Dynamics of Uncertain Opinion Formation : An Agent-Based Simulation

Opinion formation describes the dynamics of opinions in a group of interaction agents and is a powerful tool for predicting the evolution and di usion of the opinions. The existing opinion formation studies assume that the agents express their opinions by using the exact number, i.e., the exact opinions. However, when people express their opinions, sentiments, and support emotions regarding di erent issues, such as politics, products, and events, they o en cannot provide the exact opinions but express uncertain opinions. Furthermore, due to the di erences in culture backgrounds and characters of agents, people who encounter uncertain opinions o en showdi erent uncertainty tolerances. The goal of this study is to investigate the dynamics of uncertain opinion formation in the framework of bounded confidence. By taking di erent uncertain opinions and di erent uncertainty tolerances into account, we use an agent-based simulation to investigate the influences of uncertain opinions in opinion formation from two aspects: the ratios of the agents that express uncertain opinions and the widths of the uncertain opinions, and also provide the explanations of the observations obtained.


Introduction
. Opinion formation is a powerful tool for predicting the evolutions and di usion of opinions (Afshar & Asadpour ).Opinion formation describes the dynamics of opinions in a group of interaction agents (Urbig et al. ).There are two varieties of stabilized results (i.e., consensus and clusters) in opinion formation (Hegselmann et al. ).Many opinion formation models have been proposed to discuss the conditions of forming the stabilized results. .
The study of opinion formation went back to (French Jr ).According to French's study, di erent types of studies on opinion formation have been proposed (Hegselmann et ), and (v) opinion formation considering noises (Pineda et al. , ). .
The bounded confidence model assumes that each agent solely communicates with the agents who hold similar opinions and ignores the agents that have su iciently di erent opinions.The earliest bounded confidence models have been introduced independently by Hegselmann and Krause ( ) and by De uant and Weisbuch ( ).The two bounded confidence models are called the HK model and the DW model, respectively.In the HK model, agents synchronously update their opinions by averaging all opinions in their confidence sets; in the DW model, agents follow a pairwise-sequential updating mechanism.Based on the HK and DW models, interesting extended research studies regarding the HK model and the DW model have been conducted (Fortunato et al. ; Ceragioli & Frasca ; Morarescu & Girard ).
. Previous studies have significantly advanced the bounded confidence models.In this study, we propose the dynamics of uncertain opinion formation in the framework of bounded confidence.This study is motivated by the following aspects: . In the existing studies, the agents express opinions by using the exact number, i.e., the exact opinions.However, whether the opinion formation occurs in daily life or in the context of the Internet, the opinions of the agents o en exhibit uncertainty.For example, when people express their opinions, sentiments, or support emotions regarding di erent issues, such as politics, products, and events, they o en cannot provide exact opinions, but express uncertain opinions.Generally, the numerical intervals are the most basic formats of uncertain opinions (Dong et al. ; Dong & Herrera-Viedma ).Thus, it is necessary to propose the uncertain opinion formation model, which will provide a foundation for investigating the dynamics of uncertain opinion formation.
. In the practical opinion formation problem, the agents who encountered uncertain opinions o en show di erent uncertainty tolerances.The di erences in uncertainty tolerances are close to the culture backgrounds and characters of agents (Sutton et al. ).For example, the agents with decisive or perfection seeking characters are interested in exact opinions, and they hope to communicate with the exact opinions.Thus, it is necessary to investigate the dynamics of uncertain opinion formation by considering di erent uncertainty tolerances.

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The proposal can be applied to address certain opinion formation problems in the real world such as, when the government attempts to analyse the dynamics of public opinions on introducing the chemical project, some citizens may express uncertain opinions on the necessity of introducing the chemical project.Furthermore, di erent citizens who encountered uncertain opinions show di erent uncertainty tolerances.Therefore, when the proposal incorporated the above uncertainty factors, it can provide the decision support for the government to analyse the dynamics of public opinions.
. The remainder of this study is organized as follows.Section introduces the HK bounded confidence model.Then, Section proposes the uncertain opinion formation model in the framework of bounded confidence.Next, Section discuss the influences of uncertain opinions in opinion formation.In Section , the influences of uncertain tolerances in opinion formation are investigated.Finally, concluding remarks are included in Section .

