Exposure rule

When observing and interacting with opinions there is a possibility of exposure to distant opinions (for example, through chance encounters, family connections, workplaces) and those opinions which are more similar to the observer. As such influences should be possible between all agents, but the likelihood of an agent being exposed to the opinion of another should reflect the principle of homophily. That is to say, individuals are more likely to interact with those that they are similar to, widely acknowledged as a determinant of exposure in the literature [50]. Therefore agents in the model have a non-zero probability of exposure with any other agent in the population, however exposures become increasingly likely between those who are more similar in opinion (a smaller absolute opinion difference ).

To reflect this, we have a complete directed network (so any agent may be exposed to any other agent’s opinion) with weighted edges between agents (the likelihood that an exposure occurs). The likelihood of exposure is a function of the opinion difference between actors and an exposure parameter, E. Our model comprises discrete time steps, and an independent random exposure network computed on each time step. The probability that an actor u observes the opinion of v at any given time step is

(1)

This prescribed likelihood function reflects an increasing likelihood of exposure as opinions are more similar, while E plays the role of a shape parameter. Some illustrative likelihood functions for different parameters may be seen in Fig 1 to get a sense of how E impacts the chance of interacting with agents of significantly different opinions. To create the directed exposure networks, at each step we compute the probability of each one of the N² potential edges existing in the system, and we draw them as Bernoulli random variables.

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Fig 1. (a) Exposure between agents u and v is attractive since is less than prescribed tolerance T, while influence between u and w is repulsive since d_uw is greater than the tolerance threshold T.

(b) The probability that two agents interact given their opinion difference, under different levels of exposure.

https://doi.org/10.1371/journal.pone.0338486.g001

Opinion updating rule

Following an exposure, an agent will update their opinion. This will either be an attractive or repulsive update; that is, the agent’s opinion will become more or less similar to the observed opinion by reducing or increasing opinion difference on the opinion scale. This reflects the desired behavior that agents distance themselves from significantly different opinions while assimilating with similar ones.

We employ two model parameters: tolerance, T, and responsiveness, R. Tolerance is the opinion difference at which influences from exposure flip from being attractive () to repulsive (). Responsiveness is the fraction of the opinion difference by which the updating actor will change their opinion by — one extreme is to completely adopt the new observed opinion, the other is to not change opinion at all. With this we can express the opinion update rule for agent u at a time step t resulting from an exposure to the position of agent v as

(2)

Notice that opinion difference is calculated in the direction from u to v to determine the correct sense for attraction/repulsion.

Fig 1 illustrates a situation with three agents u, v, and w. Agent u interacting with v is attractive, while u interacting with w results in repulsion given that the opinion difference is greater than the tolerance threshold. The adjacent figure indicates the likelihood of exposure, as parameter E decreases or opinion difference increases the probability becomes less likely.

Group identification

The final part of the model introduces group perception and identification. We want to reflect that agents treat their own in-group differently to the out-group, which results in different levels of values E, R, and T. Therefore in our model we take each of the three parameters, tolerance T, responsiveness R, and exposure E, and split them into an in-group parameter and an out-group parameter one-by-one. We then organize our simulations into three groups, independently considering in- and out-group tolerance (keeping a common responsiveness and exposure; i.e., ), responsiveness (keeping a common tolerance and exposure; i.e., ), and finally exposure (keeping a common tolerance and responsiveness ). This will allow us to then dissect the differing roles of in-group and out-group exposure, responsiveness, and tolerance in the opinion model.

Introducing group-dependency on top of the variables in the ARM implies a potential connection between social identity and these variables. At the simplest level, the existence of in-group favouritism and out-group discrimination [33] may be applied to each variable that operates intra-group and inter-group as these biases could be present. More specifically however, there is evidence that assimilation of content can be greatly increased by feeling of closeness to another person [51] and that people are more likely to act on the opinions of their in-group [52], both of which relate to group-dependent responsiveness. The negative relationship between positive evaluation of out-group arguments and positive identification of the other as out-group [53] points to the possibility of group-dependent tolerance. And finally, it has been seen that group-dependent exposure can exist, in part due to recommendation systems, in digital spaces [13,28,54] but also self-organization in physical space [55].

