Asynchronous opinion dynamics with online and offline interactions in bounded confidence model

Nowadays, in the world, about half of the population can receive information and exchange their opinions with others in online environments (e.g. the Internet); while the other half obtain information and exchange their opinions in offline environments (e.g. face to face) (see eMarketer Report, 2016). The speed at which information is received and opinions are exchanged in online environments is much faster than in offline environments. To model this phenomenon, in this paper we consider online and offline as two subsystems in opinion dynamics, and there is asynchronization when the agents in these two subsystems update their opinions. We show that asynchronization strongly impacts the steady-state time of the opinion dynamics, the opinion clusters and the interactions between the online subsystem and offline subsystem. Furthermore, these effects are often enhanced the larger the size of the online subsystem. Keyword: Opinion dynamics, Asynchronism, Bounded confidence, Agent-based simulation.


Introduction
. Opinion dynamics is a research tool widely used to investigate the opinion evolution in many collective phenomena. The study of opinion formation goes back as far as French ( ). According to French's study, some opinion dynamics models based on di erent communication regimes had been proposed, such as DeGroot model (DeGroot ; Berger ), Friedkin and Johnsen model (Friedkin & Johnsen ) and bounded confidence model (De uant et al. ; Hegselmann & Krause ).
. Among these opinion dynamics models, the bounded confidence model has been frequently used in recent years. The bounded confidence model assumes all agents are bounded confident, i.e. each agent updates her/his opinion by averaging the agents' opinions that di er from his/her own no more than a certain confidence level. In the De uant & Weisbuch model (i.e., DW model), agents follow a pairwise-sequential updating mechanism, while in the Hegselmann & Krause model (i.e., HK model), each agent updates his/her opinion by averaging all opinions in their confidence sets. In other words, the DW model and the HK model are very similar but di er mainly in the communication regime (Urbig et al. ). Following the DW and HK models, some interesting extended studies have been conducted (Dong et al. ; Weisbuch ; Fortunato et al. ; Lorenz ; Mor et al. ; Ceragioli & Frasca ; Liang et al. ; Mathias et al. ). .
In most existing opinion dynamics models, all agents update their opinions at the same time according to the established rules, i.e. the evolution of opinions is synchronous. The general theory of asynchronous systems has been supported in the specialized literature (Bertsekas & Tsitsiklis ; Chen et al. ; Dong & Zhang ; Frommer & Szyld ; Kozyakin ). Particularly, Alizadeh & Cio i-Revilla ( ) studied the asynchronous updating schemes in the bounded confidence model in an elegant and concise way. They applied four di erent asynchronous updating schemes including random, uniform, and two state-driven Poisson updating schemes, and compared the e ect of di erent activation regimes (i.e. the timing of activation).

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With the development of the Information and Internet technology, there exists a very common asynchronous phenomenon in online and o line interactions. According to eMarketer Report ( ), in the world about half of the population can receive information and exchange their opinions with others in an online environment (e.g. the Internet), while the other half obtain information and exchange their opinions in an o line environment (e.g. face to face). The Internet technologies (e.g. Facebook, Myspace, etc.) enable online agents to spread and share information in a more rapid way than the o line agents (Bakshy et al. ; Song & Yan ; Zhao et al. ). For the above reasons, in this paper we consider online and o line as two subsystems in opinion dynamics, and assume that there is an asynchronization when the agents in these two subsystems update their opinions. Then, based on the HK bounded confidence model, we investigate the opinion dynamics with asynchronous interactions between online and o line agents. We focus on how the asynchronization in online and o line interactions impacts the dynamics of opinion formation. .
Through extensive agent-based simulations and analyses, we unveil that asynchronization in online and o line interactions strongly impacts the dynamics of opinion formation. Specifically, asynchronization lengthens the steady-state time of opinion evolution, and leads to the absorption phenomena between the online and o line subsystems. .
The remainder of this paper is arranged as follows. Section introduces the HK bounded confidence model. Section then proposes the asynchronous opinion dynamics model with online and o line interactions in the framework of bounded confidence. Next, Section discusses the influences of asynchronism and the size of the online subsystem in the proposed model. Finally, Section presents the concluding remarks.