The HK Bounded Confidence Model
. In this section, we briefly introduce the HK bounded confidence model, which is also the basic model of this study.The DW model and the HK model are very similar but di er mainly in the communication regime (Urbig et al. ).Thus, if we adopt the DW model as the basic model, a similar work will be conducted.
. Consider an opinion formation problem.Let A = {A 1 , A 2 , . . ., A N } be a set of agents, and t be a discrete time, respectively.Let X(t) = (x 1 (t), x 2 (t), . . ., x N (t)) T be the opinion profile at time t, where x i (t) ∈ [0, 1] denotes the exact opinion expressed by agent A i ∈ A at time t.Let ε be the homogeneous bounded confidence of the agents.
. Let I(A i , X(t)) be the confidence set of agent A i at time t, and let w ij (t) be the weight that agent A i assigns to agent A j at time t, i.e., where I(A i , X(t)) = {A j ||x i (t)−x j (t)| ≤ ε}, and #I(A i , X(t)) is the number of agents in the set I(A i , X(t)).
( ) The Uncertain Opinion Formation Model in the Framework of Bounded Confidence .In this section, based on the original HK model, we propose the uncertain opinion formation model in the framework of bounded confidence.In the proposed uncertain opinion formation model, the agents express their opinions by either using the exact number (i.e., the exact opinions) or using the numerical intervals (i.e., the uncertain opinions Then, the second step is to determine the weights that one agent assigns to other agents.Let w ij (t) be the weight that agent A i assigns to agent A j at time t, and w ij (t) is given by where # Ĩ(A i , X(t)) is the number of agents in the set Ĩ(A i , X(t)).
. Finally, the third step is to determine the updated opinions for each agent.Due to the di erences in culture backgrounds and characters of agents, people who encounter uncertain opinions o en show di erent uncertainty tolerances.So, in this study the agents are divided into two types: the agents with the uncertainty tolerances, and the agents without the uncertainty tolerances.The agents with the uncertainty tolerances refer to the agents who can directly communicate both the exact opinions and uncertain opinions.The agents without the uncertainty tolerances refer to the agents who only communicate with exact opinions.Thus, when confronting uncertain opinions, the agents without the uncertainty tolerances will provide accurate estimations of the uncertain opinions.For notational simplicity, let A u be the set of agents with the uncertainty tolerances, and let A o be the set of agents without the uncertainty tolerances, where . Specifically, let the agent A i ∈ A u , and A j ∈ Ĩ(A i , X(t)).Then A i will directly update his/her opinion based on the opinion , the agent A i will provide the accurate estimation f ij (t) as the opinion of A j , where Notably, the accurate estimation f ij (t) in our simulation is randomly and uniformly selected from the uncertain opinion [x L j (t), xU j (t)].Next, A i will update his/her opinion based on the accurate estimation ] be as before, then In the Appendix A, we compare the HK model with the proposed model, and analyze the proposed model to illustrate certain desired properties.

