2.1 Definition and interpretation of opinions

The dynamics evolve over rounds indexed $t=1,2,\ldots$ . At the start of each round our agent holds an opinion $a$ from the set of possible opinions $\mathcal{A}$ . We refer to the opinion held by the agent in round $t\in \mathbb{N}$ as $a_t\in \mathcal{A}$ . The agent is subsequently exposed to an experience. We call the outcome of an experience in round $t$ holding opinion $a\in \mathcal{A}$ : $X_a^t\in \{0,1\}$ . When $X_{a}^t$ takes the value one, the agent’s opinion $a$ aligns with (is reinforced by) the experience in round $t$ . Conversely, when $X_a^t$ takes the value zero, the agent’s opinion $a$ does not align with the experience in round $t$ . Specifically $X_a^t$ for all $a\in \mathcal{A}$ and $t\in \mathbb{N}$ are random variables:

(1) \begin{equation} X_a^t = \begin{cases} 1, \quad &\text{with probability } \theta _a\\ 0, &\text{with probability }1-\theta _a, \end{cases} \quad \text{with }\theta _a \in (0,1), \forall a\in \mathcal{A}. \end{equation}

We call the probability that opinion $a$ aligns with an experience, opinion $a$ ’s reliability, $\theta _a\in (0,1)$ for all $a\in \mathcal{A}$ . Experiences $\{X_a^t\,:\,t\in \mathbb{N}\}$ are i.i.d. for each opinion $a\in \mathcal{A}$ . We note the conscious decision to not fix $X_a^t=1-X_b^t$ or $\theta _a =1-\theta _b$ for $a\neq b$ and $a,b \in \mathcal{A}$ . While opinions $a\neq b$ from $\mathcal{A}$ are indeed concerning the same subject, we allow for overlap in the sets of experiences which may align with different opinions. Consider, for example, that dissatisfaction with one political party does not necessarily imply satisfaction with an alternative. The agent only interprets experiences using the opinion they hold. This means they do not observe $X_b^t$ for $b\neq a_t$ . We assume this for simplicity. If the agent did observe $X_b^t$ for $b\neq a_t$ would imply that agents interpret all of their experiences with each possible opinion. Furthermore, they do not know the respective opinion’s reliabilities $\theta _a$ for $a\in \mathcal{A}$ . We suppose that the agent receives utility $p\in \mathbb{N}$ when an experience aligns with their opinion ( $X_a^t = 1$ ) and loses utility $l\in \mathbb{N}$ when it not( $X_a^t = 0$ ).

The analogy of viewing an opinion as a lens through which to interpret experiences may be elaborated into a set of views which comprise something like a narrative about a particular subject. The narrative we follow may provide: explanations for experiences, a guideline on how to act, heuristic answers to questions, and an ideal to strive for. We present an example of our interpretation from the folklore of Robin Hood.

The binary expression of the opinions in this example is support for Robin or the Sheriff. The opinions on the other hand are complex objects. In the case of political preferences our “narrative” interpretation aligns with the description of an ideology summarized by Weber (Reference Weber2019). We use “opinion” rather than “narrative,” in order to avoid (direct) connection to how narratives within media might influence opinions.

2.2 Agent belief

The agent holding opinion $a\in \mathcal{A}$ has a belief which measures the strength of their conviction of their opinion. This takes the form of a belief density (function) $b(x)$ and a corresponding belief distribution (function) $B(x)$ . These are related in the usual way:

(2) \begin{equation} B(x) = \int _0^x b(s)\text{d}s. \end{equation}

In particular the belief distribution denotes the subjective probability (as in the Bayesian interpretation of probability) they place on the true reliability of the opinion they hold being below a given value,

(3) \begin{equation} B(x)\,:\!=\, \mathbb{P}(\theta _a\leq x)\quad x\in [0,1]. \end{equation}

The agent uses their belief distribution to attain an estimate for the true value of $\theta _a$ . In the absence of other evidence, the agent uses their prior belief. After any number of experiences, the agent adjusts their belief accordingly.

2.2.1 Prior belief distribution

We model the agent to start with a prior belief distribution which is what they believe the distribution function of $\theta _a$ to be for all opinions $a\in \mathcal{A}$ that they have no other information about. The agent has prior belief density $b_0(x)$ and distribution $B_0(x)$ meaning that they initially believe that the probability relating to the reliability of opinion $a_{0}$ is such that:

(4) \begin{equation} \mathbb{P}(\theta _{a_{0}}\leq x) =B_0(x) = \int _{0}^{x} b_0(s)\text{d}s. \end{equation}

If the agent switches their opinion at some time $t_0$ to opinion $a_{t_0}\in \mathcal{A}$ then they revert to their prior belief $b_0(x)$ regardless of which opinion they are switching to. This models the agent’s forgetting of experiences with an opinion they may have held in the past. The formulation (4) generalizes the formulation of belief in Meylahn et al. (Reference Meylahn, den Boer and Mandjes2024) by allowing general belief distributions instead of only the Beta-distribution.

In agent-based models with more richness, agent features may play an important role in determining the prior an agent has for each of the opinions. Examples of characteristics that may play a role are age, education attained and current employment. We assume that their prior beliefs are all the same for simplicity.

