Modeling confirmation bias and peer pressure in opinion dynamics

People tend to accept claims that adhere to their prior beliefs, i.e., being within their identity scope, and ignore dissenting claims [1, 2, 3]. This mechanism, confirmation bias, causes the emergence of echo chambers and group polarizations in public discourse [4, 5]. Political polarization is a prominent case where Republicans reject the statements supporting Democrats as false and vice versa, which threatens the democracy [6]. In addition, individuals are likely to reshape their opinions, attitudes or behaviors according to the position of the majority [7, 8]. This mechanism, peer pressure, appears to be a primary driver of opinion evolution [9, 10]. It is sometimes strategically utilized by partisan organizations to get more votes in election, for instance, deploying vast social bots [11] and information gerrymandering [12]. Thus, to comprehensively understand the opinion polarization and to forecast the winner in competing processes require a model of opinion dynamics integrating the conjugate effect of confirmation bias and peer pressure.

The development of online networks has radically changed the way that people consume information and exchange opinions, which results in substantial impacts on opinion dynamics [13, 14, 15, 16]. Individuals could easily seek out contents coherent with prior beliefs on large-scale online discourse, which amplifies confirmation bias [17]. In addition, the wide availability of dissenting content on the web makes individuals exposed to peer pressure every day [18]. Under such circumstance, opinion dynamics has attracted great attentions recently [19]. Classical models including Friedkin and Johnsen model [20, 21, 22], Sznajd model [23] and voter model [24] were further explored on large-scale social networks. These models sufficiently considered interpersonal influence including peer pressure between discordant pairs and showed the consensus state where all agents finally share the same opinion.

In addition to group consensus, empirical studies showed fragmentation and polarization of opinions, which has aroused great concerns in diverse fields [25, 26, 27]. For instance, political scientists observed markedly increased political polarization in the United States, which threatens democracies [28]. Sociologists noted the aggregation of individuals trusting false news and misinformation which have been recorded as one of main threats to human society by World Economic Forum [29, 30]. Various classical mathematical models were proposed to explain the ubiquitous phenomena [31, 32, 33]. Deffuant et al. proposed bounded confidence model (BCM) where individuals just accept contents within their identity scope to mimic confirmation bias and found the occurrence of several opinion clusters when identity scope is small [34]. Vicario et al. incorporated negative updating rule of opinions among discordant pairs of users into BCM and observed the coexistence of two stable final opinions [35]. Wang et al. introduced an agent-based model integrating external political campaigns and opinion dynamics, which reproduced the 2016 American presidential election well and yielded the intriguing moderate clusters in addition to two polarized clusters [36].

While confirmation bias and peer pressure are not necessarily new phenomena, few attempts have been made to explore how the interplay between the two mechanisms affects opinion evolution. Besides, moderate cluster but not polarized clusters usually determines the winner in competing processes of opinions. Presidential election is a prominent case where the party who wins the majority of moderates would win. Unveiling the facets of moderates is a meaningful problem which is of vital significance to design efficient strategies of guiding public opinions.

In this paper, we propose a modified bounded confidence model with the conjugate effect of confirmation bias and peer pressure to describe the continuous opinion dynamics on large-scale social networks. Firstly, we consider population with homogeneous susceptibility to peer pressure. Simulations results show intriguing non-monotonous changes in the number of opinion fragments, which first increases and then reduces to unique one with identity scope increasing. Then we consider heterogeneous population where impressionable individuals who are sensitive to peer pressure and ’confident’ individuals who are immune to peer pressure coexist. Results show critical phenomena about the critical fraction of impressionable individuals leading to public consensus: the value is very small when identity scope is relatively large, while explosively increases to a large one once identity scope decreases to a threshold. In addition, we highlight that majority of impressionable individuals would become moderates. It indicates the emergence of an important but easily influenced group: impressionable moderates. This result implies the insight that targeting impressionable moderates might be an efficient strategy to guide public opinions even when the system has reached steady state.

