We provide an agent-based model to explain the emergence of collective opinions not based on feedback between different opinions, but based on emotional interactions between agents. The driving variable is the emotional state of agents, characterized by their valence, quantifying the emotion from unpleasant to pleasant, and their arousal, quantifying the degree of activity associated with the emotion. Both determine their emotional expression, from which collective emotional information is generated. This information feeds back on the dynamics of emotional states and individual opinions in a nonlinear manner. We derive the critical conditions for emotional interactions to obtain either consensus or polarization of opinions. Stochastic agent-based simulations and formal analyses of the model explain our results. Possible ways to validate the model are discussed.
In the past decades, the significance of emotions in opinion formation and decision making has been recognized by the scientific community, and its study is mainly pioneered by the field of behavioral economics and empirical psychology. Experimental research on individual behavior shows that emotions fuel information sharing [
The modelling of emotions and opinions has mainly focused on their interaction at the individual level, modelling how affective and cognitive mechanisms influence each other [
We apply the principles of a modelling framework of collective emotions in which the collective state arises through interactions via a common information field and not through one-on-one interactions [
Emotions are psychological states of high relevance for the individual that imply cognitive and physiological effects. They are closely related to our behavior and how we interpret our own actions [
In terms of measurement of emotions, the circumplex model of core affect [
The emotions of humans do not exist in isolation and often collective emotional states are triggered or emerge in a crowd. Collective emotions are defined as emotional states shared by large amounts of people at the same time [
The availability of data produced by the digital society motivated the study of collective emotions in online communities and social media. Collective emotions have been analyzed through sentiment analysis of real-time group chats [
While such results from data-driven modelling of emotions were quite convincing, recent proposals to formally relate
To link emotions and opinion polarization, recent computational models simply rephrased the dimension of valence as opinion, to then study cusp catastrophes of state changes depending on arousal [
The main focus of our model is to explain the evolution of opinions based on
Here, we only recap the core dynamics of our agent-based framework of emotional influence, schematically shown in Figure
Schematic representation of emotional influence in our agent based model [
Agent
The emotional information generated this way by all agents participating is contained in the information field
We further define
This way, the total information field
Dynamics of valence and arousal: As indicated in Figure
Here,
This approach matches the observed dynamics in experiments of emotion interaction in online media [
Regarding the sign and value of the coefficients
To prevent a valence explosion, in this dynamics
For the dynamics of arousal, we have considered contributions up to 2nd order, i.e.,
Our model requires a small initial positive bias,
For
We note that in both cases, fluctuations play an important role in establishing an active regime. They first push agents to a positive arousal which is then amplified by the positive feedback, until it reaches the threshold. This then generates emotional expressions that establish a communication field which in turn feeds back on the agent’s valence and arousal. In our model,
We now have to specify how the emotional interaction described above influences the dynamics of
Specifically, our model shall explain the
Given that opinions are continuous variables,
To specify the
For our discussion, we will use terms up to 3rd order from the power series. Neglecting individual differences and stochastic influences for the moment, we can express the opinion dynamics as
To discuss the possible stationary solutions,
I.e., to obtain nontrivial values for the opinion,
Plot of equation (
For
Bifurcation diagrams: (a)
To understand the impact of
For intermediate values of
In order to couple the opinion dynamics to the emotional interactions of the agents, we need to determine how the coefficients
In our model,
The important parameter for the coupling between emotions and opinions is
In conclusion, the dynamics of the opinions read now
Relations to the bounded confidence model: To further understand this dynamics, let us neglect all small or higher-order terms and focus on the core dynamics, which reads for an individual agent
For the
This leads, in
Very similar to the bounded confidence model, we should expect scenarios that can lead to
We first present the results of agent-based computer simulations, to verify that the model works as expected. The parameter values not explicitly mentioned in the following are chosen, in accordance with [
Figure
(a, b) Opinion trajectories. (c, d) Opinion distribution at
As the first observation, we find indeed the
As the second observation, we see that the number of agents with positive and negative opinions can differ significantly dependent on the parameter
Eventually, we can also obtain scenarios in which consensus is reached, i.e., instead of a bimodal opinion distribution, we find a unimodal distribution. This is illustrated in Figure
Opinion trajectories for the case of
To fully understand the role of the coefficients
These two coupled equations can be solved numerically using a 4th order Runge–Kutta method. The results are shown in Figure
Phase portrait
The range of these unstable solutions can be obtained by setting
These values are clearly indicated in Figure
So where do agents end up with their opinions dependent on their initial conditions, if we only consider the deterministic dynamics? This is answered by the
The only difference in the expression is in the values
It is important to notice that the dynamics captured by the phase portrait, shown in Figure
In this paper, we have provided a model to formally link the dynamics of
Following established measures from social psychology, the dynamics of emotions is characterized by two agent variables,
Our modelling approach fits a general framework to model
Validation scenario: Our investigations point to real-world mechanisms in the formation of opinions which, in addition to rational considerations, are very much dependent on sentiment. Because the measurement of emotions is simpler than a corresponding measurement of opinions (see Section
In order to validate the link between emotion dynamics and opinion dynamics, we also need to estimate the Polarization from the final amount of likes/dislikes: This dichotomoy constrains opinions as strictly positive or negative, not neutral. Data, e.g. from Polarization from the coexistence of positive and negative expressions of opinions: Using sentiment analysis methods, we can estimate the polarization of a discussion in terms of the simultaneous presence of positive and negative expressions [ Polarization from the existence of network components: The online communication between users can be represented as a social network, on which we can perform a community analysis, to detect communities with different opinions [
Extension to multi-dimensional opinion space: Our model so far considers that opinion is a
We emphasize that a multi-dimensional opinion space exacerbates the problem of
It is an open question how to define a
The manuscript does not have any data.
The authors declare that they have no conflicts of interest.
All authors acknowledge funding from the Swiss National Science Foundation (CR21I1_146499). D.G. acknowledges funding from the Vienna Science and Technology Fund through the Vienna Research Group Grant “Emotional Well-Being in the Digital Society” (VRG16-005).