The Influences of Uncertain Opinions in the Opinion Formation
. In this section, we assume that all the agents are with the uncertainty tolerances, i.e., A u = A. We then investigate the influences of uncertain opinions in the opinion formation from two aspects: ( ) the influences of the ratios of the agents expressing uncertain opinions, and ( ) the influences of the widths of uncertain opinions.
We consider four indexes in the investigation as follows: (i) The number of clusters (N C).N C is the number of di erent opinions in the stabilized results.Larger N C values indicate more di erent opinions among the agents in the stabilized results.In particular, N C = 1 represents that the opinions of agents reach a consensus.
(ii) The ratios of the extremely small clusters in all clusters (r ESC ).Extremely small cluster (ESC) is the cluster which includes a few of agents.Let S = {s 1 , s 2 , • • • , s N C } be the set of clusters in the stabilized results, where s v denotes the vth cluster, v = 1, 2, . . ., N C. Let #s v be the number of agents in the cluster s v .Based on the study (Urbig et al. ), if #s v ≤ εN 2 , then s v is an extremely small cluster.Let #ESC be the number of the extremely small clusters in the stabilized results, and let r ESC = #ESC N C .(iii) The ratios of the agents expressing the uncertain opinions in the stabilized results (r s ).Assume that the stabilized results are formed at time t.
. ., N } be the set of the agents expressing the uncertain opinions at time t.And let #U O be the number of agents in the set U O, and let r s = #U O N .(iv) The average widths of uncertain opinions in the stabilized results (W U ). Assume that the stabilized results are formed at time t.Let U O and #U O be as before.Then, W U is determined by Before investigating the influences noted above, we define certain variables in the simulation.Let r ∈ [0, 1] be the ratio of the agents expressing uncertain opinions.Let λ ∈ [0, 1] be the maximum width among all of the initial opinions, i.e., λ = max{x U i (0) − x L i (0), i = 1, 2, . . ., N }.Let λ ∈ [0, 1] be the average widths among all of the initial opinions, i.e., λ The influences of the ratios of the agents expressing uncertain opinions .
We investigate the influences of the ratios of the agents expressing uncertain opinions based on three criteria N C, r ESC and r s .In the simulation, let N = 500 and λ = 1.The initial opinions X(0) of N agents are randomly and uniformly generated.Specifically, without loss of generality, we assume that the former N × r agents A i (i = 1, 2, . . ., N × r) express uncertain opinions, and the latter N × (1 − r) agents (i = N × r + 1, N × r + 2, . . ., N ) express the exact opinions.Then, for the former N × r agents A i , their uncertain opinions xi (0) (i = 1, 2, . . ., N × r) are the subintervals that are randomly selected from [ , ], and for the latter  In the simulation, we find that with the increase in the ratios of the agents expressing the uncertain opinions and the decrease in the bounded confidences, the interactions among the agents will decrease.As a result, more clusters will appear.
. In the simulation, we find that the number of the isolated opinions at t = 0 will increase with the increase in the ratios of the agents expressing the uncertain opinions and the decrease in the bounded confidences.Consequently, the larger ratios of the extremely small clusters in all clusters will appear.Figure shows two findings: (i) The average r s values increase as both r and ε increase, and (ii) The average r s values are larger than the r values.This implies that with the increase in the ratios of the agents expressing the uncertain opinions and the bounded confidences, the ratios of the agents expressing the uncertain opinions in the stabilized result will increase.Meanwhile, the ratios of the agents expressing the uncertain opinions in the stabilized result will be larger than those in the initial time.These two observations can be explained as follows: It is clear that the larger ratios of the agents expressing the uncertain opinions in the initial time will lead to the larger ratios of the agents expressing uncertain opinions in the stabilized result.Meanwhile, when interacting with the uncertain opinions, the exact opinions in the initial time will gradually become uncertain.Consequently, the larger ratios of the agents expressing the uncertain opinions in the stabilized result will be yielded.

The influences of the widths of uncertain opinions
. We investigate the influences of the width of uncertain opinions based on three criteria N C, r ESC and W U .
In the simulation, let N = 500 and r = 1.We uniformly and randomly generate N exact numbers y i (0) (i = 1, 2, . . ., N ) in [ , ].Based on the generated exact number, we randomly generate the uncertain opinions xi (0) = [x L i (0), xU i (0)] (i = 1, 2, . . ., N ), where xL i (0) = max{0, y i (0) − δ i /2} and xU i (0) = min{y i (0) + δ i /2, 1}, and δ i (i = 1, 2, . . ., N ) is the exact number that is randomly selected from [ ,λ].Clearly, the widths of all of the generated uncertain opinions are smaller than λ.Furthermore, the larger λ values indicate that all of the generated opinions are with the larger average widths.Next, using Eqs.( ) and ( ) proceeds with the evolution of opinions.We set di erent λ and ε values and run each simulation times, obtaining the average N C, r ESC and W U values under di erent parameters, which are shown in Figures -.

