Grand scale opinion dynamics

ABSTRACT We investigate a set of models in opinion dynamics with a relaxation of bounded confidence against a background of noisy communication. Instead of a rigid threshold of disagreement, agents experience diminishing interaction effect as their disagreement grows. We re-explore concepts, such as consensus and social stability, and derive explicit necessary and sufficient conditions for agent pairs to remain (stochastically) stable in their opinion differences. We then generalize the results to arbitrarily large societies with heterogeneous agents, i.e. agent-level fixed effects. We apply this model environment and the stability results to two existing models and a brand new model. Finally, we generalize to asymmetric updating mechanics, where we provide strong results and conjectures for stability as well. 1. Introduction The notion of bounded confidence is one that, by now, most researchers working on opinion dynamics are acquainted with. The premise of this assumption is simple: once individuals disagree beyond a certain threshold, their interactions seize to yield attraction. With this formalization, bounded confidence is a method of capturing confirmation bias and skepticism in social interactions. Via direct mathematical analyses and agent-based model simulations, researchers are then elegantly able to explore how bounded confidence affects the development of public opinion, in particular whether society reaches (an often rigid) consensus or polarizes. Multiple recent approaches to opinion dynamics focus on further broadening the notion of bounded confidence. This is in the form of relaxing the tight constraint that bounded confidence imposes (smoothing the hard cutoff value at a specific level of disagreement), the presence of noise in the system (which also changes how we define concepts such as consensus and polarization), and the inclusion of heterogeneous interactions (specific covariates for individual agents). In this article, we aim to extend these efforts by setting up a general model environment that allows for noisy opinion dynamics, heterogeneous agents, and smoothed bounded confidence. We shall explore how the definitions of consensus and sociological stability change; the interplay of noise and bounded confidence smoothing, for which we then provide generally applicable results; and applications of our model environment (in the form of two existing models and one new model). Before further motivating our contribution, a quick review of some core essentials of the opinion dynamics literature is beneficial. Note, the overview is non-exhaustive but provides the necessary groundwork underlying the motivation of this work. 1.1. Background Numerous efforts have been made by physicists, mathematicians, and computational sociologists to create mathematical models exploring the formation and evolution of opinions. Well accepted by the opinion dynamics crowd of the academic community (see for example Grabisch & Rusinowska, Citation2020) is that there is a clear methodological distinction between the models based on the underlying “opinion space.” A clear split arises between the models where opinions are treated as discrete objects (taking values in countable spaces) and those where opinions are treated as continuous objects (taking values in uncountable spaces). Many of the early developed models in opinion dynamics assume a discrete opinion space. In its most rudimentary setup, this discrete opinion space solely allows for two opinion values, representing an individual in a binary way. The most famous of these models is the “Voter Model” (Holley & Liggett, Citation1975) – where agents are selected at random to voice their opinion, then spontaneously and without failure convincing the neighboring agents around them. Another famous example of a binary model of opinion dynamics is the “United we Stand, Divided we Fall” model (Sznajd-Weron & Sznajd, Citation2000) – where pairs of agreeing agents will be able to convince their neighbors, while a disagreeing agent pair creates repelling effects on their neighbors. These models illustrate interesting social concepts and have placed the foundation for a lot of mathematical research as well. The drawback, from the perspective of many researchers, is that treating opinions as discrete objects mean that a significant amount of information is lost in translation (Xia et al., Citation2011). Opinions are essentially complex and abstract objects, and modeling opinions as continuous objects could provide a significant advantage through a broader range of modeling freedom. A significant portion of researchers have therefore dedicated time and effort to the development of models based on opinions as continuous and multidimensional objects. A notable model bridging from the discrete to the continuous opinion space is the “Continuous Opinions and Discrete Actions” (CODA) model (Martins, Citation2008) – where agents obtain information by observing discrete actions of other agents, subsequently using Bayes’ rule to update their own latent, continuous opinions. This model captures the gap between the states of the individuals and the observations of the individuals. Another important model direction concerns the conformism assumption prevalent in many early opinion-dynamics models. Although a baseline tendency to conform might be a reasonable assumption, there are many factors influencing and potentially breaking this trend. Full conformism is the exception rather than the rule, and it is clearly not the case that information is transferred flawlessly. This builds the case for Bounded Confidence models. Bounded confidence states that only those agents that agree sufficiently will exert an effect on each other. The most prominent examples of bounded confidence models are the famous Weisbuch et al. (Citation2003) and the Hegselmann and Krause (Citation2002). In the Deffuant–Weisbuch model, agents randomly meet in pairs and interact symmetrically if they agree sufficiently. In the Hegselmann–Krause model, agents take in all opinions simultaneously and then listen to those within a bounded radius of confidence. The bounded confidence assumption provides a promising lead for the construction of our model environment, as there is significant evidence for confirmation and selection biases being present due to the preference of congenial information (in support of the current belief of an individual) over uncongenial information (challenging the current belief of an individual). See, among many other articles, the work by Nickerson (Citation1998) and Hart et al. (Citation2009). Bounded confidence is often seen as a model representation of confirmation bias, but it can capture more than just that. Big deviations from one’s current perspective are cognitively expensive, as individuals are generally both emotionally and rationally anchored in the perspective that they currently (Furnham & Boo, Citation2011). Even a conformist individual will incur a significant cost to changing their point of view significantly. Thus, adapting to a substantially differing viewpoint requires a lot of effort, inducing heavy resistance. This is commonly known as skepticism (Choi et al., Citation2009). Although the notions of skepticism and confirmation bias are not fully disjoint, they refer to different concepts. Skepticism, on itself, is not a bias, whereas confirmation bias naturally