The Hegselmann & Krause Bounded Confidence Model
. In this section, we briefly introduce the HK bounded confidence model. Since the DW model and the HK model are rather similar, if we adopt the DW model as the basic model, a similar asynchronous opinion dynamics model will be conducted.
. Let A = {1, 2, . . . , N } be a set of the agents. Let x t i ∈ [0, 1] be the opinion of agent i at time t, and thus X t = (x t 1 , . . . , x t i , . . . , x t N ) T be the opinion profile at time t. Let be the homogeneous confidence level of the agents. .
The process of the HK model consists of three steps as follows: The first step is to determine of the confidence set. The confidence set I(i, X t ) of the agent i at time t is determined as: Then, the second step is to calculate of the weights that one agent assigns to other agents. Let w t ij be the weight of agent i assigns to agent j at time t, i.e., where #I(i, X t ) denotes the number of agents in the confidence set I(i, X t ).
. Finally, the third step is to determine the updated opinions for each agent. The updated opinion x t+1 i is modeled as a weighted arithmetic mean of opinions in the confidence set, i.e.,

The Asynchronous Opinion Dynamics Model in Online and O line Interactions
. In this section, we propose the asynchronous opinion dynamics model in online and o line interactions based on the HK bounded confidence model. In the same way as the HK model in Section , let A = {1, 2, . . . , N } be a set of the agents, x t i ∈ [0, 1] be the opinion of agent i at time t, and be the homogeneous confidence level of the agents. . Based on existing studies (Bakshy et al. ; Song & Yan ; Zhao et al. ), we assume that the speed of updating opinions for the online agents is much faster than that for the o line agents, and let T be the degree of asynchronization between the online and o line subsystems, where T = 1 and T ∈ N . Then, let T on = {0, 1, 2, . . . } and T of f = {0, T, 2T, . . . } be two sets of discrete time, where T of f ⊆ T on . When time t ∈ T on and t / ∈ T of f , only the online agents will update their opinions, and when t ∈ T of f , both the online and o line agents will update their opinions. And thus T = 1 represents synchronization and T = 2 asynchronization. Obviously, the larger the T value is, the more asynchronization between the agents in online and o line subsystems. .
Next, we propose the asynchronous opinion dynamics model with online and o line interactions in the framework of bounded confidence based on the following two cases: Case A: t + 1 ∈ T on and t + 1 / ∈ T of f . In this case, for any agent i ∈ A on , he/she only communicate with other online agents at time t. And the confidence set I A (i, X t ) of the agent i ∈ A on is determined as: Then, the weight w t ij of agent i assigns to agent j at time t can be calculated as: In addition, any agent i ∈ A of f , he/she does not communicate with other agents at time t and thus he/she will not update his/her opinion at time t + 1, i.e. x t+1 i = x t i . Above all, in this case, the updated opinion x t+1 i is calculated as: Case B: t + 1 ∈ T on and t + 1 ∈ T of f . In this case, the agent i ∈ A can communicate with both the online and o line agents at time t. Thus, the confidence set I B (i, X t ) is determined as: Then, w t ij of agent i assigns to agent j at time t is determined as: In this case, the updated opinion x t+1 i is calculated as: Based on Cases A and B, for any agent i at time t + 1 ∈ T on , the updated opinion x t+1 i is calculated as: where w t ij is determined by Equations and in the case of i ∈ A on and t + 1 / ∈ T of f , or is determined by Equations and in the case of i ∈ A and t + 1 ∈ T of f .

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Similar to the HK model, the confidence level plays a key role in our model. We set N = 1000, with half being online agents and the other half o line agents, and the initial opinions of all agents are uniformly and randomly distributed in [0, 1]. We use Equation to proceed with the evolution of opinions, and analyze the steady-state consensus ratio from independent realizations. As shown in Figure  .
In the simulation, when X t+1 − X t ≤ δ, we consider that the opinions of all agents reach the stable state, where X = max 1≤i≤n |x i |, and we set δ = 10 −3 . Notice that a di erent norm of the vector X, such as X = n i=1 |x i |, does not influence the main results in this paper. Meanwhile, let x i , x j be the opinions of agents i, j when the opinions reach the stable state. We assign the agents i, j to a same cluster when |x i − x j | < d, and we set d = 10 −2 (the pseudo-code of calculating opinion clusters is included in Appendix B).

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Although we only represent the results when T ∈ [1, 100] and N = 1000 in the paper, the results are similar when setting T > 100 and di erent N values (e.g., N = , and ). . Figure illustrates how the number of opinion clusters changes along with the confidence level and the degree of asynchronization. Increasing the confidence level yields an increase in communication among the agents, this translates to a decrease of the number of opinion clusters. Meanwhile, the number of opinion clusters increases as T increases from T = 1 to T = 10, and such an e ect is more evident for the large size of online agents (p = 80%) and for low confidence levels. The asynchronization delays the update time of o line agents, and thus the agents will form more opinion clusters because of fewer communications between the online and o line agents. However, with further increments of T , the evolution of online agents' opinions will rapidly reach a stable state among them before the o line agents start to update opinions, which stops the increase of the number of opinion clusters. .   Figure ) and nearly stops increasing when the degree of asynchronization exceeds as per Figure . The number of pure o line clusters increases from T = 1 to T = 10 because of fewer communications between the online and o line agents. However, as T increases above this threshold, it is observed that the online agents rapidly reach a stable state among them, which stops the increase in the number of pure o line clusters. When > 0.13, we barely observe pure o line clusters except a small area in the upper right-hand corner of the third subfigure of Figure     . However, we find that pure online clusters are hardly observed in the simulation, and this phenomenon implies that online clusters can always attract a certain number of o line agents. Next, we investigate the absorption of the online and o line subsystems to further study the interactions between them.