2.2.2 Belief distribution update

We call the consecutive rounds in which the agent held opinion $a$ , a run with opinion $a$ . We refer to the most recent switching time as $S_t$ , which identifies the round in which the current run started:

(5) \begin{equation} S_t\,:\!=\,\min \{n\,:\,a_{m}=a_{t}, \forall m\in [n,t]\}. \end{equation}

We model the agent to “forget” previous runs with an opinion. In constructing their belief distribution of an opinion, they only use their most recent run’s history which started at time $S_t$ . This means that they only use the experiences gained since they most recently switched to the opinion they are currently holding to formulate their belief distribution of that opinion. We define the agent’s current experience history until the end of round $t\in \mathbb{N}$ as $H_t$ . That is the set of experiences observed up until (and including) round $t$ during their current run:

(6) \begin{equation} H_t = \{X_{a_{n}}^n \,:\, n \in [ S_t,t] \} \quad \text{for }t\in \mathbb{N}. \end{equation}

This information along with their prior belief is used by the agent to formulate a belief distribution at time $t\in \mathbb{N}$ :

(7) \begin{equation} B_0\times H_t\to B_t(x). \end{equation}

With this incorporation of an agent’s ability to “forget” it may be noted that more complicated mechanisms are possible, and we do not insist that ours is the best, however, we do suggest there is value in having at least some form of forgetting in a model of opinion dynamics which includes learning.

2.2.3 Point estimate of opinion reliability

We model the agent to use this belief to attain a point estimate $\widehat{\theta }_{t}\in [0,1]$ of the reliability, $\theta _{a_{t}}$ . A common example is the mean value of the belief density,

(8) \begin{equation} \widehat{\theta }_{t} =\int _0^1 x b_t(x) \text{d}x, \quad \in [0,1]. \end{equation}

Alternatives to the mean value are the upper or lower confidence intervals and many more. The way in which the history is incorporated is part of the modeller’s choice. A logical choice for a Beta-distributed prior belief is Bayesian updating. Alternatively if the prior is a point mass on the estimate, simple exponential smoothing might better serve the task.

Note that each agent has only one belief distribution at any time which is their belief in the opinion they currently hold. We assume, as simplification, that the agents do not continuously compare the alternative interpretations (resulting from different opinions) of an experience. This is what would be required to track the reliability of each opinion in order to explicitly compare these. Should the number of possible opinions be large, agents which compare between all opinions often would be faced with a heavy computational effort.

2.3 Threshold decision making (choosing an opinion)

The agent is faced with deciding whether to place their trust in the opinion which they held in the previous round. They do so by a satisficing procedure (cf. Simon Reference Simon1956; Artinger et al. Reference Artinger, Gigerenzer and Jacobs2022); checking if the current opinion is good enough. The agent decision-making process we use has mechanical similarities with the “state system” in the “consumats” model proposed by Jager and co-authors (see e.g., Jager et al. Reference Jager, van Asselt, Boodt, Rotmans and Vlek1995, Reference Jager, Janssen and Vlek1999; Janssen and Jager, Reference Janssen and Jager1999). In this consumer behavior model, agents who are satisfied with their consumption do not spend cognitive resources to find alternatives as long as they deem their current options to be good enough. It is only the dissatisfied consumer who investigates other avenues of consumption as motivated by their dissatisfaction.

In choosing an opinion to hold the agent asks themselves whether they expect positive utility from the opinion they are currently holding. In other words they check the truth of the inequality:

(9) \begin{equation} p\widehat{\theta }_{t}-l(1-\widehat{\theta }_{t})\geq 0. \end{equation}

This inequality may be rearranged and so equivalently the agent asks themselves whether:

(10) \begin{equation} \widehat{\theta }_{t}\geq \frac{l}{p+l} \,=\!:\, \theta _{\text{crit}}\in (0,1). \end{equation}

Here we have defined $\theta _{\text{crit}}$ , the minimum reliability point estimate the agent requires an opinion to have in order to continue holding that opinion. If the agent chooses to switch opinions (because they are sufficiently dissatisfied), they choose a new one from the set of opinions excluding the opinion they are switching from. The choice of the agent at time $t\in \mathbb{N}$ can now be defined:

(11) \begin{equation} a_{t} = \begin{cases} a_{t-1},\quad &\text{if }\widehat{\theta }_{t}\geq \theta _{\text{crit}},\\ b \in \mathcal{A}\setminus a_{t-1}, &\text{otherwise}. \end{cases} \end{equation}

This generalizes the decision making in the single agent model presented by Meylahn et al. (Reference Meylahn, den Boer and Mandjes2024) from trusting or not trusting to holding one of the opinions in $\mathcal{A}$ . The protocol used to choose which of the alternative opinions the agent chooses is up to the modeller. For a set $\mathcal{A}$ of only two opinions the choice is straightforward; simply the other opinion. If there are three or more opinions the choice could make use of a direct comparison between the remaining options to decide which opinion to switch to. These mechanisms of switching in the solo agent model serve as a baseline. In the multiple agent model there are more sophisticated and believable mechanisms by which an agent might choose which opinion to switch to.

This is a simple satisficing model in which the agent is not concerned with continuous comparisons between opinions. Instead the agent has a desired level of reliability and retains the opinion they are holding if they believe it to satisfy this level.

3.1 A network of solo agents

The framework does not prescribe a population structure but assumes that one is given. That is, either the modeller uses a network empirically sourced or makes use of an appropriate random graph model to generate a network. The interested reader may consult the work of Robins, et al. (Reference Robins, Pattison and Elliott2001) relating to models of network generation. We suppose that there exists a population of agents of finite size $N\in \mathbb{N}$ . The population is embedded in a network $G=(V,E)$ , with $|V| = N$ vertices representing agents and a set of social ties represented by the edges, $E$ . The edges are directional in nature; an edge $(u,j)\in E$ indicates that agent $u$ exerts social influence on agent $j$ .

