As a kind of bounded confidence group opinion dynamics, the HK model has attracted the research interests of many scholars since it was put forward by Hegselmann and Krause [6] in 2002. After that, scholars used different methods to extend this model. [26] integrated the Hegselmann–Krause model into the process of product improvement and explored the relationship among the evolution of opinion, product scheme, and design iteration; [27] proposed a bounded confidence model which considered the opinion priority of leaders and divided the group into “opinion leaders” and ordinary people groups; based on the HK model, [28] considered social learning and explored the agent’s opinion evolution when they are influenced by the truth or a conflicting source. In addition, some researchers discussed the consensus and convergence in the HK model [29, 30]. The aforementioned research provided some insights to the following scholars on the HK model, and these research studies broaden the research boundary of opinion dynamics.

As a kind of sociological and psychological theory, social influence theory has been given more and more attention by scholars [31]. Social influence comes from two aspects. On one hand, an agent will be influenced by the role of another. Under the influence of others, an individual will more or less change in one way or another. On the other hand, the surrounding environment will form a kind of normative force, promoting members’ psychology to form a kind of group pressure and making its members be in line with the group [4, 32]. In terms of social influence, [33] explored the effect of social influence based on the Durkheimian opinion dynamics by constructing a social influence function with the social ranking (SR) and the distance; [34] also introduced individuals’ social influences in social networks to highlight the heterogeneity of individuals in opinion evolution; [35] considered the individuals’ influential power and explored the authority effect and the Matthew effect in opinion dynamics. In addition, some researchers also explored individuals’ social influence on themselves in opinion dynamics [36].

As an indicator of people’s estrangement, affinity is an important factor that influences people’s attitudes and behavior. In opinion dynamics, [37] proposed a model to simulate the process of opinion formation and explored the role of mutual affinity between interacting agents; [38] developed a statistical method to find affinity relations in a large opinion network; [39] took the mutual affinity between agents into the model and explored the process of opinion formation in an open community; [40] used the opinion dynamics to explore the partisan conflicts in recent America and discuss the change of affinity among co-partisans; [41] explored the nonlinear interaction of ideological affinity with the psychological reaction of agents in the frame of a multiparametric mathematical model of opinion dynamics. It can be seen from the aforementioned research that affinity has gradually gained the attention of scholars and has been applied to sociology, opinion dynamics, and other fields.

In addition, in the process of evolution, social noise can also influence people’s behavior or opinions [42]. At present, [43] explored the HK model under the influence of noise and found that the fragmentation of HK dynamics tends to vanish when persistent noise was present; [44] explored the influence of noise on the HK model and found that the system presented an orderly to disordered phase transition with the increase of noise in the HK model; [45] also found that noise was helpful in adjusting the split phenomenon of viewpoint through the study of HK model noise; [46] used an agent-based modeling and a computational social science approach to explore the opinion evolution with various noise levels; [47] introduced the noise into the Degroot framework and proposed a new Degroot-type social learning model.

The aforementioned scholars explored opinion dynamics from different perspectives. Few studies have explored the affinity and social noise in opinion dynamics based on social influence theory systematically. To bridge this gap, this study takes the affinity and noise into the HK model and constructs an affinity and social noise Hegselmann–Krause model. In this model, we explore opinion evolution and propose the evolutionary rules and algorithm. After that, experimental analysis is carried out, and some conclusions are discussed.

Within the limited confidence threshold, we assume that there is a set A, A = [1, 2, 3, … , i, j, N], the set A has N individuals, and the opinion of individuals i and j is assigned as x i t and x j t and x i t and x j t ∈ [ 0,1 ] , respectively; if | x i t − x j t | ≤ ε , then: x i t + 1 = ∑ j : | x i t − x j t | α i j x j t , (1) where the aforementioned model [Eq. (1)] is the Hegselmann–Krause model of opinion dynamics[6], x indicates the individual’s opinion, α _ij is the influence weight of individual i on the individual j, α _ij ∈ [0, 1], and ∑ j = 1 N α i j = 1 . If the α _ij is larger, it indicates that one individual’s influence on other individuals is more obvious; ɛ is the opinion threshold, and the smaller the ɛ is, it is difficult for an individual’s opinions to influence each other. In addition, the aforementioned model is a simplified model; it ignores the order in which ideas are presented and is suitable for simultaneous group updates, such as a group meeting and an exchange within a forum.