The absorptions of the online and o line subsystem .
Assume .
In the following, we define L on and L of f to measure the absorption of the online subsystem and the o line subsystem, respectively. For low confidence levels (e.g., = 0.03, 0.13), when T increases, L on increases and L of f decreases for p < 50%, while their direction reverse for p > 50%. However, such an e ect is not evident for large confidence levels (e.g., = 0.23) due to the numerous communications among agents that are generated in these cases.  .
According to eMarketer Report ( ), in the world about half of the population are online agents, consequently we pay more attention on the case of p = 50% in the following Figures -. L on highlights the centre area of Figure . This observation can be explained as follows. When T is small, it is di icult for the online agents to form some online opinion clusters; when T is large, the online agents will rapidly reach a stable state and thus the number of o line agents that are influenced by the online agents will decrease accordingly. Meanwhile, if the confidence level is small, the online agents only communicate with a very limited number of the o line agents, and if the confidence level is large, the o line agents can simultaneously attract strongly the online agents. So, when and T are both in the middle size, the online subsystem shows a stronger absorption capacity. .
Meanwhile, highlights the upper right-hand corner of Figure . This observation can be explained as follows.
When the confidence level is large ( = 0.23) and T is small (T ≤ 10), all agents can always reach a consensus, and thus L on is . . However, with further increments of T , we find that all agents will be gradually divided into two opinion clusters: one is an online opinion cluster and the other one is an o line opinion cluster, with the number of o line agents in the online opinion cluster being high. So, the online subsystem shows a stronger absorption capacity on the upper right corner of Figure   . Figure shows that L of f starts decreasing, and then stabilizes as T increases. The main reason for this observation is that the interaction between the two subsystems decreases as T increases, which leads to a decrease of L of f . With further increments of T , the online agents will reach a stable state before the o line agents start to update opinions. Thus, the interaction between the two subsystems barely changes, and then L of f stabilizes.

Conclusions
. In this paper, we propose asynchronous opinion dynamics with online and o line interactions in a bounded confidence model. In the proposed model, the asynchronous updating mechanisms between the online and o line agents are analyzed in detail.
. We unfold that the asynchronization strongly impacts the steady-state time, the number of opinion clusters and the interaction between the online and o line agents, and that as the size of the online agent increases these e ects are enhanced.
. We show that online agents have a stronger absorption capacity than o line agents, which leads to the appearance of pure o line clusters. Thus, we suggest that governments should provide more supports to promote interactions with some o line agents; otherwise, some of the o line agents could end up being isolated from society. .
With the development of Information and Internet technology, asynchronization between online and o line agents is a very popular phenomenon in the evolution of real-life public opinions. In order to make our research more realistic and reliable, we plan to develop further studies to improve the understanding of asynchronization in opinion dynamics in other relevant models, and to extend the study in a complex network context. Figure reveals the impact of T and on the steady-state time T * . As T increases, the steady-state time T * increases under the di erent values. Compared with synchronization (i.e. T = 1), the o line agents update their opinions more slowly than the online agents, and thus the evolution of opinions needs a longer time to reach a stable state. Particularly, the di erence of the steady-state time between the synchronization case and the asynchronization case with large T is very obvious.
A clue can be found in Figure showing the relationship between the size of the online subsystem and the steady-state time: The smaller the online subsystem is, i.e. the smaller p is, the longer it took in time for the stabilization of opinions when ≤ 0.11. However, this observation cannot always be obtained when > 0.11. We can clearly see this thread from Figure : When the value of is small ( = 0.33), the steady-state time decreases when the size of the online subsystem increases, while the result does not hold when = 0.13. As shown in the middle subgraphs of Figure , it took more time for the stabilization of opinions when p = 80% than for p = 60%.
The main reason for this observation is that the steady-state time T * is simultaneously a ected by p and . When is small ( ≤ 0.11), the steady-state time T * is mainly a ected by p, and we find that there is the above regularity between the size of the online subsystem p and the steady-state time T * . However, as the value of increases, the influence of on the steady-state time T * grows, which leads that there is not an obvious regularity between p and T * .