Group identification within our model is determined by the distribution of the opinions of the agents and similarities, or clustering, within it. In reality, however, a multitude of social and political dimensions contribute to group identity such as political parties, ethnic groups, professional groups, and so on [22]. For the purposes of this article we rely on a simple modeling of the group identification process, while acknowledging the existence of more complex underlying mechanisms. In our model, the attribute that is communicated in exposures, i.e., the opinions, is also the same attribute on which agents perceive agroupments and identify into perceived groups. Furthermore, our model does not accommodate means of accounting for possible persistence in the perceived group identity, as agents identify into groups purely on the basis of their current opinion. A concrete example of a context where these assumptions are violated, is provided by many political settings in Europe, where multiple opinion dimensions are necessary to distinguish political groups from each other [56]. Our model provides, we argue, the first tool to address co-evolution of groups and opinions, and we offer in the conclusion paths to enriching it in view of matching different empirical settings.

Persistence mechanisms are well known to the opinion dynamics literature regarding individual positional opinions. One traditional approach consists in explicitly including a parameter moderating the relative weight between new influences (acting on a given time step) and past positions held by the agent (e.g., [57]). A second popular approach is the use of self-influence: when the influence of agents j over agent i is computed, typically encoded in an influence matrix with elements w_ij, the self-influence w_ii accommodates the relative weight between persistence of the previously held position (in the immediately past time step) and the influence of other agents (e.g., [58]).

The operationalization of the persistence of groups (i.e., the degree to which agents resist changes in groups), as they are formalized in our model, does not have immediate extensions starting from the individual position persistence mechanisms described. This lack of natural extension relates to group indices being categorical, as opposed to continuous positional quantities. We also remark that the persistence of groups that do not depend on the positional opinions of agents simulated in opinion dynamics models, might also be of importance to capture relevant groups in empirical settings (e.g., demographic groups). S3 Fig of the Supporting Information document, S1 File, provides an illustration of a simulation using our model, but fixing groups at the start.

Comparison with existing opinion dynamics models

The model proposed in this section exhibits both similarities and differences relative to other opinion dynamics models. Two fundamental features of our model are derived from Axelrod’s Attraction-Repulsion Model (ARM [44]), upon which this work is predicated: (1) bias mechanisms and (2) distancing effects, modeled by the E,R,T parameters. Through the exposure parameter E, the ARM incorporates a bounded confidence effect by which agents preferentially interact with others who are sufficiently close in the opinion space, a recurring mechanism in opinion dynamics models since the seminal works of Axelrod (1997) [47], Deffuant et al. (2000) [46] and Helgselmann and Krause (2002) [45]. In these models, agents interact only if their opinions differ by less than a fixed threshold, while in the ARM, the confidence bound is a continuous function of opinion distance, allowing for interactions far across the ideological spectrum—albeit with low probability. At the same time, the ARM integrates distancing effects (modelled by the tolerance parameter T) by which agents are repelled away from those holding discordant opinions. These distancing effects have been documented in the literature since (at least) the early 2000s [59,60]. Finally, the responsiveness parameter R allows us to modulate the extent to which agents are willing to reassess their opinions after interactions, reminiscent of opinion persistence mechanisms in other models [58].

The primary distinction between the proposed model and existing approaches lies in the manner through which groups are imposed and operationalized within the computational rules governing opinion updates at each temporal iteration. Whereas in most models groups manifest as emergent macro-properties, the present framework treats group identification as a prescribed computational element at each step, thereby rendering group-based rules explicitly specifiable. For example, many models (encompassing the bounded confidence models mentioned above) take interest in the study of how pair-wise opinion dynamics may induce the fragmentation of the population into ideologically homogeneous clusters. In contrast, in our model, group identification and attribution for agents constitutes a variable that serves as an argument within model parameters conceptualized as functions. Group indices for each agent are categorical values that are arguments of model parameters: responsiveness R, tolerance T, and exposure E. This functionality enables the modeling of differential tolerance values for in-group versus out-group exposures, operationalizing social psychological principles that motivated the ARM proposition [44] and addressing recent empirical observations [36].