We define the opinion held by agent $j\in V$ in round $t$ as $a_{t}(j)\in \mathcal{A}$ . In each round every agent (a) holds an opinion, (b) has an experience which corroborates or contradicts their opinion and (c) observes the opinions held by their neighbors. The outcome of the experience had by agent $j\in V$ , holding opinion $a\in \mathcal{A}$ at time $t\in \mathbb{N}$ is the random variable:

(12) \begin{equation} X_a^t(j) = \begin{cases} 1\quad &\text{with probability }\theta _a,\\ 0&\text{with probability } 1-\theta _a, \end{cases}\quad \text{with }\theta _a\in (0,1), \forall a\in \mathcal{A}. \end{equation}

Similarly to the solo agent model, agent $j\in V$ receives utility $p_j\in \mathbb{N}$ when an experience agrees with their opinion and loses utility $l_j\in \mathbb{N}$ when an experience disagrees with their opinion. We think it is a reasonable starting point to assume that $\theta _a$ for $a\in \mathcal{A}$ is the same for all agents because it is a property of the opinion rather than the agents.

Now each agent $j\in V$ is equipped with a prior belief density $b_0^j(x)$ and distribution $B_0^j(x)$ relating to the reliability of opinion $a_{0}(j)$ as before:

(13) \begin{equation} B_0^j(x)\,:\!=\,\mathbb{P}(\theta _{a_{t}(j)} \leq x) = \int _0^xb_0^j(s)\text{d}s,\quad x\in [0,1]. \end{equation}

Agents may switch between opinions and so we refer to agent $j$ ’s most recent switching time:

(14) \begin{equation} S_t(j)\,:\!=\,\min \{n\,:\,a_{m}(j)=a_{t}(j), \forall m\in [n,t]\}, \quad \text{for }j\in V \text{ and }t\in \mathbb{N}. \end{equation}

Subsequently, agent $j$ ’s current experience history until the end of round $t\in \mathbb{N}$ is $H_t(j)$ . That is the set of experiences observed up until (and including) round $t$ during their current run:

(15) \begin{equation} H_t(j) = \{X_{a_{n}(j)}^n(j) \,:\, n \in [ S_t(j),t] \}, \quad \text{for }j\in V,\text{and }t\in \mathbb{N}. \end{equation}

This information along with their prior belief is used by the agent to formulate a belief distribution for the opinion they are holding at time $t\in \mathbb{N}$ :

(16) \begin{equation} B_0^j\times H_t(j)\to B_t^j(x). \end{equation}

This belief, in turn gives an updated estimate $\widehat{\theta }_{t,j}$ (assuming use of the mean value),

(17) \begin{equation} \widehat{\theta }_{t}(j)= \int _0^1xb_t^j(x) \text{d}x, \end{equation}

representing the strength of their conviction in the opinion they are holding.

3.2 Social influence

The process of social influence we present aligns with the internalization process of opinion change presented by Kelman (Reference Kelman1961). That is, the agents are using the information provided by their network to facilitate their decision making, rather than simply trying to comply or identify with one another. An agent $u\in V$ influences agent $j\in V$ if there is an edge $(u,j)\in E$ . Each agent $j\in V$ has a set of social ties which we call the neighborhood of agent $j$ :

(18) \begin{equation} N(j)\,:\!=\, \left \{u\,:\, (u,j)\in E\right \}, \end{equation}

that is the set of agents who are said to influence agent $j$ . For ease of notation we assume that the edges between agents are unweighted. Agent $j\in V$ observes the opinions held by the agents in their neighborhood $N(j)$ . We thus model agents to communicate only which opinion they hold to their neighbors and not their belief, i.e. the strength of their conviction in that opinion. This assumption is based on the idea that it is much easier to convey a discrete choice of opinion to a social connection than it is to elucidate the nuances involved in the procedure by which such a choice was made. This provides agent $j\in V$ with their network influence set for time $t\in \mathbb{N}$ :

(19) \begin{equation} I_t(j) \,:\!=\, \{a_{t}(k)\,:\, k\in N(j)\}. \end{equation}

In formulating the influence of $N(j)$ on agent $j\in V$ we aim to follow empirical literature which shows that peer-to-peer influence has its effect not in cognitive elements of belief but rather affective elements influencing decision making (Johnson and Grayson, Reference Johnson and Grayson2005; Ozdemir et al., Reference Ozdemir, Zhang, Gupta and Bebek2020). In doing so, we posit that the role of peer-to-peer influence is similar in the context of brand loyalty and the expression of opinions. Indeed the methods of marketing are commonly used in politics (Newman. Reference Newman2002) which is one of the main arenas of opinions dynamics. The affective elements may of course still influence decision making, but the channel they follow does not effect the rational calculating elements associated with decision making. Instead the agent’s threshold is adjusted based on the information gained from their network. Consider an agent’s thoughts: ‘If it is good enough for my neighbors, why should it not be good enough for me?’

