Improved Finite Time in Eliminating Disagreement of Opinion Dynamics via Noise

We aim to refine the estimation of the finite stopping time when the disagreement in an opinion group is eliminated by a simple but novel noise intervened strategy. It has been proved that, by using this noise intervened control strategy, the divisive opinions would get synchronized in finite time. Moreover, the finite stopping time when resolving the disagreement has been clarified. The estimation of the finite stopping time will effectively reveal which factors and how they determine the consequence of intervention. However, the upper bound for the estimation of the integrable stopping time when noise is oriented has been quite conservative. In this paper, we investigate the finite stopping time of eliminating the disagreement by completely oriented noise and a much more precise formula for the estimation of the finite stopping time is obtained finally via direct calculation.


Introduction
In the recent decades, modeling and analysis of opinion dynamics are becoming a much attractive area.More and more researchers from various fields have paid attention to it [1,2], and several agent-based mathematical models have been established to investigate the evolution of opinion dynamics [3][4][5][6][7][8][9][10][11][12][13].In these models, the evolution of the opinions can be determined by the interaction topology of agents which are based on the preassumed graph or the bounded confidence (BC) of the agents.Using these models, the complex evolution of opinion dynamics was effectively studied and plenty of opinion behaviors were revealed, such as the basic agreement or disagreement of the opinions and the final structure of the opinion system.
One of the emerging fields in studying the opinion dynamics is how the noise influences the evolution of the social opinions.It is known that the noise is ubiquitous in opinion dynamics and intervenes during the evolution of one's opinion from a wide range of aspects, such as information flow from free media of broadcast, newspaper, TV, and so on, the contaminated transmission channels, or even one's free will.The investigations were first carried out by some simulation methods in [14][15][16][17], and with both continuous and discrete opinion models some interesting phenomena were discovered, among which a focus is that the noise in some situations could play a positive role in enhancing the consensus of opinions.Actually, the similar positive role noise plays has been found in various areas in the past decades [18][19][20][21][22].In opinion dynamics, the complete theoretical analysis of a continuous noisy opinion dynamics was first established in [23] very recently.In [23], based on the well-known Hegselmann-Krause (HK) confidence model, it was strictly proved that the noise could almost surely induce the opinions to achieve quasi-consensus in finite time, and also the critical noise strength is obtained.These results not only reveal the richer mechanisms of the opinion evolution, but also provide new insights into designing noise intervention strategy to eliminate the disagreement in a group.
Soon after, inspired by the achievements in [23], a very simple but practical noise intervention strategy was designed in [24].In their scheme, very weak noises uniformly distributed on [− 1 ,  2 ] (0 <  1 ≤  2 ) were persistently introduced to only one agent in a divisive group described by HK dynamics, and it was strictly proved that the disagreement in the group was eliminated in finite time.In 2 Advances in Mathematical Physics particular, the finite time when the disagreement got resolved was firstly estimated in this paper.The time is actually a finite stopping time, and its expectation was shown to be infinite when the noise is neutral (i.e.,  1 =  2 ).When the noise is oriented, that is,  1 <  2 , the expectation of the stopping time was shown to be finite and a upper bound was obtained.The formula of the estimation of the integrable stopping time when the noise is oriented embeds all the factors that could determine the effectiveness of noise intervention.In this sense, calculating the stopping time possesses its own significance in investigating the design of noise intervention strategy.However, the estimation of the upper bound is quite a bit conservative in [23].Consequently, a natural concern in the following is whether this conservative upper bound can be modified, at least for some special noise case.
In this paper, we will follow the path in [24] and further directly calculate the stopping time when the disagreement is eliminated under the case of completely oriented noise, that is,  1 = 0,  2 > 0, where indeed a much more precise formula is obtained.To be specific, while the expectation of finite stopping time under completely oriented noise is (2/)( * 2 −  * 1 ) + 2 using the formula obtained in [24], it is refined in this paper to be (/2)( * 2 −  * 1 ) − (/2) + 2 + , where  is the size of the group and the other parameters are fixed and nonnegative with  * 2 −  * 1 > .Since an at least decrease of (2 + /) is obtained, the stopping time is much refined here, especially when the group size is large.
The rest of the paper is organized as follows: Section 2 presents the model and the formulation, Section 3 gives the main results of the paper, and Section 4 shows some simulations, while some concluding remarks are given in Section 5.