Experiment protocol

We organize our simulations as follows. The number of agents is set to N = 100. At the top level, simulations are organized into three types, according to what parameters (T,R or E) will be separated into in- and out-groups, resulting in three main sets of simulations. For the parameters that are not split into in- and out-groups, we explore three values that exhibit different behaviors, typically two extreme values to demonstrate the range of the system’s behavior and one intermediate value that shows multiple possible outcomes when varying the group-dependent parameter. Precise values are determined by the context of the group-dependent parameter and detailed below. For the parameter that is split into in- and out-groups, we explore 20 values between 0 and 1. This yields the following sets of parameter exploration in the three top level types of simulations:

• Group-dependent tolerance (GDT) simulations:
  ;
• Group-dependent responsiveness (GDR) simulations:
  ;
• Group-dependent exposure (GDE) simulations:
  .

For a given set of parameters T, R and E, regardless of which is split into in- and out-group parameters, we run a simulation 20 times, and report on the average outcomes. Regarding the number of runs adopted, the literature knows a broad range of values in a trade-off between managing computational complexity and achieving statistical significance. Studies on simulations computing costly LLM outputs on each iteration have reported as little as three runs in exploration for sensitivity and stability [64]. The number of runs averaged when reporting average system properties knows examples ranging from tens (e.g., in similar settings in the 2002 Hegselman-Krause model [45]), to hundreds [65] and thousands [66]. Because some of our simulations run 1000 time steps, we average over 20 runs for each set of parameters to manage the computational complexity of our study.

Each time we instantiate the simulation drawing half of the opinions (50 agents) from a Normal distribution , and the other half from a Normal distribution , with . The resultant starting distribution of opinions is always in two groups and resembles those used to demonstrate the grouping mechanism in Fig 2. Initialising the experiments with two groups is of particular interest in our context since we want to understand group behaviors—with a unimodal or uniform distribution we would begin with one group or no group, respectively, both of which are not the focus of this article which studies in-/out-group exposure. Some results with different starting distributions can be found in the Supporting Information. For each set of parameters and for each drawn population at the first time step, we let the simulation run until convergence of the measured polarization of the population.

Polarization of the opinion distribution at each time step is given by the Duclos-Esteban-Ray (DER) measure [49], which allows us to compare across subsequent opinion distributions with a single value of polarization for each. The exact formula is

(3)

where x is an opinion, f(x) is the density at x, and we set . For implementing the measure we use kernel density estimation to approximate f(x) and the sample based estimator for described in [49, Sect 4]. The starting value is typically 0.21—the exact value varies depending on the precise random initialisation of opinions —which is mild polarization, between the minimum value 0 and maximum value 0.5.

The DER measure is thus constituted of a parametric family of functions, designed to accommodate desired properties of polarization measures. Axiom 1 posits that, in uni-modal distributions, polarization increases with spread. Axiom 2 posits that, if a distribution is a mixture model, a decrease of the spread of each component of the mixture increases polarization. Axiom 3 posits that, if a distribution is a mixture model, when components of the mixture model are distanced further apart, polarization increases. Axiom 4 imposes scalability and order for the measure. In other words, the parametric DER family of measures acting on a bimodal distribution is defined such that polarization increases if modes grow further apart, or if the distribution of each mode is further concentrated around the means of the modes. S4 Fig of S3 Sect of the Supporting information provides an illustration of different DER values for a bimodal 2-component Gaussian mixture model with different separations between components and spreads.

Each simulation is run for up to one thousand iterations, or terminated before that if opinions have not changed over one hundred consecutive iterations (X = 100 in Algorithm 1). Convergence is discussed in Supporting information S2, and curves of the DER value approaching convergence are displayed in S2 Fig. At each time step, each agent evaluates first the group composition of the system, then whether it is exposed to every other agent following the exposure rule, and finally updates its opinion based on the opinion update rule. Note that all updates are synchronous, so agents i and j update their opinion at the same time following the exposures. An outline of the full experiment process is presented in Algorithm 1. The results that we report in the next pages for each parameter set is the average of the final one hundred time step iterations across the twenty experiments. While the numerical simulations may evolve in up to a thousand time steps, our convergence criterion allows us to take the last hundred values to present the mean value.

Algorithm 1 Each experiment for some set of parameters follows these steps.

while do

  for in do

   ExposureRule()

  for in do

   UpdateRule()

  assign updated opinion synchronously

  if unchanged over the X previous iterations then

   break

Types of polarization achievable by the model

Before exploring the parameter space of the model, we discuss typical dynamical behaviors that are exhibited during the simulations. Convergence to consensus and drift to a complete polarization are known behaviors in the literature, which have been described in many works on opinion dynamics. Because of the split of parameters into in- and out-groups, our model can also display other dynamical patterns that can be best understood with simple examples, such as the ones illustrated with alluvial plots (see Fig 3). S3 Fig of the Supporting information illustrates the behavior of similar simulations, in which groups are fixed and not dependent on evolving positional opinions.