Define $\theta _{\text{crit},t}^*(j)$ as the network adjusted critical reliability of agent $j\in V$ at time $t\in \mathbb{N}$ :

(20) \begin{equation} \theta _{\text{crit},t}^*(j)\,:\, p_j\times l_j\times I_t(j) \mapsto [0,1]. \end{equation}

The elements this function may use are thus contained in $p_j$ , $l_j$ and $I_t(j)$ . We define the number of agents in $j$ ’s neighborhood expressing the same opinion as agent $j\in V$ at time $t\in \mathbb{N}$ as:

(21) \begin{equation} m_t(j) \,:\!=\, \left|\left\{b\,:\, \left (b\in I_t(j)\right ) \land \left (b=a_{t}(j)\right )\right\}\right|, \end{equation}

i.e. the number of agents in agent $j$ ’s neighborhood matching agent $j$ ’s opinion. Similarly we define the number of agents in agent $j$ ’s neighborhood not matching agent $j$ ’s opinion as:

(22) \begin{equation} n_t(j)\,:\!=\, |N(j)|-m_t(j). \end{equation}

Note that if it is desirable to have weighted connections between agents, the above may be replaced with total weight within agent $j$ ’s neighbohood in agreement with agent $j$ ’s opinion and the total weight remaining, respectively. The functions $\theta _{\text{crit},t}^*(i)$ can be chosen in various ways. We suggest the following properties of the network influence function:

• • If there is equal support and opposition ( $m_t(j)=n_t(j)$ ), the effect is null, $\theta _{\text{crit},t}^*(j)=\theta _{\text{crit}}(j)$ .

• • If there is more support than opposition ( $m_t(j)\gt n_t(j)$ ), then the threshold is lowered, $\theta _{\text{crit},t}^*(j)\lt \theta _{\text{crit}}(j)$ .

• • If there is more opposition than support ( $m_t(j)\lt n_t(j)$ ), then the threshold is increased, $\theta _{\text{crit},t}^*(j)\gt \theta _{\text{crit}}(j)$ .

The belief updating of the agents thus has not changed, yet their decision making is affected by the communication of their neighbors. This means that $\widehat{\theta }_{t}(j)$ is still constructed as in the solo agent model. Instead of comparing this believed point estimate to a constant threshold $\theta _{\text{crit}}(j)$ , agent $j\in V$ uses the truth of the inequality $\widehat{\theta }_{t}(j)\gt \theta _{\text{crit},t}^*(j)$ to determine whether they keep holding their opinion. Note that there are two possible triggers for an agent to switch their opinion. Their estimate may drop below their critical reliability as a result of one too many experiences which did not align with their opinion. Alternatively, a change in the opinions held by their neighbors may increase their critical reliability above their current estimate. Once an agent has decided to switch their opinion, they might make use of one of a number of switching rules to figure out which opinion to switch to:

• • Agents can make a direct comparison of the remaining opinions based on their rational beliefs of those opinions, choosing the ‘best’ one. Such reasoned behavior has been studied extensively (Fishbein and Ajzen, Reference Fishbein and Ajzen1977; Ajzen. Reference Ajzen1991), for a recent description and discussion of developments consult (Ajzen, Reference Ajzen2020).

• • The agent might imitate a neighbor. Jager and various co-authors (Jager et al., Reference Jager, van Asselt, Boodt, Rotmans and Vlek1995, Reference Jager, Janssen and Vlek1999 Janssen and Jager, Reference Janssen and Jager1999) suggest that imitation is most likely to occur when agents are faced with high uncertainty about their options and are not willing to spend a lot of cognitive resources.

• • Or the agent can adopt the majority opinion within their neighbohood. This corresponds to the expected choice under the random imitation described in the previous option.

The choice of switching rule is up to the modeller using the framework and may well be heavily influenced by the topic of the opinions. For a more detailed discussion on agent decision making the interested reader may consult the survey of Balke and Gilbert (Reference Balke and Gilbert2014).

This formulation of social influence is in stark contrast with the mechanism of social influence in the dual agent model presented by Meylahn et al. (Reference Meylahn, den Boer and Mandjes2024). In their study, agents use the observation of their neighbors’ action rationally to adjust their belief, our agents simply use the heuristic of changing the threshold which they use for the decision making. This saves a lot of computation making the model tractable for more than two agents.

3.3 Summary of framework

Agents choose an opinion to hold in each round $a_{t}(j)\in \mathcal{A}$ . Subsequently they experience agreement or disagreement. The effect of this experience is an updated belief distribution $B_t^j$ based on $H_t(j)$ . In order to choose whether to switch opinion in the following round, they compare a point estimate from their belief $\widehat{\theta }_{t}(j)$ with a critical reliability $\theta _{\text{crit},t}^*(j)$ . This critical reliability is adjusted according to the opinions expressed by the agent’s neighbors captured in $I_t(j)$ . Their own opinion expression is also how the agent influences their neighbors. Figure 1 serves to illustrate how the elements of the framework fit together. In the single agent model, the agent’s opinion boils down to a ‘strong enough’ belief. In the multi-agent model, an agent’s opinion is the result of a process combining cognitive processing of experiences in the agent belief and the affective forces of social influence.

Figure 1.

A graphical illustration of the opinion dynamics framework proposed.

4.1 Belief in the opinion

The agents in our model have a prior belief density in the form of a Beta-distribution with shape parameters $\alpha ,\beta \in \mathbb{N}$ ,

(23) \begin{equation} b_0(x) = \frac{x^{\alpha -1} (1-x)^{\beta -1}}{\int _0^1 y^{\alpha -1}(1-y)^{\beta -1}{\textrm{d}}y}. \end{equation}

At time $t\in \mathbb{N}$ , each agent $j\in V$ is only aware of the most recent history $H_t(j)$ , which pertains to the opinion they currently hold. As such the agent keeps track of the number of confirming experiences during their most recent history $H_t(j)$ up until time $t\in \mathbb{N}$ using:

(24) \begin{equation} c_t(j) = \sum _{n=S_t(j)}^t X_n. \end{equation}

Similarly they use

(25) \begin{equation} r_t(j)=\sum _{n=S_t(j)}^t (1-X_n), \end{equation}

to keep track of the number of experiences during their most recent history $H_t(j)$ up until time $t\in \mathbb{N}$ which refute their current opinion. Furthermore, the agents make use of Bayesian belief updating to keep:

(26) \begin{equation} B_t^j(x) = \mathbb{P}\left (\theta _{a_{t}(j)}\leq x\mid H_t(j)\right ), \quad \text{for }t\in \mathbb{N}. \end{equation}

With the convention that if $H_t(j) = \emptyset$ , then $\mathbb{P}(\theta _{a_{t}(j)}\leq x) = B_0^j(x)$ . The agent $j$ thus holds a belief density $b_{t}^j(\theta )$ at time $t\in \mathbb{N}$

(27) \begin{equation} b_{t}^j(x)=\frac{x^{c_t(j)}(1-x)^{r_t(j)}b_0^j(x)}{\int _0^1 y^{c_t(j)}(1-y)^{r_t(j)}b_0^j(y){\textrm{d}}y}, \quad x\in [0,1]. \end{equation}

As point estimate the agents use the mean of their belief distribution at time $t\in \mathbb{N}$ . Conveniently, for the combination of Beta-distributed prior belief and Bayesian updating the mean of the belief distribution is given by

(28) \begin{equation} \widehat{\theta }_{t}(j) =\frac{\alpha + c_t(j)}{\alpha +\beta +c_t(j)+r_t(j)},\quad \forall t\in \mathbb{N}, \text{ and all }j\in V. \end{equation}

4.2 Threshold and network influence

In equations (21) and (22) in §3.2 we have defined the number of agents in agent $j$ ’s neighborhood in agreement with agent $j\in V$ at time $t\in \mathbb{N}$ as $m_t(j)$ and the number of agents in disagreement as $n_t(j)$ , respectively. We define the network influence function $\theta _{\text{crit},t}^*(j)$ representing the influence of agent $j\in V$ ’s neighborhood on their decision making as a mapping $\theta _{\text{crit},t}^*(j)\,:\,p_j\times l_j\times I_t(j)\mapsto [0,1]$ by

(29) \begin{equation} \theta _{\text{crit},t}^* (j) = \frac{p_j +f_j(I_t(j))}{p_j+l_j}, \quad \text{for }t\in \mathbb{N}, \end{equation}

with the crucial element $f_j(I_t(j))$ defined:

(30) \begin{equation} f_j(I_t(j))\,:\!=\,\frac{n_t(j)-m_t(j)}{\kappa _j |N(j)|},\quad \text{for }t\in \mathbb{N}, j\in V, \end{equation}

where $\kappa _j\in (0,\infty )$ is agent $j$ ’s stubbornness parameter. A large $\kappa$ represents agents who are not very influenced by their neighborhood. In fact if $\kappa$ is large enough for an agent, their opinion switching happens independently from their neighborhood.Footnote ³ It should be noted that even the most stubborn agents in our model may change their opinion based on their individual belief updating. For the most stubborn agent possible ( $\kappa _j=\infty$ ) the network influence becomes negligible ( $f_j \to 0$ ) and so their criteria remains unchanged $\theta _{\text{crit},t}^* = \theta _{\text{crit}}$ . It is possible that this agent’s estimate of their opinion’s reliability drops below this threshold and so they can still switch their opinion. This highlights that the stubbornness in our model is not a stubbornness of a particular opinion but rather a stubbornness with respect to influence from others.

It is worth mentioning that if $\kappa$ is small enough for all agents then the model reduces to one in which agents adopt the opinion being held by the majority of their neighborhood.Footnote ⁴

We use a natural starting point for utility parameters. By setting $l_j=p_j=1$ for all $j\in V$ we have that every agent’s starting threshold is $\theta _{\text{crit}} = 1/2$ . This means that an agent in the absence of a network influence will continue to hold their current opinion if their belief satisfies: $\widehat{\theta }_{t}(j)\gt 0.5$ . The resulting decision making is made by checking the inequality of $\widehat{\theta }_{t}(j)\gt \theta _{\text{crit},t}^*(j)$ :

(31) \begin{equation} \frac{\alpha + c_t(j)}{\alpha +\beta +c_t(j)+r_t(j)}\geq \left ({1+\frac{n_t(j) - m_t(j)}{\kappa _j|N(j)|}}\right )/{2}. \end{equation}

A graphical representation of this model is presented in Figure 2. This example has three agents $V=\{1,2,3\}$ with connections $E=\{(1,2),(2,1),(1,3),(3,1)\}$ . The question mark icons represent a check of the inequality (31). A transition from trusting one opinion to trusting the other should take place whenever this inequality is not true.

Figure 2.

A graphical illustration of an example network agent model.

4.3 Observable outcomes of a simulation

In the model we have presented, agents are able to converge onto an opinion. For an agent $j\in V$ holding opinion $a\in \mathcal{A}$ , with enough time (and thus sampling of experiences) the agent’s estimated reliability may tend towards the true reliability, $\widehat{\theta }_{t}(j)\to \theta _a$ and if $\theta _a\gt \theta _{\text{crit},t}^*(j)$ it is possible that the agent holds this opinion indefinitely. If at some time $t_0\in \mathbb{N}$ this is true for all the agents in the network, the process has reached a steady state, as anymore switching of opinions becomes more unlikely. For the purposes of the simulation, we use a proxy for steady state: 100 simulated rounds in which no agent switches their opinion. We are interested in whether there is consensus in such a steady state or if there is some level of discordance. We define the probability of consensus as the probability of each agent holding the same opinion at a steady state time $t_0\in \mathbb{N}$ :

(32) \begin{equation} C\,:\!=\,\mathbb{P}\left ( a_{t_0}(j) = a_{t_0}(1), \forall j \in V \right ). \end{equation}

The choice of reference to the first agent’s opinion $a_{t_0}(1)$ is arbitrary as all agents are required to be in agreement.