We introduce an agent-based model of continuous opinion dynamics which integrates the conjugate effect of confirmation bias and peer pressure in this section. Consider a large-scale social network with $N$ agents. Initially, each agent has an opinion $x_{i}$ that is uniformly distributed in [0,1]. At each time step, social platforms allow agents to rapidly receive opinions from all their neighbors. According to the well-known confirmation bias, agents just think that opinions close enough to their beliefs are reasonable and that corresponding neighbors are supporters of their cognition. Complying with previous studies [36], we define $\delta$ as identity scope to describe the phenomena. Specifically, neighbor $j$ is recognized as a supporter of agent $i$ if $|x_{i}-x_{j}|<\delta$. Otherwise, neighbor $j$ is recognized as an opponent. In this way, any agent could divide its neighbors into two classes: supporters and opponents (see figure 1(a)). Another important mechanism in opinion dynamics is peer pressure that individuals are likely to reshape their cognition, attitudes or behaviors when being affected by others, especially by opponents [37, 38]. Here we adopt the thought of threshold model to mimic peer pressure, where we define threshold $\alpha$ as susceptibility to peer pressure. If the fraction of supporters is more than the threshold $\alpha$, agents would affirm their cognition and readjust opinions by just interacting with neighbor supporters, which is shown as confirmation bias in figure 1(b). Otherwise, agents would doubt their own cognition due to lack of support and heavy peer pressure from opponents, which urges them to update opinions according to all neighbors’ position (see figure 1(c)).

Specifically, our dynamical model goes through the following steps:

• 1)

  At each time step, agent $i$ divides its neighbors into supporters and opponents, according to the following rule: if $|x_{i}-x_{j}|<\delta$, neighbor $j$ is recognized as a supporter. Otherwise, $j$ is an opponent.

• 2)

  At each time step, one of the two possible interactions would happen.

  • (a)

    (Confirmation bias) If the fraction of supporters is more than $\alpha$, agent $i$ updates its opinion from $x_{i}$ to $\tilde{x_{i}}$ by interacting with supporters:

    $$\tilde{x_{i}}=\mu x_{i}+(1-\mu)\sum\limits_{j\in\Omega_{i}^{S}}\frac{x_{j}}{|\Omega_{i}^{S}|}$$

    where $\mu$ reflects the stubbornness and $\Omega_{i}^{S}$ represents the set of neighbor supporters of agent $i$.

  • (b)

    (Peer pressure) Otherwise, agent $i$ updates its opinion from $x_{i}$ to $\tilde{x_{i}}$ by interacting with all neighbors:

    $$\tilde{x_{i}}=\mu x_{i}+(1-\mu)\sum\limits_{j\in\Omega_{i}}\frac{x_{j}}{|\Omega_{i}|}$$

    where $\Omega_{i}$ represents the set of neighbors of agent $i$.

Opinion evolution on large-scale social networks has been widely concerned [39, 40, 41]. People are more likely to accept claims within their identity scope and ignore the dissenting claims [42, 43]. On the contrary, people might alter their decisions if receiving vast opposing views [44]. Both confirmation bias and peer pressure were proved as core factors in opinion dynamics, separately [45, 46]. However, it remains unclear how the interplay between these two mechanisms affect opinion evolution.

In this work, we propose an agent-based model of opinion evolution which takes into account both confirmation bias and peer pressure. By performing simulations in homogeneous population, we find the non-monotonous conjugate effect of these two mechanisms. To be specific, the number of opinion fragments first increases and then reduced to one with identity scope growing, which is completely different from the insight of classical bounded confidence model [34]. The counter-intuitive phenomena can be explained by the following microscopic reason: agents with smaller identity scope consider that more opinions deviate from their beliefs and thus receive more peer pressure which urges them to adapt to the public.

In addition, we consider the heterogeneity of susceptibility to peer pressure. Specially, the heterogeneous population is divided into impressionable individuals and individuals immune to peer pressure. We explore the detailed effect of impressionable individuals. First, we find intriguing critical phenomena that critical fraction of impressionable individuals resulting in public consensus explosively increases from a tiny value to a large value when identity scope goes through a threshold. This result indicates that only when identity scope is relatively large can a few impressionable individuals efficiently eliminate polarization of public. Moreover, we highlight that while impressionable individuals’ initial opinions are randomly distributed, more than 80% of them finally become moderates when system reaches the steady state. It reflects an important but easily influenced group: impressionable moderates.

Our work utilizes simple mechanisms to reveal the non-monotonous conjugate effect of confirmation bias and peer pressure and explains the counter-intuitive phenomena from microscopic level. Besides, the study about heterogeneity of susceptibility to peer pressure shows a powerful method of eliminating the polarization of public when identity scope is relatively large, which is deploying a few impressionable individuals. In real online networks, we could deploy a small number of social bots to replace impressionable individuals. Moreover, our model shows the emergence of impressionable moderates. The important but easily influenced group would naturally become targets of interest to all parties in competing processes. The strategies of targeting impressionable individuals deserves further study.

This work is supported by Program of National Natural Science Foundation of China Grant No. 11871004, 11922102, and National Key Research and Development Program of China Grant No. 2018AAA0101100.