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Fig 3. Illustration of qualitative types of polarization achievable by the model, displaying positions of agents in each of the two initial groups, and their evolution for the first fifteen time steps of a simulation.

Vertical position is equivalent to the opinion position; i.e., position between 0 and 1 in opinion space. Groups are rectangular areas centered at the opinion mean of the group, with length representing the size of the group (not the opinion area they cover) hence a longer rectangle when the whole population is in one group (as in Fig 3a). Colour reflects the opinion value too. Four qualitative polarization types are illustrated: (a) consensus, (b) stable polarization, (c) unstable polarization due to group drift, and (d) unstable polarization due to group fragmentation.

https://doi.org/10.1371/journal.pone.0338486.g003

To achieve consensus, actors in a group must be exposed to those in another group (therefore of differing opinion) and be sufficiently tolerant so as to be attracted towards their opinion. Over a series of iterations we will eventually arrive at some opinion shared by the whole population. Fig 3a illustrates such a situation in which the two starting groups gradually arrive at a consensus opinion and combine to become one group.

Stable polarization occurs when out-group exposure is so low that the out-group opinions are rarely seen, and when they are observed they are outweighed by the in-group opinion. Thus the two groups neither attract nor repel. In this case the groups remain polarized but not to the extremes of the opinion space since each group is in its own bubble (Fig 3b).

Our model can also achieve unstable polarization states. These are either intermediate polarization states that will eventually converge to stable polarized states, or, crucially, states that are unstable because of group fragmentation. We show this through two qualitative examples. The first occurs when a group is exposed to another but does not tolerate the difference in opinion, so actors in this group drift to the extremes by repulsion — an example of this can be seen in Fig 3c. Fig 3d illustrates a second and different situation, in which small groups of agents float in the space between the two groups due to fragmentation. The in-between agents then either change group, rejoin the previous group, or stay in the middle, during the next iteration. Maximum polarization is never quite achieved with the middle ground populated by some agents despite the two starting groups being repelled by each other.

Effects of Group-Dependent Tolerance (GDT Simulations)

Regarding GDT simulations, Fig 4, we observe that at fixed R, higher exposure tends to increase the threshold at which we can have low polarization. While at fixed E, higher responsiveness results in more polarization as agents quickly find the extremes of the opinion scale from where it is hard to be attracted to any out-group opinion unless  = 1. This aligns with the main findings reported by [44] without groups. If R is low enough, then the boundary at which leads to a highly polarized state or a lowly polarized state becomes very sharp.

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Fig 4. Group-Dependent Tolerance (GDT) simulations.

The most important factor in determining polarization outcomes of the population is as it decides whether two groups will be attracted to each other or separate. This is dependent on E being high enough that out-group opinions are observed often enough to impact the in-group.

https://doi.org/10.1371/journal.pone.0338486.g004

typically does not affect the general behavior (for common E and R values) unless it is so low that it causes a fragmentation of the in-group from within. This can be observed under low responsiveness (R = 0.01) and some level of out-group exposure () where we do not arrive at maximum polarization for  = 0.05 due to the middle ground between the two extremes being occupied by some agents (as illustrated in Fig 3d). At higher R, the middle ground agents’ opinion changes so drastically that it arrives at the extremes in one iteration, so they break out of the middle ground.

These observations can be appraised in Fig 4, in which the primary conclusion is that out-group tolerance is the main driver of behavior change in our model. Ultimately, it governs whether or not two groups can come together to consensus by attraction to each others’ opinions. It is not necessary that opinion difference with the whole out-group is below the tolerance threshold but that a sufficient number of out-group agents are within the limit to result in net attraction between groups. As such, the distance between groups being more or less than the out-group tolerance is critical to the resultant behaviors.