is. When it comes to the psychology of how individuals process incoming persuasive attempts, researchers from the academic field of behavior change (a subfield of psychology) acknowledge skepticism as one of three main factors giving rise to resistance. The other two are reactance and inertia. Reactance occurs when the effect of interaction causes a repulsive rather than an attractive effect of interaction. This is caused by an inherent tension in the communication styles of agents, in which an individual feels emotionally compelled to rebel against the other party by moving in the direction of a polar-opposite stance (i.e. disagreeing for the sake of disagreeing). This is hypothesized to be the result of a desire to establish authority or to counter a perceived loss of freedom on the side of recipient of a message (Brehm, Citation1966). Notably, in the opinion dynamics literature, models with repulsive interactions are only scarcely available, see Altafini (Citation2013). Then, there is the concept of inertia, which applies to the rigidity of individuals that do not obtain sufficient amounts of differentiating external input. At very low levels of disagreement, there is a lacking incentive for individuals to distinguish the opinion of the other from their own, hence the interaction has practically no effect (Kuppens et al., Citation2010). One could see this as the reverse of bounded confidence, with a lower bound on disagreement being a relevant factor. In light of imperfect communication, if noise is present in the interaction, low levels of disagreement will induce negligible changes relative to the noise present in the interaction. 1.2. The scope of this article For the scope of this article, there is a set of three points that we want to draw explicit attention to. First, we consider the bounded confidence assumption to be an attractive starting point for the formulation of a model environment. However, bounded confidence is much too strict as a formalization of confirmation biases and skepticism. It imposes an immediate cutoff of interaction effect at growing levels of disagreement, instead of a continuously decreasing effect as disagreement increases. Where the idea of diminishing effect of interaction in light of growing disagreement is very reasonable, the strong assumption that this cutoff is spontaneous and immediate could be relaxed. Second, some researchers active in the field have expressed their desire for a modeling framework where linguistic and interagent factors can be incorporated (Zha et al. Citation2021). This means that the dynamics of opinion exchanges depend not solely on the current opinions of the agents (the dynamic, endogenous factor) but depends also on factors characterizing the language of interaction (in a broad sense) and mutual compatibility between the agents on this language (so, a static, exogenous factor). Practically, this means creating a framework that allows for an appropriate amount of heterogeneity at the level of agent pairs. Instead of having to fit the compatibility between every single-agent pair without any overarching model framework (which requires a number of operations that scales quadratically with the number of agents), it is more feasible to fit a “personality” for each agent and then generate a model for the compatibility between two personalities (which would require a number of operations scaling linearly with the number of agents). This also provides room for modeling agent types that induce reactant attitudes. Third, the central scope of research for the majority of opinion dynamics models is put on establishing properties about convergence of public opinion over long timescales. Frequently, the goal is to derive whether consensus is achieved in a given societal setup. This means that concepts such as consensus, fragmentation, and polarization are characterized through asymptotic states of the system where people either fully agree, or cluster into distinct groups of individuals that separately achieve full agreement. The absence of free will, freedom of interpretations, or noise in the interaction prohibits the evolution of a society beyond this rigid state where full uniformity has been achieved. In reality, however, consensus is not absolute and definitely not rigid. New information arrives constantly and individuals will always evolve (no matter how slightly) in their opinion. This motivates the inclusion of noise into the dynamics as done in Pineda et al. (Citation2009) and Pineda et al. (Citation2013) and also provides one of the fundamental starting principles for our model. To be clear, the impact of noise in the model cannot be understated. Unless there are strongly influential individuals that are absolutely anchored in their opinion, it is an easy consequence that public opinion as a whole can evolve. Thus, without an external anchor, the public opinion of an entire society will experience drift. With these points in mind, this article outlines a model environment for opinion dynamics that incorporates multidimensional and continuous opinions, heterogeneity at the agent-level, the inclusion of noise, and the possibility for updating dynamics with an explicit dependency on disagreement between agents. We will assign latent properties to agents that will impact their interactions with other individuals (their individual-level fixed effects). As it becomes significantly more difficult to talk about concepts like rigid consensus within these models, we introduce theory from Markov chains in order to formulate a notion of sociological stability and provide full sufficient and necessary conditions for stochastic stability of the system when it is symmetric. We then move on to expand the model for asymmetric compatibilities, meaning that we allow for leader-follower hierarchies to exist within the environment, for which we provide initial results. The mathematical methodology underlying this article, that of stability for Markov chains on uncountable spaces as outlined in the book by Meyn and Tweedie (Citation2012), provides us with ample freedom to generalize the model environment as much as possible (continuous, multidimensional opinions, heterogeneous agents, arbitrary noise terms, and so forth). To the best of our knowledge, only the article by Baccelli et al. (Citation2017) lies close in analytical spirit. They, as well, perform stability analyses using Lyapunov-Foster criteria for opinion dynamics systems and heterogeneous agent pools, albeit on countable, one-dimensional spaces. The continuous and multidimensional extension of their model, which we call the decaying confidence model, falls within our results. Note that we prominently speak about a model environment instead of a single model. This is a pedantic distinction, but an important one. Since many models of opinions, dynamics could fit as specified instances of our model environment (or comparable variations of it), the results and working methods of this article would become relevant for those researchers. Simultaneously, many researchers from social sciences with a stronger theoretical background might find inspiration in our model environment. They could then formulate models for which the results in this article are directly applicable. With this in mind, we provide a threefold of applications of our environment, two of which are adaptations of existing opinion dynamics models. For these applications, we explicitly implement our general results. 