We define the proportion of discordance as the number of discordant edges divided by the total number of edges. A discordant edge $(u,v)$ is one in which the opinions of the agents are different: $a_u\neq a_v$ . We label the set containing the discordant edges at time $t\in \mathbb{N}$ , $E_D(t)$ :

(33) \begin{equation} E_D(t)\,:\!=\, \{(u,v)\mid a_{t}(u)\neq a_{t}(v),\text{with } (u,v)\in E\}. \end{equation}

As such we can define the asymptotic proportion of discordance as:

(34) \begin{equation} D\,:\!=\, \mathbb{E}\left [\liminf _{t\to \infty }\frac{|E_D(t)|}{|E|}\right ]. \end{equation}

Considering that we are performing a simulation study we can only approximate the quantities of interest with empirical measures at the end of simulation runs. To this end we consider the empirical value:

(35) \begin{equation} \widehat{C} = \frac{z_c}{Z_{\text{sim}}}, \end{equation}

where $z_c$ is the number of simulation runs in which $a_{t_0}(j) = a_{t_0}(1), \forall j \in V$ at termination time $t_0$ , and $Z_{\text{sim}}$ is the total number of simulations that were run. Similarly, for the proportion of discordance:

(36) \begin{equation} \widehat{D} = \frac{\sum _{n=1}^{Z_{\text{sim}}}|E_D(t_0, n)|}{Z_{\text{sim}}\times |E|}, \end{equation}

where $E_D(t_0, n)$ is the set of discordant edges at the time when the $n$ -th simulation iteration terminates $t=t_0$ . In other words, we sum the number of discordant edges at the termination time of each simulation run and divide this by the total number of edges. This gives the average proportion of edges that were discordant at termination time across all simulations.

5.1 Number of agents

We vary the total number of agents $N\in \{20,30,40,50\}$ . Additionally, for each of these values of $N$ , we vary the nearest number of neighbors taking values $l\in \{4,6,8\}$ . We present the results grouped according to the number of nearest neighbors. Within Figures 3a, 4a, and 5a, depicting the probability of consensus, we observe that as the population grows, the probability of consensus decreases. By comparing between these figures we see that as the number of neighbors increases the probability of consensus increases. Both of these results are conceivable. A larger population (keeping the number of neighbors constant) is likely to make it difficult for an opinion to spread throughout the entire network. Similarly, the more neighbors the agents have (keeping the population size constant), the easier it should be for an opinion to spread throughout the population. Quite logically, we observe the inverse effect in Figures 3b, 4b, and 5b, on the level of discordance. We would like to draw the reader’s attention to the fact that both of these effects (comparing between population size) becomes smaller as the stubbornness $\kappa$ increases to $5$ . We believe this is conceivable because a greater $\kappa$ means that the agents in the population become increasingly independent of one another and thus are less effected by the network they form a part of.

Figure 5.

Results of the sensitivity analysis inspecting the number of agents in the system with 8 nearest neighbors. Parameters: $d=8$ , $w=0.2$ , $t_s=5$ , $\alpha = 4$ , $\beta = 2$ , and $\theta _0=\theta _1 = 0.6$ .

5.2 Probability of rewiring

In order to see how the probability of rewiring affects the outcome of the model, we take the probability of rewiring each edge from the set $w\in \{0,$ $0.05,$ $0.10,$ $0.15,$ $0.20,$ $0.25\}$ . We present the probability of consensus for these probabilities of rewiring in Figure 6a and the level of discordance in Figure 6b. Again we can see the effect of the network decreasing as $\kappa$ increases in Figure 6a by the fact that the probabilities of consensus seem to merge at roughly $\kappa =4$ . Sensibly, we see that the probability of consensus is higher for networks with more rewiring. We posit that this is because of the greater number of cross population connections which make the formation of two (or more) polarized groups more difficult.

We also observe an interesting phenomenon in Figure 6b. Namely, that level of convergence for different probabilities of rewiring all seem to intersect just after $\kappa =2$ . Furthermore, these have an inverse ordering before and after this crossing point. Our explanation for this effect follows: Before the crossing point, a higher probability of rewiring makes polarization harder, and so the population tends toward consensus which has the lowest level of discordance. The setting with low level of rewiring has much more structure which allows more easily for polarization. After the crossing point, a high enough $\kappa$ creates more room not only for polarization but also fragmentation. As the agents become more independent, the cross population connections at greater $w$ create a higher level of discordance. More structured populations with low $w$ have more of a balance between polarization (with relatively low discordance) and some fragmentation.

Figure 6.

Results of the sensitivity analysis inspecting the probability of rewiring. Parameters: $N=20$ , $d=4$ , $t_s=5$ , $\alpha = 4$ , $\beta = 2$ , and $\theta _0=\theta _1 = 0.6$ .

5.3 Prior belief

The parameters $\alpha$ and $\beta$ of the beta-distributed belief are a measure of optimism in the agents in which a larger ratio $\alpha /\beta$ indicating greater optimism. To identify the effect of this prior belief distribution, we choose values $(\alpha ,\beta )\in \{(4,4), (4,2), (3,3), (3,2), (2,2), (2,1), (1,1)\}$ . This takes into consideration that $\alpha \geq \beta$ must hold. This is required to ensure that $\widehat{\theta }_0\gt \theta _{\text{crit}}=0.5$ and the agents do not switch immediately away from an opinion they have just switched to. First, we consider those prior beliefs in which $\alpha =\beta$ . Thereafter, we consider the prior belief combinations in which $\alpha \gt \beta$ .