Where pixelation occurs in Figs 4, 5, and 6, it is due to volatility in the final DER polarization of the population across simulations, for which there are two reasons. The first is that fragmentation, as in Fig 3d, occurs and results in a variety of possible polarization values. The other is that the population has some probability of range of stable outcomes and so the average DER value across experiments is less predictable – for example, in Fig 4 when  = 0.4, seventeen out of the twenty simulations result in a DER value over 0.4000 with the population split between two roughly equal groups at the extremes of the opinion space, however one simulation run ends (stably) with eighty-four agents of opinion 1 and the remaining sixteen agents of opinion 0 giving a resultant 0.2499 DER value, since one group is far larger than the other. This typically occurs at a boundary between zones of behavior or due to model parameter combinations that result in large opinion shifts of agents and so either end of the polarization spectrum may be achieved. The standard deviation of final state DER polarization across experiments reflects this variability and is therefore higher in areas where more pixelation occurs.

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Fig 5. Group-Dependent Responsiveness (GDR) simulations.

The size of response in attraction or repulsion governs a more gradual change of behavior when compared to the sharp boundaries for T in Fig 4. Out-group treatment is again most important when exposure to different opinions occurs regularly.

https://doi.org/10.1371/journal.pone.0338486.g005

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Fig 6. Group-Dependent Exposure (GDE) simulations.

Over-exposure to in-group opinions and under-exposure to out-group is a scenario that mirrors concerns around filter bubbles. In our model, polarization is more likely with higher implying that polarization occurs from exposure to differing opinions that is not tempered by either high tolerance of opinions or larger exposure to in-group.

https://doi.org/10.1371/journal.pone.0338486.g006

Effects of Group-Dependent Responsiveness (GDR Simulations)

Responsiveness determines the intensity of the agents’ reactions in how they update opinions. When agents have low responsiveness to others they creep towards average opinions or slowly separate, while when the parameter is high they react strongly by pushing out to extremes of the opinion scale if repulsed or even overshooting attraction to a new opinion in the case when a different opinion value is observed twice, R is greater than 0.5, and no other opinions are observed — a behavior that does not align with reality but is presented here for comprehensiveness. We now explore varying responsiveness dependent on group identification under different model conditions.

At low exposure, E = 0.01, increasing tolerance tends to reduce polarization of the population. Furthermore, in the cases T = 0.1 and 0.4, around  = 0.5 we can observe a smooth transition between higher polarization for greater and lower polarization for smaller . We propose that this occurs due to the larger changes introduced by greater , meaning that agents tend to react more strongly to the frequent exposures to in-groups, akin to increased noise in the system so it cannot settle.

At intermediary exposure, E = 0.05, the system presents the most variety of behaviors in the relation between and as we vary tolerance. Across tolerance levels, an > 0.5 typically leads to polarization although this becomes less certain as tolerance increases. When considering we also see a tendency towards lower polarization when the parameter is lower for T = 0.01, it is unclear if this is also the case for since pixelation increases in these scenarios so the simulation outcomes are less clear. Accepting some exceptions increased responsiveness to both in-group and out-group typically results in increased polarization.

Finally, at high exposure, E = 0.25, almost all cases lead to maximum polarization no matter the level of group-dependent responsiveness. The exception being the case of low out-group responsiveness with high general tolerance (< 0.2 and T = 0.4) which can result in lower polarization, this outcome arises due either to agents not overreacting to repulsive out-group or not overshooting attraction to in-group.

Effects of Group-Dependent Exposure (GDE Simulations)

At low tolerance, the impact of increasing is to increase repulsion between groups as repulsive influences become a higher proportion. As tolerance increases, lower polarization becomes possible, although responsiveness also needs to remain at low levels. Note that the experiment under R = 0.01, T = 0.4 (low responsiveness, high tolerance) is the only experiment shown in Figs 4, 5, and 6, that arrives at complete consensus no matter what the group-dependent parameters are.

In-group exposure can be a tempering factor to reduce polarization when agents are not overly responsive. This occurs because the polarizing exposure to out-group opinions is countered by higher exposure to in-group opinions which result in an opinion update at each iteration that is dominated by movement towards in-group average.

Exposure to certain groups of users dependent on whether they are similar (in-group) or not (out-group) draws clear parallels with the much discussed topic of filter bubbles. Note however that the mechanism considered in our model integrates only a limited range of features that a Recommender System might include (namely homophily), notably excluding content-based recommendations or behavioral trace data (e.g., what users click, like, share). Yet, we observe that increasing the probability of observing out-group opinions by raising typically increases the level of polarization in the population. Similar to tolerance, the out-group parameter is more explanatory for the polarization behavior of the model; this backs up our understanding that polarizing behavior tends to occur through negative reaction to out-groups.