Basic Model and Preliminaries
In this part, we will introduce the basic divisive opinion model in line with that in [24].It is known that the disagreement phenomenon of opinion dynamics can be well characterized by the HK dynamics: where  is the size of the group with V = {1, . . ., } and () is opinion values at time  which takes values in (−∞, ∞)  .Further, is the neighbor set of  and  > 0 represents the confidence threshold of the agents.Here, | ⋅ | can be the cardinal number of a set or the absolute value of a real number accordingly.
The noise-free HK model (1) has proved that the opinion will reach static state in finite time, with agreement or disagreement status.
Proposition 1 (see [9]).For every 1 ≤  ≤  of ( 1 If  *  =  *  for all , , the opinions reach consensus and the group realizes agreement, or the opinions get separated and the disagreement occurs otherwise. To describe the divisive opinion system,   distinct standpoints  *  (1 ≤  ≤   ) are assumed in the divisive system in [24].Here, for simplicity, we suppose   = 2 and the bidivisive system can be described as Following the intervention scheme in [24] where noise intervention is added to agent 1, the noise intervention model of ( 1)-( 3) here is also, for  ∈ V, where the noises {(),  ≥ 1} are independent and uniformly distributed on a completely oriented interval and  {⋅} is the indicator function with  {=1} = 1 when  = 1 and  {=1} = 0 otherwise.Moreover, to describe the behavior of model ( 2)-( 5) conveniently, a definition of consensus in the noisy case is introduced.
With the above preliminaries, the -consensus of system (2)-( 5) and the estimation of the finite stopping time were obtained in [24].Proposition 3. Define  = inf{ :  V () ≤ } and let  ≤ / 2; then { < ∞} = 1 and the system will reach -consensus at  + 1.Moreover, It can be found from (7) that the upper bound of the stopping time is determined by the group size , the noise strength , and the initial opinion difference  * 2 −  * 1 .Though (7) gives a quite concise estimation for the stopping time , the main results displayed in the next section will show that this upper bound can be refined largely.

Main Results
To be straightforward, we first give the main result.Compared to (7), we will see that the estimation of E is much refined here.In [23], E was estimated using Wald's equation.In Theorem 4, we estimate E by directly calculating the expectation of some stopping times, and the upper bound decreases by [( * 2 − * 1 )(2/−/2)+2+/2]− 2 − .Since  ≤ /2, a more detailed examination can obtain that the decrease is at least  2 , which is a much huge improvement, especially when the social group size  is large.
In Theorem 4, the noise strength is still taken as  ≤ /2 as in Proposition 3.However, this does not mean the noise intervention fails to work when noise strength exceeds /2.Actually, the restriction of noise strength bound only facilitates the proof of Proposition 3, and simulations show that when noise strength exceeds /2, the disagreement can still be eliminated (see Section 4).
To prove Theorem 4, some lemmas are needed.

Simulations
In this part, we will present some simulation results to show the how the opinions reach consensus under the driven of noise.Take  = 20,  = 1.According to Proposition 3, take noise strength  = 0.02 < 0.025; then Figure 1 shows that the divisive opinions reach consensus under the drive of noise.The time scale here obviously decreases compared to that in [24].Second, we take the noise strength  = 0.04, which exceeds the upper bound 0.025; then Figure 2 shows that the disagreement can also be eliminated, and the consensus time gets interestingly shorter.In addition, though we only analyze the divisive systems with only two subgroups, we provide simulation result of three subgroups to further show how the separated subgroups get seriatim merged (see Figure 3).

Conclusions
In this paper, we mainly aimed at analyzing the finite stopping time when the disagreement was eliminated by a simple noise intervention strategy.Compared to the results obtained in the previous studies, our result in this paper largely gets refined.The analysis method used in this paper would provide another tool for subtly investigating the stopping time when the system gets consensus and is expected to get further developed in the near future.