1.3. Setup of the paper The remainder of this article is structured as follows: In Section 2, we will provide the mathematical setup of the workhorse model framework. In Section 3, we will implement it via three examples of models on opinion evolution (out of which two are recently published models, and one model is brand new). In Section 4, we will explore the stability of the system for two-agent and arbitrary-agent setups. The results of this exploration are applied to the three models introduced in Section 3. In Section 5, we will expand the model to allow for asymmetry, and provide an initial result for stability in the two-agent setup. Finally, we conclude with a discussion in Section 6. The relevant mathematical literature on Markov chains on uncountable spaces is included in Appendix A, while full proofs and details on the results are included in Appendices B, C, and D. 2. The model environment 2.1. Formal construction – societies and social networks We define a society as a finite set of agents , a matrix of probabilities and an arbitrary characteristic space . The state space, i.e. the opinion space, will be for any integer that contains the dynamic (non-constant) opinions. The space will consist of static (constant) traits. Time runs discretely, so , with every interaction yielding a step in time. Thus, every agent will be assigned: an opinion process , representing an evolving opinion; a personal characteristic , representing a static property; an interaction vector representing a social neighborhood. Within this article, we assume . Via this coding approach, we use a set of covariates to capture individual-level fixed effects. Given the characteristic of one agent and of another, we define the compatibility between agents and via their inner product for the angle between and and a scaling factor. Parallel characteristics would yield an increasing effect of communication as (an exogenous bonus), whereas antiparallel characteristics might yield a diminishing effect of communication as (an exogenous penalty). We assume that opinions evolve via one-on-one encounters. Every round, there is a probability that agent approaches agent . Which agent initiates the conversation is deemed irrelevant in how the interaction proceeds, so that the probability that agents and interact in a round is . In particular, . Using both the static fits and the probabilities, we can construct a bivariate edge that represents a communication profile, representing the set of all bivariate edges between agents. This captures both how frequent two agents interact and what their compatibility is. By a social network, we mean the graph of agents and communication profiles. A few preliminary comments. First off, we reiterate that the interactions proceed in a binary manner, meaning that they involve exactly two individuals at a time. Expanding this to “multinomial” interactions, i.e. where there are interactions of groups with a different (or random) number of participants, is not too difficult. This would be notationally quite tedious and we feel that further generalizing the model in this manner could hurt the interpretability. It can also be envisioned that interactions occur one-sided (i.e. recipients and senders of messages on an online platform). Although we exclude this in the current setup as it violates our symmetry condition, it will be included in Section 5. Second, the “personal characteristics” are rather abstract objects. They allow us (or any modeler, for that sake) to introduce moderator variables within the interactions. One should think of these characteristics (depending on the model) as cultural, linguistic, geographic, or even communication-style variables that do not change appreciably due to the individual interactions but have a significant effect on how the interactions proceed. For the purpose of demonstration, we use an inner product between characteristics to generate a fit. In generality, however, can be chosen as exotic as one wishes and the static fit between agents does not have to be an inner product, but can be any symmetric function of characteristics . All that matters is that we are able to create a “measure of fit” between the agents. 2.2. Formal construction – opinion evolution Having determined the static building blocks for the model, we can move on to characterize the rules for opinion evolution. As stated, interactions proceed between exactly two agents at a time. When an interaction between agents and occurs in round , they both share their own state with the other agent. Each of these two agents then updates to new positions which are a stochastic mixture of the opinions, depending on their compatibility and their distance in opinion space . Mathematically, we posit that where we assume that are -valued random variables distributed with cumulative distribution function . The stochastic variables are instances of a white noise sequence to be specified later. The agents not involved in the interaction remain fixed. This means that the evolution of the system is characterized via a family of cumulative distribution functions on . The “exchange fractions” will be referred to as the effect of interaction, which we also allow to be negative-valued (meaning that an interaction can induce a repulsive effect). In particular, we assume that agents will have less total impact on each other as their disagreement grows, and individuals will have less attractive impact on each other as their compatibility decreases. Assumption 2.1. Pick any . Let . We assume that for all and , the map is decreasing; for all and , the map is strictly increasing. For all , the map should be eventually monotonic. In technical terms, condition (1) tells us that if , then will first-order stochastically dominate . Condition (2) tells us that if , then first-order stochastically dominates . Condition (3) of Assumption 2.1 ensures that persistently oscillating interaction effects as disagreement grows are excluded. However, it could be that the first moment of starts out positive for small , turns negative as grows, and decays to zero as , persistent up-down cycles as functions of seem unreasonable (and add significant mathematical complexity). Think of cases such as which we deem unrealistic and thus exclude. Finally, each agent has a (white) noise sequence so that . Once an agent is involved in an interaction, the noise becomes relevant and is added to the updating rule, as seen in EquationEq. (2(2) (2) ) and (Equation3(3) (3) ). For all models that we treat, we will assume an isotropic Gaussian distribution with agent-specific variance as the noise term. Yet practically all stability results are generalizable to general (zero-mean) noise terms. 