Opinion formation with time-varying bounded confidence

When individuals in social groups communicate with one another and are under the influence of neighbors’ opinions, they typically revise their own opinions to adapt to such peer opinions. The individual threshold of bounded confidence will thus be affected by both a change in individual confidence and by neighbor influence. Individuals thus update their own opinions with new bounded confidence, while their updated opinions also influence their neighbors’ opinions. Based on this reasoned factual assumption, we propose an opinion dynamics model with time-varying bounded confidence. A directed network is formed by the rule of the individual bounded confidence threshold. The threshold of individual bounded confidence involves both confidence variation and the in/out degree of the individual node. When the confidence variation is greater, an individual’s confidence in persisting in his own opinion in interactions is weaker, and the individual is more likely to adopt neighbors’ opinions. In networks, the in/out degree is determined by individual neighbors. Our main research involves the process of opinion evolution and the basic laws of opinion cluster formation. Group opinions converge exponentially to consensus with stable neighbors. An individual opinion evolution is determined by the average neighbor opinion effect strength. We also explore the conditions involved in forming a stable neighbor relationship and the influence of the confidence variation in the convergence of the threshold of bounded confidence. The results show that the influence on opinion evolution is greater with increased confidence variation.


Introduction
People often communicate their daily concerns with others. When an individual's opinion differs from that of others, he is likely to heuristically revise his opinion to adapt to his peer's behavior, with some changes in his confidence [1,2]. In social behavior, individual opinion interactions depend on various social or psychological factors, such as personality, trust level, reputation, social status, and persuasive abilities [3,4]. Research has shown that opinion and confidence are related at the outset; when most of an individual's peers have similar opinions, that individual may choose to compromise his opinion [3]. However, when those same peers have opinions that are similar to the individual's opinions, it can boost an individual's a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 'confidence variation'. The agent determines his new threshold of bounded confidence based on this confidence variation and the in/out degree of neighbor nodes; his new threshold of bounded confidence will also impact his neighbor's opinions in subsequent interactions. Thus, the threshold of bounded confidence is time-varying. The agent's opinion undergoes a unidirectional change, i.e., the agent will revise his opinion to become closer to his peer's opinion when his threshold of bounded confidence is less than his peer's threshold; however, his peer will remain unaffected unless his peer's own threshold is less than the agent's threshold. The lower confidence level makes the individual insist on his opinion; therefore, his peers' opinions will affect him less [3]. In this paper, we investigate the process of opinion evolution and the basic law of opinion formation.
The remainder of this paper is organized as follows. We begin with a definition of our model in Section 2. An analysis of opinion cluster formation and evolution is described in Section 3 and is followed by the experimental design and simulation results in Section 4.A conclusion is presented in the last section.

The time-varying bounded confidence model
In the Deffuant model [8], simple opinion dynamics were proposed for the fully connected graph in which all nodes were interconnected. Here, we apply the idea of neighbors connected by bidirectional or unidirectional links. A directed network G(t) = (V(t),(E(t)), V(t) consists of the nodes set at time t, and E(t) indicates the edges set at time t. In i,j 2 V(t), node(agent) j is the neighbor of node i only if there is a link from j to i. Therefore a t ij ¼ 1, a t ij 2 A þ ðtÞ and A + (t) is the in-degree adjacent matrix of G(t); otherwise, a t ij ¼ 0. Meanwhile b t ji ¼ 1, b t ji 2 A À ðtÞ, and A − (t) is the out-degree adjacent matrix of G(t); otherwise, b t ji ¼ 0. The agents in a network have an opinion space that is denoted by X = [0,1] for some social issues, and the opinion of Agenti at time t is x i (t) 2 X. The bounded confidence rule for Agentsi and j is kx i (t) − x j (t)k ε i (t), and the function of ε i (t) is: When t = 0, ε i (t) = ε i (0) and ε i (0) is the initial threshold value of bounded confidencewhich is a critical value that indicates how susceptible an agent is to the influence of the opinion of other agents -each agent has his own value at the beginning, and the confidence level for each agent is a variable in the [0.0,1.0]range with an initial value of 1. δ(t) = min{δ(t − 1) + Δδ Ã t, 1} is the confidence variation, and Δδ is the confidence variation increment, 0 < Δδ < 1.
is related by the in/out degree of Agenti and is used to express the influence of neighbors on Agenti's opinion, and t > 0. If the in-degree k þ i ðt À 1Þ ¼ P j a tÀ 1 ij is larger than the out-degree k À i ðt À 1Þ ¼ P j b tÀ 1 ij , thenthe confidence level of Agenti is relatively high, and the agent's opinion depends much more on his neighbors. By contrast, when k À i ðt À 1Þ is greater than k þ i ðt À 1Þ, Agenti has a lower confidence level. If k À i ðt À 1Þ and k þ i ðt À 1Þ are balanced, Agenti is open, and his confidence is not easily changed; thus the influence of his neighbors is not essential to his opinion. In this instance, the threshold of bounded confidence is time-varying because the agent changes his confidence following exposure to neighbors' opinions.
After each round of opinion interactions, the agent will revise his opinion using certain rules to update his neighbors based on his confidence level, and he will weigh the opinions of his neighbors, on average, against his own opinion at the next interaction [3,10,20]. However, we believe that it is more reasonable to consider an individual as a rational agent who can stick to his opinion at first and is thereafter influenced by his peers, such that the more closed the relationships are, the more influence they have in the evolution of the individual's opinion. The rules of opinion evolution for each agent are provided in formula (2).The opinion of Agenti at the t time step is where ω ij 2 W is the weighted relationship between Agentj and Agenti; the weighted spaceis W = [0,0.5], which indicates that the opinion of each agent will be affected half as strongly by his neighbors for the most part; and ω ij [x j (t − 1) − x i (t − 1)] is the effect strength of neighborAgent j to Agenti. M tÀ 1 i is the number of neighbors for Agenti at the t − 1 time step. At the t time step, Agent i's opinion depends on his own opinion and the average effect strength of his neighbor's opinion at the t-1time step.

