fact, as δ approaches to 0, Lemmas 5 and 7 also hold. Consider the event(16)A20=S0=1,2and  then♯1,1,♯2,2≤M.
Becausemini,j∈Vxi0-xj0≤η, in the event A2, note that x1t-x2t≤η holds for any t∈N. We get that d(t)≤2δ if the selection pair is (1,2) at time t-1 and d(t)≤2Mδ for other conditions. Thus, denote f(δ)=max⁡{2Mδ,2δ}, thus, the system achieves robust consensus in the event A20.
Similarly, define(17)A21=S0=1,2and  then♯i,i≤M+1,♯s∈Ti,iqξis<0<M,♯s∈Ti,iqξis>0<M,for♯i,i=M+1,i=1,2,q∈N.
Because♯s∈T(i,i)(q){ξi(s)>0}<M and ♯{(i,i)}=M+1, there exists at least one moment t′∈T(i,i)(k) such that ξi(t′)’s symbol is different from the ones in some other moments. Thus, agent i would not get out of the confidence range of another agent j, i=1, j=2, or i=2,j=1. Thus, with the similar method, the system achieves robust consensus in the event A21.
With a similar method, we can define an event sequence{A2k}, where (18)A2k=S0=1,2and  then♯i,i≤M+k,♯s∈Ti,iqξis<0<M,♯s∈Ti,iqξis>0<M,for♯i,i=M+k,i=1,2,q∈N.
Based on the randomness ofξi(t), we get that (19)P⋃k=0∞A2k=1-P⋂k=0∞A2kc≥1-P⋂k=0∞S0=1,2,∃♯s∈Ti,iqξis<0≥Mfor♯i,i=M+k>0.Thus, the conclusion holds when n=2.
[ 2 ] For n>2, we can generate the analysis for n=2 into n>2. Similarly, we analyse the event sequence {Ank} where (20)Ank=dT≤ηand  then♯i,i≤M+k,♯s∈Ti,iqξis<0<M,♯s∈Ti,iqξis>0<M,for♯i,i=M+k,i∈V,q∈N.
By Lemma7, we get that the event {d(t)≤η} happens i.o.; thus, d(T)≤η happens with the probability larger than 0, T<∞. Along with the same method used before, we can get that the conclusion holds for general cases.Remark 10.
The main result, Theorem8, shows that when δ is sufficiently small, the noisy DW model will achieve T-robust consensus. Furthermore, Theorem 9 tells us that the noisy DW model achievesrobust consensus with a positive probability. However, it is nearly impossible to estimate the probability P{d(T)≤η}; thus we could not provide it in Theorem 9. Besides, it is still unknown on the relation of δ and η? We denote the threshold values of δ as δ∗. By the previous result, we get that δ∗≤(1/4)η. For further study of the parameter δ∗, 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 ofδ changes as the confidence bound η 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 limt→∞d(t) changes as parameters increase. Secondly, the cluster number is studied as the noise strength δ changes. Finally, the parameter δ∗ 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 ranged(t) changes as parameters change. In Figure 6, we show how the fluctuation strength for each cluster varies as the confidence bound η and the noise strength δ change, respectively.Figure 6
Parameters aren=20, the termination time T=105. The noisy strength is δ=0.05 (a) and the confidence bound is η=0.2. (b).
(a)
Howd(T) changes as η increases (b)
Howd(T) changes as δ increasesSecondly, we simulate the cluster number and average cluster width (Figure7). 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.Figure 7
Parameters aren=20, the termination time T=105, and the confidence bound is η=0.2.
(a)
How cluster number changes asδ increases (b)
How cluster width changes asδ increases


### 5.2. Noise Strength

Then, we demonstrate the threshold values less which opinions will achieve robust consensus whent is sufficiently large. In Figure 8, we show how the threshold values of δ change when the confidence bound η 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 δ=(1/4)η. However, because of the finiteness of the termination time T, we get that the time complexity is very high when η is very small. Thus, the estimated threshold values of δ 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.Figure 8
The termination timeT=105. Dots denote the threshold experimental values, and the dotted line is the approximated linear relationship of δ and η.
(a)
Howδ∗ changes as η increases (b)
Howδ∗ changes as δ increasesFor Lemma7 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 [0,η], another in which opinions are located in [1-η,1].


## 5.1. Opinion Range

First, we show how the opinion ranged(t) changes as parameters change. In Figure 6, we show how the fluctuation strength for each cluster varies as the confidence bound η and the noise strength δ change, respectively.Figure 6
Parameters aren=20, the termination time T=105. The noisy strength is δ=0.05 (a) and the confidence bound is η=0.2. (b).
(a)
Howd(T) changes as η increases (b)
Howd(T) changes as δ increasesSecondly, we simulate the cluster number and average cluster width (Figure7). 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.Figure 7
Parameters aren=20, the termination time T=105, and the confidence bound is η=0.2.
(a)
How cluster number changes asδ increases (b)
How cluster width changes asδ increases


## 5.2. Noise Strength

Then, we demonstrate the threshold values less which opinions will achieve robust consensus whent is sufficiently large. In Figure 8, we show how the threshold values of δ change when the confidence bound η 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 δ=(1/4)η. However, because of the finiteness of the termination time T, we get that the time complexity is very high when η is very small. Thus, the estimated threshold values of δ 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.Figure 8
The termination timeT=105. Dots denote the threshold experimental values, and the dotted line is the approximated linear relationship of δ and η.
(a)
Howδ∗ changes as η increases (b)
Howδ∗ changes as δ increasesFor Lemma7 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 [0,η], another in which opinions are located in [1-η,1].


## 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 strengthδ of noises {ξi(t)} is small enough, opinions of this noisy DW model will reach T-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 achieverobust consensus with a positive probability. The definition of therobust consensus was introduced to describe the relation of opinion fluctuation and noise strength δ. 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 η, when the noisy strength δ is sufficiently small; For another, we described that the separated parts will merge together when δ is sufficiently small. Then we gave the theorem describing that the noisy DW model achieves robust consensus when δ is sufficiently small. Finally, we demonstrated threshold values of δ, 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.



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*Source: 1065451-2018-12-30.xml*