3. Applications 3.1. Smoothed bounded confidence Many models in opinion dynamics hinge on the assumption of bounded confidence, enforcing that individuals will not exchange information if they disagree Deffuant, et al. (Citation2000). This assumption is definitely plausible, given the often proven presence of confirmation bias in human interaction (see Nickerson (Citation1998) for the psychology literature on the topic), but the bounded confidence assumption is rather strong as it represents an immediate cutoff at a specific radius of disagreement. There have been suggestions for weakening this assumption, such as by Deffuant et al. (Citation2004) and by Brooks et al. (Citation2024) in the setting of the Hegselmann–Krause model. In the context of the Deffuant–Weisbuch model, weakening the bounded confidence transition creates a heterogeneous, smoothed variant of the model. Instead of a hard cutoff at some radius of disagreement beyond which individuals no longer exchange information, the decrease of the interaction effect is gradual. Formally, the opinion space is for a and there is a social network that contains interaction probabilities and compatibilities , for . Assume that agents and interact in round . They will exchange a static amount multiplied by a penalty function depending on their disagreement . Both agents incur a centered additive noise with magnitude of finite second moment, i.e. . All other agents remain fixed in their opinion. We call this the smoothed bounded confidence model. The updating equations for two interacting agents become: As mentioned, we shall assume that with for every agent . The penalty function can be chosen freely in the model, although a decaying function is most relevant given the bounded confidence assumption. Deffuant et al. (Citation2004) modeling themselves chose a Gaussian penalty function, while Brooks et al. (Citation2024) used a member of the family of generalized Gaussians for . We have plotted the family of generalized Gaussians in . Note, in particular, that if , then pointwise. Thus, in the limit of going to infinity, we obtain a noisy bounded confidence model with cutoff radius . All our simulations will be done in a one-dimensional opinion space. Before considering a heterogeneous population, we will first focus our analysis to the decay function. Every pair of agents is provided with a compatibility . Thus, all agents get along equally well. We also assume each pair interacts equally often in probability (i.e. a uniform social network). Everybody has a standard normal distribution as noise term, thus for all and . We start with a member of the generalized Gaussian family, as in Brooks et al. (Citation2024). Fixing the radius , we simulate public opinion evolution for different values in , for agents with initial opinions uniformly distributed from to . The algorithm runs for 6 million iterations. Depending on the power parameter of the Gaussian, these penalty functions give rise to different patterns of cluster formation and dissipation, at least in the short run. The lower , the more clusters arise and the more robust these clusters are. The higher , the higher the possibility of agents detaching and wandering off into another cluster, or a cluster dissipating in total. For all exponents, if an individual escapes outside of the range of a cluster, the interagent pull falls off quickly and the outlier starts to follow a random walk. In Section 4, we prove that these penalties lead to unstable systems. Continuing our exploration of the penalty function, we move on to reciprocal penalties: for . This family of functions has the similar behavior as the family of Gaussians, albeit with fat tails. We shall pick . With agents, starting with initial opinion , their evolution over 6 million iterations is plotted in . The parameter choices were not arbitrary, as it determines whether the population sticks together. Although the mean public opinion process follows a random walk (so the joint opinion cannot be stationary), note that parameter values and result in the clustering of public opinion via a bounded sample variance . As we expect, yields a lower long-term variance around which the process oscillates compared to . Still, for both exponents, the sample variance process is mean-reverting with no long-term drift. In contrast, and causes public opinion to disperse. The interagent pull cannot overcome the noise, and the sample variance keeps growing linearly over time. In Section 4, we shall explore the phenomenon observed in in a mathematically rigorous manner. We shall close our exploration of the smoothed bounded confidence model with a heterogeneous society and reciprocal-polynomial probability kernels as in (7). We set and , and allow four types of characteristics in this society: . The types will represent more extreme, opposite types (each making up 15% of the population). The types will represent more moderate, opposite characters (making up 35% of the population, each). These types interact as follows: All types strongly attract their own types. strongly repulses , slightly repulses and slightly attracts . slightly attracts and slightly repulses . slightly attracts . These characteristics are coded in the following way: The compatibility between two agents and is obtained via the inner product between the types in accordance with Equationequation (1)(1) (1) , multiplied by a factor to scale down to an appropriate magnitude. This yields the compatibilities of . Agents are initialized according to a uniform distribution on , and the simulation runs for 6 million rounds. This leads to . As we can see, the core of moderates stays centered, attracting each other while finding mutual repulsion on both sides of the spectrum. The extremists personalities form clusters on either side of the spectrum, and the groups of homogeneous character do not necessarily all cluster together (immediately). This is due to the unidimensional opinion space, so that alternating clusters of repelling and attracting types create soft “boundaries” between the clusters of agreeable types. These boundaries are soft, as agents can cross them to join another cluster of compatible agents by crossing through a cluster of incompatible agents, although this takes considerable time. 3.2. Decaying confidence In the Deffuant–Weisbuch model, either with or without smoothing, two interacting individuals will always behave according to a deterministic dynamic supplemented by an additive white noise. The smoothing of the DW model can, instead, also be performed in a different and more intuitive manner, as also done by Baccelli et al. (Citation2017) in their “stochastic bounded confidence” model. Instead of smoothing the “amount of opinion” that agents will surely exchange against growing disagreement, we smooth the probability that individuals listen to the opinion of the other agent. Hence, where the assumption for bounded confidence is that two individuals will accept information as long as they agree enough, and smoothed bounded confidence decreases the impact of accepted information as disagreement grows, we can also model an accept/reject decision as a coin toss with success rate depending on current disagreement. We refer to this as the decaying confidence model. To formalize this, let the compatibility determine the base factor that two agents and exchange in case of a successful interaction. Introduce the random variables as the “coin tosses” determining if an agent successfully listens to the another in an interaction, with probability of a success being for the penalty function and the current disagreement. As with smoothed bounded confidence, the noise terms are added irrespective of whether the “coin flip” yields a success or a failure. This leads to the updating equations: Similar to smoothed bounded confidence, we explore the decaying confidence model with a homogeneous population on a one-dimensional opinion space, with initial opinions drawn from a uniform distribution on and a penalty function a generalized Gaussian. We give every pair