Neighbor relationship for opinion formation
The neighbor relationship is important to opinion formation. For example, in a seminar in which participants exchange opinions with one another and reach an agreement, this neighbor relationship is bidirectional. In the case of media information regarding social issues that influences people's opinions, the neighbor relationship is unidirectional. In a political election, different parties have different political opinions, and most of these opinions are in opposition to one another. To explore the influence of the neighbor relationship on opinion formation with time-varying bounded confidence, a definition of the neighbor relationship is given for Agentsi and j in G(t) as follows.

Bidirectional neighbors: For Agenti and Agentj
, Agenti and Agentj are bidirectional neighbors of one another. All bidirectional neighbors are denoted as N $ (t) = {i,j 2 V(t): (i,j) 2 E(t),(j,i) 2 E(t)}. All nodes in N $ (t) are connected fully and form a strongly connected network.

Unidirectional neighbors: For Agenti and Agentj
Agentj is the neighbor of Agenti, but Agenti is not the neighbor of Agentj. All unidirectional neighbors are denoted as

Non-neighbors: For Agenti and Agentj
Agentsi and Agentj are non-neighbors of one another. All the non-neighbors are denoted by 4. Stable neighbor relationship: At the t time step, Agenti and Agentj are neighbors ofone another whose relationship is defined by one of the above definitions. If at the t + n time step, the relationship between Agenti and Agent j has not changed and continues for any t + n + l time step, and n ! 0,l > 1 then this is a stable neighbor relationship.
As the neighbor relationship plays an important role in opinion evolution, the law of opinion evolution must be further explicated.

Discussion
In the HK model, the number of the final opinion cluster depends on the homogeneous confidence level [20,22]. However, what are the main factors in our model regarding opinion evolution? What is the influence of the neighbor relationship in opinion formation?

Impacts of neighbor relationship
When individuals land in a new human social environment, they will most likely communicate with others who have similar characteristics and interests. As exchanges deepen, they form stable relationship networks.
If there is a stable neighbor relationship at the t + n time step, the opinion of Agenti at time t + n + 1 is redefined based on formula (2) as follows: where M is the number of neighbors and N is the set of neighbors of Agenti. Because of the stable neighbor relationship, M and N do not change with time.
When opinion evolution occurs at the t + n time step and there is a stable bidirectional neighbor relationship, we use Theorem 1.
Theorem 1: At the t + n time step, N $ (t) is set for any Agenti and Agentj who are bidirectional neighbors; then, at the t + n + l time step, the difference of their opinion value is (1 − Θ) l times in the t + n time step, and Y ¼ 1 (3), we deduce the differences in opinion for Agenti and Agent j (who are bidirectional neighbors) as follows: Thus, at the t + n + l time step, Agentsi and j are also bidirectional neighbors, and the difference of their opinion value is (1 − Θ) l times in the t + n time step.
The evolutionary trend of an opinion cluster formed by the nodes' opinions belongs to V l (t) in N (t) and is related to the overall effect strength of neighbors' opinions S t . If S t > 0, it will move ahead to a higher interval; if S t < 0, it will move ahead to a lower interval; and if S t = 0, it moves ahead. Moreover, if there is sufficient time, the opinion cluster will converge to consensus.
Based on the formula (3), there is: , and the trend of the opinion cluster consisting of V 1 (t) in N (t) is related to S t ; lim n!1 X = 0 when n is long enough, and the opinion in N (t) will be the same and a consensus will be reached.