Figure shows the following finding:
The average r ESC values increase as λ increases.This implies that the ratios of the extremely small clusters in all clusters will increase with the increase in the widths of uncertain opinions.The findings observed in Figures and can be explained as follows: In the simulation, we find that with the increase in the widths of uncertain opinions, the interactions among the agents will decrease, and the  The average WU values  =0.05  =0.08  =0.11 =0.14  =0.17  =0.2 '

Figure :
The average W U values under di erent λ and ε values W U values are smaller than the λ values.This implies that the widths of uncertain opinions in the stabilized results will increase with the increase in the widths of uncertain opinions in the initial time.Meanwhile, the average widths of uncertain opinions in the stabilized result will be smaller than those in the initial time.It is clear that the larger widths of uncertain opinions in the initial time will lead to the larger widths of uncertain opinions in the stabilized result.Meanwhile, in the simulation, we find that the number of uncertain opinions with the smaller widths will increase in the dynamics of uncertain opinion formation.Consequently, the average widths of uncertain opinions in the stabilized result will become smaller.

The Influences of the Ratios of the Agents with the Uncertainty Tolerances
. We investigate the influences of the ratios of the agents with the uncertainty tolerances based on four criteria (i.e., N C, r ESC , r s and W U ) mentioned in section . .Before investigating these influences, we denote the ratio of the agents with the uncertainty tolerances as β, where , the agent A i will provide the accurate estimation f ij (t) as the opinion of A j , where Notably, the accurate estimation f ij (t) in our simulation is randomly and uniformly selected from the uncertain opinion In the simulation, we randomly selected N ×β agents from A, and assume that these selected N ×β agents are with the uncertainty tolerances, i.e., the selected N ×β agents belong to the set A u and the other agents belong to the set A o .If the agent A i ∈ A o and A j ∈ Ĩ(A i , X(t)), the agent A i will provide the accurate estimation f ij (t) as the opinion of A j , where f ij (t) is randomly and uniformly generated from   In the simulation, when the ratios of the agent expressing uncertain opinions or the widths of uncertain opinions are su iciently large, we find that the number of agents in the confidence sets will increase with the increase in the agents with the uncertainty tolerances.Consequently, less clusters will appear. .

Figures (a)-(b)
show the following finding: When r ≥ 0.7 or λ ≥ 0.4, the average r ESC values decrease as β increases.This implies that when the ratios of the agents expressing uncertain opinions or the widths of uncertain opinions are su iciently large, the ratios of the extremely small clusters in all clusters will decrease with the increase in the ratios of the agents with the uncertainty tolerances.The finding in Figure (a)-(b) can be explained as follows: In the simulation, when the ratios of the agent expressing uncertain opinions or the widths of uncertain opinions are su iciently large, we find that the number of isolated opinions in the dynamics of uncertain opinion formation will decrease with the increase in the ratios of the agents with the uncertainty tolerances.In the simulation, we find that the opinions of the agents without the uncertainty tolerances will gradually become more accurate in the dynamics of uncertain opinion formation.Thus, with the increase in the ratios of the agents with the uncertainty tolerances, the larger ratios of the agents expressing uncertain opinions and the larger average widths of uncertain opinions in the stabilized result will appear.