of agents a compatibility , set the cutoff radius at and vary . For agents and 6 million rounds, the results are plotted in . Note that the clustering happens in a much sharper fashion compared to the smoothed bounded confidence setup. Although the probability of interacting positively drops off rather quickly as disagreement grows, the few instances of successful interaction immediately bridge a large part of the gap between the agents. The result of this dynamic is more robust clusters from which agents wander less frequently. Once they do wander away, the probability of successful interaction diminishes rather quickly, but this is counteracted by the effect of a single success being significant. In Section 4, we will show that stability of this model is guaranteed under similar conditions as in the smoothed bounded confidence model. Now, we proceed with the treatment of the decaying confidence setup in a heterogeneous society. We will impose the same reciprocal polynomial penalty function and the same four characteristics as in smoothed bounded confidence, so two more extremist types and two more moderate types. The results are plotted in , for agents and 6 million iterations. In contrast with the smoothed bounded confidence model, both extreme personality types are (loosely) clustered on their own side of the opinion spectrum, albeit not perfectly (we can still observe random stragglers of one personality type being on the opposite end of the spectrum). Although the clusters are much less dense, we do see a stronger attraction over longer ranges. The moderates are still centered in the middle, although these personalities now also have a slight preference for one side over another. This is not strange given that the extreme personalities each have more nicely distributed themselves on either side of the opinion spectrum. 3.3. The beta model Our current modeling environment provides us with the possibility to add more realism in the construction. In this subsection, we will motivate a new model which builds on the Beta-distribution. Recall that for if has a probability density for the Gamma-function. This distribution is encountered often when modeling the probability distribution of probabilities themselves. What makes the Beta-distribution so attractive for modeling interaction effects is the fact that the two parameters and can be interpreted as the ratio of “hits” (positive communication instances) versus “misses” (negative communication instances). Furthermore, the compact support of the distribution is particularly convenient. Personalities that induce more reactant attitudes will yield much more misses (negative effects); hence, we assume a higher parameter compared to the parameter. Conformist personalities will induce more hits (positive effects), yielding a higher compared to . We want an overall diminishing effect of communication for growing disagreement. The bigger the sum of the parameters becomes, the more the distribution will peak around (i.e. the lower the variance on the distribution will be). Thus, if both parameters scale up equally (i.e. ), the distribution will center around and decrease in width, meaning that the bulk of effects realized will go to zero. We will implement this into our model as follows. Assume agents and interact in round , with current disagreement . Draw two random variables , for to be defined later. We assume that the interaction effects of the agents are and respectively, thus stretching the Beta-distribution to . Noise is added, again, with . The agents then update as A positive compatibility should create more attractive communication, while a negative compatibility should repulse more often. Thus, we offset by and by (the exponential transformation is arbitrary but ensures that the parameters remain positive). Finally, growing disagreement should make both parameters grow equally, so we add a penalty function , for , to both parameters. This is all summarized in EquationEquation 13(13) (13) , and EquationEquation 14(14) (14) as well as below. Below, we take a homogeneous society of individuals, a baseline variance for every agent and personality matching for everybody, as well as a constant in the penalty function. Initial opinions are distributed uniformly along . We then plot the evolution of public opinion over iterations for different exponents in the penalty function, see . If we simulate the evolution of public opinion for agents with the same four types as in the example of smoothed bounded confidence and decaying confidence, now with a penalty function , then we obtain the sample paths plotted in . As we clearly see, the two extremist groups are now more strongly defined on whatever side of the spectrum they ended up populating. The clusters are tighter and the long-range attraction is also much stronger. The moderates remain squeezed in the center, staying in equilibrium between all the attractive and repulsive forces. 4. Sociological stability In our exploration of smoothed bounded confidence, in particular in , we observed that the asymptotic behavior of the penalty function has a dramatic impact on the clustering of the opinion processes in the model. Specifically, in the smoothed bounded confidence model, it seemed that clusters appeared as long as the penalty function has a thinner tail than . This hints at a bifurcation point that there is a “critical decay point” of the system, beyond which the system switches between stability and instability. This intuition turns out to be mostly correct, and such a principle actually holds much more generally. Since the opinion processes of the agents are Markov processes (and so are the processes of opinion differences), the formation of stable clusters is mathematically equivalent to the existence of an invariant measure for the opinion difference processes. This means that, if the opinions are distributed relative to each other according to the invariant measure, the next-round opinion differences will again be distributed in the same manner. After a small exploration of the interplay between stability, consensus, polarization, and fragmentation, we will proceed with stating the explicit conditions for stability in the general model as described in Section 2. We begin with a society consisting of only two agents before moving to symmetric societies with arbitrarily many individuals. The underlying mathematical machinery is that of Markov processes on uncountable state spaces, the essentials of which are outlined in Appendix A. The detailed proofs of the theorems can be found in Appendices B and C. 4.1. Stability, consensus, polarization, and fragmentation The sociological interpretation of a concept such as “stability” might not seem immediately evident. Although it closely relates to concepts such as consensus, polarization, and fragmentation, it is not synonymous with any of them without further specification of the model context. Broadly, social stability most closely relates to the establishment of predictable collective norms, values, and beliefs. This does not mean that full consensus is reached, especially when expanding to asymmetric models. If there are fixed agents (say media parties or influencers) present in the model at considerable distances from each other, and each individual listens primarily to a strict subsets of influencers, the system can be stable while it converges to a steady state where the public