Analysis of Δδ forthe time-varying threshold of bounded confidence
Individuals have different confidence levels and will thus choose to trust some individuals and distrust others; as the trusted persons are grouped together to form a group, they will have similar confidence levels.
For all agents, there is an Agenti whose trust threshold value is ε i (t + n) at the t + n time step. Based on formula (1), we have Theorem 3: At time t, a set N(t) with M nodes and i 2 N(t) exists, and the confidence variation increment of Agenti is 0 < Δδ 1; at time t + n, Agenti and his neighbor have a stable neighbor relationship. The in and out degrees are k + and k − for all nodes in set N(t), such that K ¼ k þ k þ þk À . When K < ε i (0), the time-varying threshold of bounded confidence for Agenti is decreased progressively at δΔε i . When K > ε i (0), the time-varying threshold is increased progressively at δΔε i . When K = ε i (0), the time-varying threshold does not change overtime. Thus, at this juncture, For any agent at the t + n + 1 time step, we can infer as follows based on formula (4): If 0 < Δδ 1, the change trend of [ε i (t + n + 1) − ε i (t + n)] depends on Δε i . Additionally, the total difference of time-varying bounded confidence is X i ½ε i ðt þ n þ 1Þ À ε i ðt þ nÞ ¼ À Dd

Results
Experimental hypotheses on the initial conditions are provided to better investigate the dynamic behavior of an opinion cluster in simulation and instanceanalysis.

Experimental hypothesis
1. There are200 agents in total, and each is coded for identification purposes from 1 to 200.
2. An agent's opinion and his thresholds of bounded confidence obey a uniform distribution of [0,1.0], where the weight factor ω satisfies the uniform distribution of [0, 0.5].
3. There is no isolated node in the network, and all agents are willing to exchange opinions.
5. An opinion evolution period is called a time step in each experiment.
Next, we will reveal how opinion clusters are formed with experimental methods.