5.3.1 Priors with $\alpha = \beta$

In these cases, switching early is quite likely as the initial estimate is on the cusp of the critical level. The general trend we observe in Figure 7a is that the lower $\alpha$ and $\beta$ lead to lower probability of consensus than higher vales of $\alpha$ and $\beta$ . This is explained by the fact that greater values of $\alpha$ and $\beta$ mean that the change in their estimate from one round to the next is comparatively small at the start of a run with an opinion. As an illustration, consider an agent with 5 affirming experiences during the warm-up. If this agent has the $\alpha = \beta =1$ prior, their estimate at the end of the warm-up is $\widehat{\theta }=0.86$ , which may allow them to retain this opinion in the face of a disagreeing neighborhood. If instead this agent had the $\alpha =\beta =4$ prior, their estimate at the end of the warm-up would be $\widehat{\theta }=0.69$ . This lower estimate can withstand less disagreement and so it makes sense that greater initial values of $\alpha =\beta$ lead to more consensus. We see the opposite effect on the level of discordance in Figure 7b.

Figure 7.

Results of the sensitivity analysis inspecting the prior belief distribution of the agents with $\alpha = \beta$ . Parameters: $N=20$ , $d=4$ , $w=0.2$ , $t_s=5$ , and $\theta _0=\theta _1 = 0.6$ .

5.3.2 Priors with $\alpha \gt \beta$

The case with $\alpha \gt \beta$ exemplifies a greater optimism of an agent in their opinion at the start of a run. The greater level of optimism is represented by a ratio of 2 to 1 in the settings of $(\alpha ,\beta )\in \{(2,1),(4,2)\}$ . The remaining settings $(\alpha ,\beta )\in \{(3,2),(4,3)\}$ also exhibit optimism but to a lesser extent. In Figures 8a and 8b, respectively, we see that the more optimistic settings lead to less consensus and more discordance than the somewhat less optimistic settings. Furthermore, we see that the more optimistic settings (both having a ratio of 2:1) do not differ greatly from one another. The slight difference we do see is that there is less consensus and more discordance when $(\alpha ,\beta ) = (4,2)$ than $(2,1)$ . This is because the greater values of $\alpha$ and $\beta$ mean that the belief estimate changes less in the first couple of rounds, and so there is a greater chance of staying with the starting opinion. Within the less optimistic settings, we note that the slightly more optimistic of the two ( $\alpha =3, \beta =2$ ) leads to less consensus and more discordance than ( $\alpha = 4,\beta = 3$ ). In summary, as optimism increases, the probability of consensus decreases and the proportion of discordance increases.

Figure 8.

Results of the sensitivity analysis inspecting the prior belief distribution of the agents with $\alpha \gt \beta$ . Parameters: $N=20$ , $d=4$ , $w=0.2$ , $t_s=5$ , and $\theta _0=\theta _1 = 0.6$ .

5.4 Opinion reliability

To elucidate the effect of the level of reliability of the opinions, we choose $\theta _0=\theta _1\in \{0.55, 0.60, 0.65,0.70, 0.75\}$ . We plot the probability of consensus for these settings in Figure 9a and the level of discordance in Figure 9b. We see that a greater reliability of both opinions lead to less consensus and more discordance. This is conceivable as when the reliability is greater, convergence of their point estimate of the reliability to this true parameter allows for neighborhoods with more disagreement (and thus greater $\theta _{\text{crit},t}^*$ ). As the reliability increases, agents are less dependent on their network to agree with them (thus decreasing $\theta _{\text{crit},t}^*$ ) in order for their point estimate of an opinion’s reliability to converge.

Figure 9.

Results of the sensitivity analysis inspecting the effect of the reliability of the opinions $\theta _0 = \theta _1$ . Parameters: $N=20$ , $d=4$ , $w=0.2$ , $t_s=5$ , $\alpha =4$ , and $\beta = 2$ .

Figure 10.

Results of the sensitivity analysis inspecting the effect of the warm-up period. Parameters: $N=20$ , $d=4$ , $w=0.2$ , $\alpha = 4$ , $\beta =2$ , and $\theta _0=\theta _1 = 0.6$ .

5.5 Warm-up length

We vary the warm-up length $t_s$ taking values $t_s \in \{0,10,20,30\}$ . In Figure 10a, which plots the probability of consensus for differing warm-up lengths, we see that a shorter warm-up period leads to more consensus. Similarly in Figure 10b, we see that longer warm-up periods lead to more discordance. This result is conceivable as in the early stages of an interaction with an opinion, the estimated reliability is changing a lot more than toward the end. In other words, having a longer warm-up period allows for an agent’s belief distribution to “settle” before having to compete with a network adjusted threshold $\theta _{\text{crit},t}^*$ .

5.6 Discussion

We highlight the fact that in the model, the agents are exposed only to attractive forces between each other. That is we follow the assumption of assimilative social influence between agents and yet we have rich results illustrating a range of possible outcomes from polarization to consensus. We believe that especially for agent-based simulation modelers who would like to include an opinion dynamic within a greater context, this model may prove to be useful because of the diversity of its outcomes which interact in a conceivable way with the parameter settings. We acknowledge that stubbornness does facilitate the reaching of polarization and fragmentation states. We highlight that a high stubbornness is not a requirement for avoiding consensus which is clear from the simulation results showing that even when $\kappa =0.5$ , the greatest probability of consensus observed in Figure 5a is at $C(\kappa )\approx 0.85$ . We thus hypothesise the origin of the disagreement between neighboring agents in steady states to be rooted in the differences in agent experiences (and interpretations thereof) and not in the incorporation of a stubbornness parameter.