opinion distribution is multimodal. One might still speak of a degree of fragmentation or polarization in such context, yet that is entirely dependent on the choice of the modeler. At the same time, a multimodal state of public opinion does not necessarily imply polarization or fragmentation, as long as the “cliques” of the system (making up the modes of the opinion distribution) are in close proximity, again as specified by context. What we can say is that a social system cannot attain a state of structural consensus without stability. Although in an unstable system there can be temporary states of consensus, these are bound to break. Similarly, structural fragmentation can only be present in an unstable system. Since sociological fragmentation deals with the relative alienation of individuals or subcultures, these respective parties must be too weakly connected to the “main” society as a whole, otherwise any outlier is bound to return to the bulk that makes up the society. In our model, fragmentation means that there are stable clusters: groups of agents that, when considered in isolation, yield stable patterns and thus collective norms and belief systems. Together, however, they are not strongly bound, making the system collectively unstable. Polarization, in our environment, forms mostly from the context. A society that is unstable due to weak social bonds or non-interacting clusters will generally not actively polarize, instead it will fragment (instability due to lack of cohesion). On the other hand, a social system unstable due to consistent repulsive interactions between groups will cause clusters of agents to push each other to infinite distances, which seemingly does capture polarization (instability due to active repulsion). At the same, a stable system with a multimodal equilibrium distribution of opinions can still represent a polarized society. This requires the modeler to envision a notion of “distance” in opinion space, to be able to quantify what “strong opposition” actually means. That the concepts of consensus, polarization, and fragmentation become fluid in our methodological setup is not surprising. These traits of social systems, both in their conceptualization and their origin, have been and continue to be debated in academic circles. From Durkheim to Axelrod, how social influence leads to the macrostates of human systems remains an open discussion (Flache et al., Citation2017). 4.2. Stability for two-agent interactions We start our analysis by considering a society with only two agents with compatibility . We collect the opinion processes of the individual agents into one -valued process . Each round consists of a pairwise interaction between these two agents. Our interest lies in determining whether these agents cluster. Thus, define the opinion difference process via for all , and also define . Starting from (2) and (3), algebra shows that forms an -valued Markov process with update rule For the results on stability, we do not need the entire family of cumulative distribution functions , just the moments of as functions of and . Define Note that this representation is valid for any choice of . As the compatibility is a constant and there is only one pair of agents, we suppress the dependency in notation. In order for the opinion difference process to be stable, we only require that it does not grow to arbitrary large magnitudes with positive probability. As such, it is the asymptotic behavior of the that is of interest to us. We will start by treating the pathological cases, in particular the cases where the interaction effect does not vanish as the distance in opinion space grows bigger. For this, define If as , it is evidently not yet ensured that . Proposition 4.1. Let be the Markov process described by Equations (15) and (16). Assume that the asymptotic interaction effect is non-vanishing, meaning that Then admits an invariant measure if and only if, eventually (for large Δ): In case that it exists, is unique and converges to it in distribution as . Although Proposition 4.1 solely concerns a (for us) pathological scenario, the results are interesting in their own right nonetheless. Note, in particular, the mean of the interaction effect is asymptotically allowed to become slightly negative. Thus, the opinion differences are allowed to evolve with slightly positive drift, as long as there is sufficient variance of the interaction effect to offset this. One speaks of a detailed balance between the mean interaction effect and its variance. We can now proceed to the main result. For what follows, we shall assume that the asymptotic interaction effect is truly diminishing. Thus: We introduce the asymptotic quantities : Just as we saw in Proposition 4.1, it can happen that converges to zero from below (i.e. ). This does not directly imply instability. In order to quantify this, we will need to define the decay parameter Thus, resembles the best polynomial decay rate for the function. Note that it is possible for to be zero, for example if decays as . Still, it turns out that only the polynomial order is relevant when it comes to stability criteria. The choice of these objects might be rather abstract, so we will revisit it after stating our main theorem. Theorem 4.2. Let be the Markov process described by EquationEq. (15(15) (15) ) and (Equation16(16) (16) ) and assume that Condition (20) is satisfied. If , then admits a finite invariant measure. If and , then admits a finite invariant measure if and only if (22) (22)If , then does not admit a finite invariant measure. If , then admits a finite invariant measure if and only if and eventually (23) (23) If admits an invariant measure, it is unique and will converge to it in distribution. The detailed proofs of Theorem 4.2 and Proposition 4.1 can be found in Appendix B. Here, we will provide some intuitive remarks. First off, this result rather succinctly reveals the relation between clustering opinion processes and the speed at which interaction becomes weaker as disagreement grows. The constants are very specific indicators of the asymptotic significance of the interaction effect compared to the random noise that the agents incur in every interaction (whether this is noise due to imperfect interpretation of signals, or evolution of thought). Intuitively, we interpret the significance of the -factors as follows: If , then the intuition is that has a fat enough tail and the interaction remains attractive on average. For large disagreements, the decay should be strictly slower than , so that the product with blows up; If , then the leading decay term of should exactly be , leading to a finite value of the limit; If , then should have thin tails. The intuition here is that decays quicker than , so that the product with vanishes in the limit; If , then the interaction is asymptotically repulsive on average. Yet if the decay is not too rapid (less than ) and there remains a balance between the mean and variance of the interaction effects, stability can still be attained. At the bifurcation scenario (when or is a finite constant), the delicate interplay between compatibility of the agents and the variance of each of the opinion processes becomes evident. In the context of smoothed bounded confidence, we illustrate this in . Second, consider scenario (2). The condition as is purely technical and coincides with . It ensures condition (50) holds. Although we also