Experiment result
Opinion evolution is simulated in SWARM programming [33]. Fig 1 shows the evolution of opinions, and the confidence variation plays a prominent role under the four conditions, which is shown in the figures. Fig 1 shows the opinion evolution over 1,000 time steps in Conditions I-IV. The times differ for opinion cluster formation, and the interval distribution of the largest opinion cluster changes from high to low due to different confidence variations. Condition IV changes dramatically.
The largest opinion clusters in the four conditions change downwards under the influence ofthe effect strength of neighbors' opinions. To investigate the influence of the neighbor relationship on the agent's opinion, we choose Agent117(the agent identified as 117), who has been observed in the biggest opinion cluster in Conditions I-IV in the experiments.
We analyze the effect strength of his neighbors' opinions in Conditions I-IV, which is shown in Fig 2A; all the lines in the figure first increase sharply, then decrease to less than zero, and finally increase slowly until they are steady at zero. The special situation is studied in Fig  2B for Condition III, where the neighbor relationship of Agent 117 is steady at the seventh time step, which is known from his in-degree and out-degree adjacent matrixes. However, at the 16th time step, his opinion changes to the max value, meeting in the average neighbor opinion value, and the effect strength of his neighbors' opinions is more than zero at this time step. Then, his opinion decreases gradually along with his neighbors' opinions effect strength to less than zero and converges over 10,000 time steps.  We find that there are different neighbor relationship sets in-degree and out-degree adjacent matrixes, such as bidirectional neighbors sets N 7 i$j ¼ f19; 46; 72; 103; 140; 191; 196g, N 7 i$j ¼ f29; 42; 109; 144g, N 1 i$j ¼ f35g, and N 186 i$j ; the nonneighbors set N 6 i$j ¼ f6g; and one unidirectional neighbors set N 186þ The sign V represents the set of all nodes, the superscript of the set N is the label (which is referred to here as the number of nodes), and the subscript is the relationship between agents. The numbers in sets serve to identify the agents. And we have got the sum of effect strength of opinions of agent 117's neighbours in 1000 time steps from the data source as follows, and please see data file [S1 Dataset] for the specific calculation: 117;1000 þ S 4 117;1000 þ S 1 117;1000 þ S 186 117;1000 ¼ À 0:001036445 The opinion values of agents in set N 186 i$j change less because their neighbors with small opinion values have a greater effect regarding the strength of their opinions, such that the largest opinion cluster changes downwards.
Based on Theorem 2, we confirm that a group of random data must exist regarding opinions as well as a threshold and weight factor for those agents that make the opinion cluster move ahead to a higher level. Therefore, we simulate them in SWARM, and the results are as follows. Fig 3 shows that the largest opinion cluster moves ahead to a higher level with the influence of neighbor effect strength on other random data for the opinion of agents. Fig 4 displays the changes in the threshold of bounded confidence in Conditions I-IV, In Conditions II-IV, the confidence level converges in different groups, and the convergent value is the average value of the group, i.e., when they have the same threshold of bounded confidence, they have a stable relationship and are in the same group, which is consistent with the results and theoretical analysis embodied in Theorem 3. Fig 4 presents the thresholds of bounded confidence changes in the four conditions; in Condition IV, the best cluster of the trust threshold value is larger than the others.
When the confidence variation is larger, we find that the largest opinion cluster moves faster from high to low and the number of opinion clusters decreases.
Definition2: For all agents in a group who change their opinion, the ultimate opinion values are called a group consensus opinion.
To analyze how a group consensus opinion forms and the number of consensus opinions in a group, we demonstrate convergence over time in Fig 5. The process of convergence is shown in 5,000 time steps and at the beginning of convergence. Those agents with similar opinions in the group begin to share the same opinions as their neighbors; the number of opinions are 14,5,5, and 2 in the four conditions at 5000 time step. However, in Condition I, the number 14 will be decreased in theory over longer time steps, and it is more difficult to converge to the consensus opinion in Condition I than in the other conditions. The confidence variation is the main reason to accelerate convergence, and when the confidence variation is larger, convergence is more rapid. However, there are certain phenomena that are not initially clear between Conditions II and III. In Condition II, it converges a little faster and earlier than is the case in Condition III. We explain this phenomenon in Fig 5B, the opinion sample variance, where the degree of sample variance in the Condition II is shown to be lower than in Condition III; we infer that this sample variance is the primary reason for the different convergences in Condition II.
A special case analysis of Δδ = 1 Δδ = 1 is a special case because it only concernsthe network structure. For , the in/out degree of the real network is crucial for an agent's opinion. Ifwe suppose that the in and out degrees are 1, then we have the following experimental results. Fig 6A and 6B show the opinion evolution and confidence level overtime. The trust threshold value is always 0.5,all agents are neighbors at the outset, and opinion clusters and the group consensus opinion both converge to 0.5, as shown in Fig 6C. Theorem 3 shows that if  Time-varying opinion formation it will converge to the consensus opinion. Because ε i (0) = 0.5 at the beginning, then k þ i ðtÀ 1Þ k þ i ðtÀ 1Þþk À i ðtÀ 1Þ ¼ 0:5, and the network is connected completely which is also known form the adjacent matrixes. Fig 6D shows ; when there are more increments for Δδ, the agent's opinion has greater influence.
The real network structure plays an important role in opinion evolution, although this structure will be explored in another paper.

Conclusion
We advance an opinion formation model with time-varying bounded confidence that is based on how people communicate in daily life; the model involves confidence variation and the degree of neighbor node. We provided the mathematical definition of the model and figures regarding opinion evolution and time-varying bounded confidence that were obtained under experimental conditions. We explored the evolutionary trend of opinion clusters by defining the neighbor relationship, and we found that opinions in groups converged exponentially into a stable neighbor relationship. The evolutionary trend of opinion clusters relates to the average opinion strength of the neighbor. If the average opinion strength is greater than zero, the opinion cluster will transform to a high interval. If the average opinion strength is less than zero, the opinion cluster will transform to a low interval. If the strength of the opinion equals zero, then neighbors share the same opinions, i.e., group consensus opinions. Moreover, the confidence variation increment is the main factor in shortening the convergence time. We explored time-varying bounded confidence using 0 < Δδ 1, and found that the threshold of bounded confidence converges at a rate to consensus; meanwhile, K ¼ 1 M P i ε i ð0Þ. Δδ = 1 is a special condition involving the real network structure. Our work is preliminary, and more in-depth research will be conducted in the future.
Confidence variation will affect the number of opinion clusters, the convergence of the trust threshold value and the degree of the opinion sample dispersion. When Δδ is larger, the stable neighbor relationships will form faster, and the effect on opinion evolution will be greater.
Opinion clusters are also related to the characteristics of individuals. Agents with a high opinion value but a low trust threshold value find it difficult to incorporate their opinion cluster into other groups, even when the confidence variation increment is sufficiently large. Notably, the role of individual characteristics in consensus will be discussed in future work.