6.1 Heterogeneous agent stubbornness

The value of $\kappa$ plays an important role in determining how sensitive the agents are to the opinions held by their neighbors. We conducted simulation runs in which the stubbornness of each agent was drawn from a Gaussian distribution centered on $\mu \in \{1.5, 2.5, 3.5, 4.5\}$ . The standard deviation of this distribution was varied taking values $\sigma \in \{0.5, 1.0, 1.5\}$ . In fact, we use the truncated Gaussian distribution on the support $\{0.5, 5.5\}$ . We present the results from a set of simulation runs in which the stubbornness parameter for each agent was taken from the uniform distribution $\mathcal{U}_{[0.5, 5.5]}$ . The resulting probability of consensus is depicted in Figure 11a, and the proportion of discordance is depicted in Figure 11b.

Figure 11.

Results of experiment with agent stubbornness drawn from a Gaussian distribution with mean depicted on the horizontal axis and standard deviation shown in the legend. Parameters: $N=30$ , $d=6$ , $w=0.2$ , $t_s=10$ , $\alpha = 4$ , $\beta = 2$ , and $\theta _0=\theta _1=0.6$ .

In Figure 11a and Figure 11b, we see that simulations with greater $\sigma$ behave more like the simulation in which the stubbornness is uniformly distributed than the other simulation runs. This means that when $\mu$ is relatively low, there is more discordance and less consensus for these runs compared to runs with lower $\sigma .$ The opposite holds for greater $\mu$ . In words: A population which has, in general, a lower stubbornness, greater variability between agents helps diversify opinions. Conversely, in population with, in general, a greater stubbornness (stronger sense of individuality), greater variability between agents hinders diversity of opinions. This showcases a subtle (though possibly expected) paradox: In populations that are in general individualistic (greater $\mu$ ), a greater diversity of agent characteristics (greater $\sigma$ ) results in lower diversity of agent opinion. Reason for this is that with a greater $\sigma$ there are more agents who also have a lower stubbornness. The agents with a lower stubbornness drive the system to more agreement. Thus, an increase in diversity of agent stubbornness causes a decrease of diversity in agent opinions.

6.2 Opinions with different reliability

This experiment bears the flavor of models of learning in populations. The agents in question are given a homogeneous $\kappa$ , but the true reliability of the opinions is set unequal: $\theta _0\gt \theta _1$ . Specifically, we simulated the pairs $(\theta _0,\theta _1)\in \{(0.65,0.60),(0.70,0.60),(0.75, 0.60)\}$ , which showcase a growing difference between the more reliable opinion and the other. We also experiment on the effect of a constant difference in the reliability between two opinion’s reliabilities by the pairs $(\theta _0,\theta _1)\in \{(0.65,0.60),(0.70,0.65),(0.75, 0.70)\}$ , which highlight the effect of a greater general reliability while keeping the nominal difference between the two opinion’s reliability constant.

6.2.1 Growing difference

We depict the probability of consensus in the growing difference experiment in Figure 12a. The corresponding portion of discordance is depicted in Figure 12b. In Figure 12a, we see that a greater difference in the opinion’s reliability fosters a greater probability of consensus. We also note that as the stubbornness of the population grows, the less likely it becomes that the population reaches consensus on the “inferior” opinion. This becomes so extreme that if $\kappa$ is great enough, if there is consensus in the population, this is on the opinion with the greater reliability. Similarly in Figure 12b, we see that a greater difference in reliability implies a lower portion of discordance in expectation.

Figure 12.

Results of the experiment in which the difference between $\theta _0$ and $\theta _1$ is growing. Additionally plotted in solid lines is the probability of consensus on opinion $0$ keeping in mind that $\theta _0\gt \theta _1$ . As $\kappa$ increases the solid lines join their simulation’s counterpart: $\theta _0 = 0.65, \theta _1 = 0.6$ in blue, $\theta _0 = 0.7, \theta _1 = 0.6$ in purple, and $\theta _0=0.75,\theta _1 =0.6$ in red. Parameters: $N=30$ , $d=6$ , $w=0.2$ , $t_s=10$ , $\alpha = 4$ , and $\beta = 2$ .

6.2.2 Constant difference

We plot the probability of consensus in the subexperiment with constant difference between $\theta _0$ and $\theta _1$ in Figure 13a. We plot the corresponding portion of discordance in Figure 13b. We see a similar trend in Figure 13a to the one present in the sensitivity analysis: Lower reliability leads to more consensus of opinion. As in the experiment with a growing difference between opinion reliability, we see that greater stubbornness in the population leads to a greater chance of agreeing on the “better” opinion. In Figure 13b, we see again (as in the sensitivity analysis) that a greater reliability leads to more discordance.

Figure 13.

Results of the experiment in which the difference between $\theta _0$ and $\theta _1$ is constant. Additionally plotted in solid lines is the probability of consensus on opinion $0$ keeping in mind that $\theta _0\gt \theta _1$ . As $\kappa$ increases the solid lines join their simulation’s counterpart: $\theta _0 = 0.65, \theta _1 = 0.6$ in red, $\theta _0 = 0.7, \theta _1 = 0.65$ in purple, and $\theta _0=0.75,\theta _1 =0.7$ in blue. Parameters: $N=30$ , $d=6$ , $w=0.2$ , $t_s=10$ , $\alpha = 4$ , and $\beta = 2$ .