consider , condition (22) is never satisfied if . This is in contrast with Proposition 4.1 or the case , where a slightly negative mean drift was potentially allowed. Furthermore, remark that the famous “curse of dimensionality” makes its entrance in scenario (2): the higher dimensional the opinion space, the stronger the compatibility between agents needs to scale up against the noise in order to create agents that remain clustered. This is explicitly evident from the constraint (22).Footnote1 Finally, for the two-agent model, symmetry in the interaction is not a pure necessity. We can easily extend the proof to the situation where , meaning that the interaction effects and are independent, but not identically distributed. In fact, the symmetric system results are a direct consequence of the asymmetric system results. We do this in subsection 5.2 and Appendix A. 4.3. Stability for -agent interactions The 2-agent result is an analytically insightful illustration of the general dynamics that we observed in , but it can be lifted to general societies (with arbitrarily many agents) as well. Assume to be working with an agent pool and define, for any pair of agents , the opinion difference process . If these two agents interact in a given round , then the update rule is If only agent is involved in an interaction (with third agent ), then updates as Rather quickly, we see that grand process of opinion differences is a Markov process in which satisfies , for all ; , for all ; , for all . Introduce the map (in line with the definition of the two-agent society, but now allowing for heterogeneity of trust) and the asymptotic quantity . To keep matters insightful, we restrict to the scenario where there is no asymptotic repulsion (in terms of growing disagreement), meaning that as grows large, for all agent pairs . What seems intuitive, is that for the entire society to remain stable (as in, agents remain close to everybody else in the society), all of the agents should be consistently obtaining information from all of the other agents in order to maintain a shared truth. To illustrate, in , a line between two agents implies that these agents would be stable in the 2-agent scenario (see previous section). As long as the entire group remains connected sufficiently, this should ensure stability of the society as a whole. To formalize this connectedness, define the auxiliary variables . Set if the two agents would be stable in a two-agent society with and (so, they interact with positive probability and fall under scenario (1) of Theorem 4.2). Set otherwise. Define the relationship graph for our society with vertices and edges formed by the variables . The graph provides a map of what agent pairs are strongly connected (scenario (1) of Theorem 4.2). Then, the following result holds: Theorem 4.3. Let be the Markov process with update rules (24) - (26). If the relationship graph is connected, then will admit an invariant measure. The proof for this result follows the same principles as the 2-agent scenario, now with a test function consisting of the sum of all squared differences (which is proportional to the sample variance of our public opinion). The full proof can be found in Appendix C. The intuition of this result is clear: if the public opinion grows so much that the sample variance of all opinions becomes sufficiently large, public opinion has to contract again, so that the variance again decreases in expectation. One should also note that societies allowing for consistent negative relationships (i.e. the existence agent pairs for which ) and weaker stable relationships (agent pairs in situation (2) of Result 1) might be stable, albeit conditions for those societies become significantly more complex. 4.4. Applications We will now apply the two-agent results (Theorem 4.2) to the smoothed bounded confidence, decaying confidence, and beta models. Assume two agents have a fit and recall that we had defined and . 4.4.1. Application – Smoothed bounded confidence The smoothed bounded confidence model (Subsection 3.1) most closely described the Deffuant–Weisbuch model with smoothly decaying interaction effects and added noise. It is captured by EquationEq. (4(4) (4) )–(Equation6(6) (6) ), with a deterministic interaction effect that depends on the product of the static fit and the penalty function . This makes applying Theorem 4.2 straightforward. We find the moments of the interaction effect when individuals disagree an amount to be given by The stability of the connection between two agents depends solely the mean interaction effect not decaying too quickly. We require , meaning and must decay not faster than . Define . We thus have stability ensured if , as we then fall into scenario (1) of Result 1, or if . In the latter case, we find by EquationEq. (22(22) (22) ) the connection to be stable only if the mean asymptotic interaction weighs up against the variance of the two agents. Since if , we then get Furthermore, a negative choice of or approaching from below will always yield instability. This is due to the fact that are deterministic and hence cannot compensate a negative average effect with any variance. Note, as well, that the penalty function choice of a generalized Gaussian will therefore always yield unstable societies. Our result aligns with the simulations outlined in , where we see that the parameter values create stability and break stability. 4.4.2. Application – decaying confidence The decaying confidence model (Subsection 3.2), as also in part modeled by Baccelli et al. (Citation2017) and analyzed via similar methods as we employ here assumed a decreasing probability of successful interactions as the disagreement grows. It was captured by EquationEq. (8(8) (8) )–(Equation10(10) (10) ). If two agents have a fit τ and disagreement ∆, we find the respective moments: Note that the penalty function comes into all the interaction effect moments with equal exponent. As before, define . Thus, for the connection between the individuals to be stable, we require either so that we arrive at scenario (1) of Theorem 4.2; or we require , in which case and . Since , we find if and only if , which we have assumed in its construction. Hence, we can invoke Theorem 4.2 to obtain the sub-condition: If none of these conditions hold, then the system will not be stable. 4.4.3. Application – the beta model The beta model (Subsection 3.3) gave an interaction structure that allows for more freedom but also yields more complexity. Analyzing the stability of the model, however, is still very much feasible. In particular, we had formulated the model via equations (11), (12), and (13) which involved a penalty function . In a slightly different fashion than for the two models before, the penalty function should not grow too quickly (not more than quadratically, to be specific). Therefore, redefine . Let , meaning that grows slower than . We surely find the existence of a stable relation between two agents if , in which case the mean interaction effect is attractive and not too strongly decaying. If , then the interaction effect is negative on average. Yet, stability is still found from (23) if and Note that implies . Another way to see this, mathematically, is that . If instead ( grows as , asymptotically) then the system is stable if and only if if meets condition (22). This translates to: That is a technical lemma. We easily see that lower trust quickly decreases the left-hand side, and so does a higher value of or a higher variance between the agents. We find the results of this section in alignment with . The calculations for this model are more involved than for the other applications and can be found in Appendix D. 5. The asymmetric model The model, in its current form, already provides significant freedom in modeling agent-specific interactions. The availability of analytical results for conditions on when a society is stable or not is powerful. However, the symmetric nature of the model (i.e. both agents involved in an interaction move equal distances with equal likelihoods) excludes the possibility of leader-follower hierarchies. To remedy this, we have to remove the condition of symmetric compatibilities. The fit from agent to agent , given by , then no longer has to equal the reverse fit . Equations (2) and (3) still apply, now reading: with multivariate Gaussians and . Since is not necessarily equal to , this means that agent and could interact in an asymmetric manner. 