Based on the HK model, according to the group opinion dynamics’ characteristics, we focus on the affinity and social noise in the opinion evolution. Affinity always exists in the relationship among people. If the relationship between two individuals is good, the affinity is large; on the contrary, if the relationship between two individuals is bad, the affinity is small. To take into account the affinity, we use the r i j t to indicate the degree of opinion influence: r i j t = max R i j − | x i t − x j t | − ε i , 0 max | x i t − x j t | , φ , (2) where R _ij ∈ [0, 1] is a real number of N × N nonnegative and asymmetric matrix of R, and it indicates the affinity degree between people. If the R _ij is smaller, the relationship between them is worse; instead, if the R _ij is larger, the relationship between them is good. x i t is the opinion value of i, and x j t is the opinion value of j. r i j t is the degree of opinion influence, which indicates the influence degree of individual j on i in the t round. φ is a number that is not equal to 0. To avoid the denominator being equal to 0, we set φ as a minimal number, here φ = 0.001. Especially, if | x i t − x j t | > ε i & R i j < δ , the opinion influence weight is 0, and to measure the degree of opinion influence weight, we define ρ _ij is the opinion influence weight, which is given by Eq. 3. Here, the δ is the affinity threshold, and this parameter shows that if the opinion difference between two individuals does not exceed the opinion influence threshold, the affinity between individuals can also affect the evolution of opinion. ρ i j = r i j t , o t h e r w i s e 0 , i f | x i t − x j t | > ε i & R i j < δ (3)

By standardizing the influence degree of the aforementioned opinions, the following Φ i j t + 1 indicates the opinion influence weight of individual j on individual i. Φ i j t + 1 = 0 , ∑ κ ∈ N ρ i k t = 0 & i ≠ j ρ i j t ∑ κ ∈ N ρ i k t , ∑ κ ∈ N ρ i k t ≠ 0 1 , ∑ κ ∈ N ρ i k t = 0 & i = j (4)

Furthermore, to take into account the social noise, the extended HK equation in opinion dynamics can be constructed as follows. x i ∗ t = μ i x i t + 1 − μ i ∑ j ∈ N i , x t Φ i j x j t + ξ i t , (5) where μ _i indicates the initial opinion persistence, ξ _i is the social noise, and it obeys the random uniform distribution. Moreover, the social noise is independent and identically distributed (i.i.d.) with Eξ ₁ (1) = 0, E ξ 1 2 ( 1 ) > 0 [48,49]. μ _i is the persistence of individual i, and 1 − μ _i indicates the sensitivity of individuals to the opinions of the public around them. The initial persistence generally follows the normal distribution, that is to say, most people’s persistence of opinions is neutral, and the persistence of few individuals is extreme. When the mean value of the normal distribution is large, it is easy to form multiple opinion clusters, but not easy to form a consensus. On the contrary, if the majority of people’s opinions is amiable, individuals’ opinions are easily influenced by other individuals and finally form a consistent opinion.

The conditions for the evolution of opinion are as follows: N i , x t = j ∈ ν | x j t − x i t | ≤ ε , ε ∈ 0,1 . (6)

At the t + 1, the opinion evolution equation is as follows: x i t + 1 = 1 , x i ∗ t > 1 x i ∗ t , x ∗ ∈ 0,1 , ∀ i ∈ N , t ≥ 0 0 , x i ∗ t < 0 (7)

To clearly explain the algorithm, we have shown the calculation process of the affinity and social noise Hegselmann–Krause model in Figure 2.

Based on the aforementioned model construction, the affinity and social noise Hegselmann–Krause model is built, and the specific parameter note is shown in Table 1.

Generally speaking, in real life, the persistence of opinion is neutral, and the opinion of small people is extremely close to 0 or 1. Therefore, we can assume that the distribution of people’s opinion persistence belongs to normal distribution. Default settings for the parameters used in the experiments are shown in Table 2.

In the proposed model, the first analysis is performed by simulating the influence of affinity degree on opinion evolution. In terms of parameter setting, we set the affinity degree R _ij to 0.1, 0.5, and 0.9. At this moment, the threshold of the affinity is 0.8 and the opinion numbers are 100. Also, we keep other parameters unchanged, and the simulation result is shown in Figure 3.

As shown in Figure 3A, we find that the opinion evolution is related to the affinity degree. In Figure 3A, when R _ij = 0.1, the opinion will be stable at 200 steps, and the opinion will form six opinion clusters; when R _ij = 0.5, the opinion will be stable at 50 steps, and the opinion will form five opinion clusters in Figure 3B; as the affinity continues to increase, when the affinity degree increases to 0.9, at this moment, the opinion will form one opinion cluster in Figure 3C. Based on the aforementioned simulation results, we can see that affinity can improve the opinion to tend to an agreement, and the result is also consistent with the social influence theory. Meanwhile, the aforementioned results indicate that affinity among people has a positive effect on opinions. If the affinity among individuals increases, the opinion will be easier to reach an agreement.