Conclusions
. In this study, we investigate the dynamics of uncertain opinion formation based on the bounded confidence model.In the proposed model, the agents express their opinions by using either numerical intervals (i.e., the uncertain opinions) or exact numbers (i.e., the exact opinions).Furthermore, based on di erent communication regimes, the agents are divided into two types: the agents with uncertainty tolerances and the agents without uncertainty tolerances.
. We use an agent-based simulation to obtain the following findings: (i) With the increase in the ratios of the agents expressing uncertain opinions and the widths of uncertain opinions, more clusters and larger ratios of the extremely small clusters in all clusters will appear.Meanwhile, when the ratios of the agents expressing uncertain opinions or the widths of uncertain opinions are su iciently large, less clusters and smaller ratios of the extremely small clusters in all clusters will appear with the increase in the ratios of the agents with the uncertainty tolerances; (ii) When all the agents are with the uncertainty tolerances, the ratios of the agents expressing the uncertain opinions in the stabilized result will be larger than those in the initial time.But the average widths of uncertain opinions in the stabilized result will be smaller than those in the initial time; (iii) When there exist certain agents without the uncertainty tolerances, both the ratios of the agents expressing the uncertain opinions and the average widths of uncertain opinions in the stabilized results will be smaller than those in the initial time.
. The proposed model can be applied to address certain opinion formation problems in the real world.For example, when the government attempts to analyse the dynamics of public opinions on introducing a chemical project, certain citizens may express uncertain opinions on the necessity of introducing the chemical project.Furthermore, di erent citizens who encountered uncertain opinions have di erent uncertainty tolerances.
. Generally, people express their opinions, sentiments, or support emotions regarding di erent issues in a social network.However, in this paper, the influences of di erent social network structures on uncertain opinion formation are not considered.Therefore, it would be an interesting future topic to investigate the dynamics of uncertain opinion formation by considering di erent social network structures.
Example is used to illustrate that (iii) is not satisfied in the proposed model.In Example , there are four agents A 1 , A 2 , A 3 and A 4 , where A 1 , A 2 , A 3 , A 4 ∈ A o .The agents' initial opinions are given by and the bounded confidence is assumed as: ε = 0.2.
Based on Eq. ( ), we find that two isolated fully connected groups exist that are formed at time t = 0. Specifically, agents A 1 and A 2 formed one isolated fully connected group, and agents A 3 and A 4 formed the other isolated fully connected group.
The initial accurate estimations of A 1 , A 2 , A 3 and A 4 are given as follows:  Assume that x 1 (t ) = x 2 (t ) = 0.2425 and x 3 (t ) = x 4 (t ) = 0.0575, Because ε = 0.2 and d(x i (t ), xj (t )) ≤ ε, for i, j = 1, 2, 3, 4 and i = j, then the four agents A 1 , A 2 , A 3 and A 4 formed an isolated fully connected group at time t .
Then, two desired properties in the proposed model are discussed.The first property (see Property ) indicates that the opinions of the agents without uncertainty tolerances will gradually become more accurate in the dynamics of uncertain opinion formation.The second property (see Property ) indicates that all of the agents will hold an exact opinion when a consensus among the agents is achieved.
N ) are the exact numbers that are randomly selected from [ , ]. Next, using Eqs.( )-( ) proceeds with the evolution of opinions.We set di erent r and ε values and run the simulation times, obtaining the average N C, r ESC and r s values under di erent parameters, which are shown in Figures -..
Figure shows two findings: (i) The average N C values increase as r increases and (ii) the average N C values decrease as ε increases.This implies that the larger ratios of the agents expressing the uncertain opinions will lead to more clusters.The opposite results will be obtained with the increase in the bounded confidences.These two observations from Figure can be explained as follows:

Figure : Figure :
Figure : The average N C values under di erent r and ε values

Figure :
Figure : The average r s values under di erent r and ε values Figure shows the following finding: The average N C values increase as λ increases.This implies that the number of clusters will increase with the increase in the widths of uncertain opinions.

Figure :
Figure : The average N C values under di erent λ and ε values

Figure :
Figure : The average r ESC values under di erent λ and ε values [x L j (t), xU j (t)].Next, we use the methods in sections .and .to generate the initial opinions X(0), and use Eqs.( )-( ) to calculate the N C, r ESC , r s and W U values.Finally, we set ε = 0.1 and di erent β, r and λ values, and run the simulation times, obtaining the average N C, r ESC , r s and W U values under di erent parameters, which are shown in Figures -.

Figure
Figure : (a) The average N C values under di erent β and r values; (b) The average N C values under di erent β and λ values Figure : (a) The average r s values under di erent β and r values; (b) The average W U values under di erent β and λ values
The two desired properties are provided as follows:Property .Let xi (t) = [x L i (t), xU i (t)] be as defined previously, and let d i (t) be the width of the opinion xi (t) , where d i (t) = xU Without loss of generality, we assume that A i ∈ A o .Based on Eqs. ( )-( ), if A i expresses the initial opinions using an exact number, A i will continue expressing the exact opinions at any time.If A i expresses the uncertain opinion [x L Aj ∈ Ĩ(Ai, X(t)),j =i w ij (t)f ij (t)−w ii (t)x L i (t)− Aj ∈ Ĩ(Ai, X(t)),j =i w ij (t)f ij (t) = w ii (t)(x U i (t) − xL i (t)).Because 0 ≤ w ii (t) ≤ 1, then xU i (t + 1) − xL i (t + 1) ≤ xU i (t) − xL i (t).Clearly, d i (t + 1) ≤ d i (t).This completes the proof.Property.Let xi (t) = [x L i (t), xU i (t)] and d i (t) be as defined previously.Assume that a consensus among the agents is achieved at time t.