Different thresholds of affinity may have different influences on opinion evolution. To further explore the influence of the threshold of affinity on opinion evolution, we carried out a related simulation about the threshold of affinity. In the simulation, we set the affinity as 0.5, and the thresholds of affinity are 0.5, 0.8, and 0.9. Keeping other parameters unchanged, the simulation result is shown in Figure 4.

As shown in Figure 4, opinion evolution has different results when the threshold of affinity takes different values. In Figure 4A, when δ = 0.5, at this moment, the affinity is equal to the threshold of affinity, the opinion will be stable at 100, and the opinion will form one opinion cluster finally. As shown in Figure 4B, when the threshold increases, for instance, if δ = 0.8, we can find that the opinion will be stable at 80, and the opinion will form six opinion clusters, compared with Figure 4A. This result indicates that the larger difference between affinity and the threshold will prevent opinions from forming a consensus and lead to a split of opinion. In Figure 4C, when the δ = 0.9, at this moment, the difference between affinity and the threshold is increasing continuously, and the opinion will form more clusters. It indicates that if the difference between affinity and the threshold is larger, the opinion will be more dispersed.

In the ASNHK model, social noise is an important factor that can affect opinion evolution. In terms of social noise, as an external factor, social noise can give individuals some pressure or influence and promote the evolution of individual opinion. To explore the influence mechanism of social noise, we set the affinity to 0.5, set the threshold to 0.8, and kept other parameters unchanged. Moreover, we set the interval of social noise distribution to ξ _i ∈ [ − 0.01, 0.01], ξ _i ∈ [ − 0.02, 0.02], and ξ _i ∈ [ − 0.03, 0.03] uniformly. The simulation result is shown in Figure 5.

As shown in Figure 5, we find that the opinion will form different evolution results when the social noise increases to different values. In Figure 5A, when there is no noise in opinion dynamics, the opinion will be stable at 95 steps and form five opinion clusters. Based on Figure 5A, when the interval of social noise distribution increases to [ − 0.01, 0.01], the simulation result is shown in Figure 5B. As shown in Figure 5B, the number of final opinion clusters is 4. This result indicates that social noise can urge the people to form more consensus opinions. Furthermore, in Figure 5C, when the interval of social noise distribution increases to [ − 0.02, 0.02], the number of the final opinion cluster is reduced from 4 to 3, and the opinion gets more close. As the social noise increases continuously, for example [ − 0.03, 0.03], the simulation result is shown in Figure 5D. From Figure 5D, we can find that the final opinion is disorderly, dispersed, and inconsistent. The possible explanation is that social noise has a critical influence. When social noise is greater than the critical value, an individual’s opinion will get dispersed. It also indicates that a phase transition exists under the influence of social influence. The aforementioned conclusions are also consistent with those of the research of Su et al.[43, 45].

In the process of opinion evolution, because people have different characteristics, people react differently to opinions, that is to say, people’s character has certain heterogeneity [50]. According to the heterogeneity, people can be divided into three types roughly: closed, easygoing, and open. We set the opinion thresholds ɛ _i ∈ [0.01, 0.05], [0.2, 0.3], and [0.4, 0.5] separately. ɛ _i ∈ [0.01, 0.05] indicates that people are closed; ɛ _i ∈ [0.2, 0.3] indicates that people are easy-going; and ɛ _i ∈ [0.4, 0.5] indicates that people are open. To explore the influence of different heterogeneity in the opinion evolution and keep other parameters unchanged, we perform the related simulations on the personnel heterogeneity. In addition, to make the evolution of ideas clearer, we also use the histogram to display the result of opinion evolution. The specific result is shown in Figures 6, 7.

As shown in Figures 6, 7, we can find that different heterogeneity can cause different evolution results. In Figure 6A, when the opinion threshold ɛ _i ∈ [0.01, 0.05], it indicates that people are closed in the community. At this moment, the opinions will form nine clusters, and the convergence steps will stabilize at 49. We also find that the final opinion will distributed in [0, 1] almost uniformly in Figure 7A; From Figure 6B, when the opinion threshold increases continuously, for example, ɛ _i ∈ [0.2, 0.3]. This threshold indicates that people are easygoing in the community. The evolution result forms four opinion clusters, and convergence steps will stabilize at 200. The final opinions will be distributed around 0.5 in Figure 7B. In Figure 6C, when the opinion threshold ɛ _i ∈ [0.4, 0.5], it indicates that people are open in the community, the opinions will form three opinion clusters, and the steps will stabilize at 200. The final opinions are distributed around 0.2 and 0.8 in Figure 7C. At this moment, people’s opinions are easier to form an agreement.