5.1. Moderates and zealots with decaying confidence To demonstrate what dynamics the inclusion of asymmetry can create, consider the decaying confidence model from Subsection 3.2. We assume there to be two fixed parties and three types of dynamic agents. The two parties communicate with personalities , respectively, and never update themselves. The three dynamic agent groups consist of two zealot groups (personalities ) each with a preference for a specific party, and disapproval of the other zealot group. The final dynamic agent type is that of a moderate (personality ) that does not listen to the parties specifically, but to the zealots and weakly to their own group (as moderates are themselves not strongly, vocally opinionated). If we then tabulate the static fits fitting this description, we obtain (row and column is how much an agent with personality listens to an agent with personality , i.e. the fit ) . We assume the baseline effect constant and that the penalty function (due to smoothed bounded confidence) is of the form The variance parameter is for any party-agent (they, thus, remain completely fixed in their opinion), while it is for every other agent. Below, we plot multiple sample paths of public opinion with agents for interactions. What one can note, is that we can now create stable clusters that individuals will flock to and wander between. Extremists get a tendency of sticking to their parties and slightly overshooting the core opinion of that party (due to the repulsive effect that the polar-opposite extremists have), while moderates wander between the two groups more freely. This is shown in . 5.2. Stability for the asymmetric two-agent processes Without too much hassle, we are able to extend the proof of stability for two-agent interactions to the asymmetric model. The result of Theorem 4.2 is slightly reformulated, while the proof proceeds in a manner exactly analogous. Start by defining Note the implicit dependence on via the distribution of . Introduce and furthermore We then obtain the following generalization of Proposition 4.1. Proposition 5.1. Let be the opinion difference process, defined via for . Assume that then admits an invariant measure if and only if For the generalization of Theorem 4.2, we need to reintroduce the -factor for the asymmetric model. In particular: This brings us to the following result. Theorem 5.2. Let be the opinion difference process and assume that Either of the following scenarios will hold: If , then admits a finite invariant measure. If and , then admits a finite invariant measure if and only if (35) (35)If , then does not admit a finite invariant measure. If , then admits finite invariant measure if and only if and eventually (36) (36) If admits a finite invariant measure, then it is unique and will converge to it in distribution. 5.3. Stability hypothesis for asymmetric societies Although we are able to derive conditions required for the stability of two-agent interactions in the asymmetric model, the approach to proving -agent stability for symmetric societies breaks down when the interactions no longer transpire symmetrically. For now, we can only conjecture what the sufficient conditions could be for general -agent societies to be stable if asymmetry is present. Recall from paragraph 4.3 the undirected relationship graph that connects pairs of agents that would remain stable in two-agent interactions introduced in Section 4. For the relationship graph, we said that two agents are connected by an edge if their 2-agent interaction in isolation would be stable. Formally, this is if they fall under scenario (1) of Result 4 (meaning ). However, the strength of the asymmetric model is that it allows for hierarchies of leaders and followers. Yet, this also means that it is possible for agents to be following clusters of leaders, while these leadership clusters are not mutually listening to each other. In that case, the lack of unified direction of these clusters can still cause a dissipating public opinion when we consider society as a whole. Below, we illustrate this scenario by simulating the opinion processes of three agents in a smoothed bounded confidence setup, with penalty function The resulting picture is rather sensitive to the choice of parameter in the exponent. Two of the agents move in a random walk, not listening to any other agent, while the third agent is strongly connected to both (as the parameter of the penalty function is , thus we satisfy the condition of Theorem 5.2). In , the orange and green paths resemble the opinion processes of the agents following a random walk, not being connected to anything in opinion space (neither an anchor, nor to each other). The blue agent is strongly connected to both of the other agents and traces out the area between the two paths of the agents until the green and orange paths diverge sufficiently. At that point, the blue agent sticks to either of the two agents (in this case, the orange agent) with no guarantee of connecting to the other agent again. Nothing keeps the green and orange agents together, hence this eliminates the possibility of having a stable public opinion. To fix this and formulate a coherent conjecture, we create the notion of a leadership graph that we denote by . Within this directed graph, we create an edge from agent to agent if agent is a follower of agent . We say that this is the case, if solely the one-directional interaction effect from agent to agent would already be strong enough to maintain a stable connection. In line with Theorem 5.2, this means that . Finally, we introduce the notion of a leadership core that we denote by . An agent falls in the leadership core if there is a directed path in the leadership graph to from any other agent in the society. In , we have created a society where the core is an empty set. There are no paths from agents 3, 4, and 5 to agents 1, 2, and 6; nor are there paths from agents 1 and 6 to 2, 3, 4, and 5. One way to resolve this (although there are many), is for agent 1 to also follow agent 2, so that there is a directed path from every agent to agents 3, 4, and 5. Thus, as shown in , we find , which is non-empty. There is one other way in which stability can be ensured, applicable in the asymmetric model when the society contains agents which are totally fixed in their opinion. If every agent in the society has some directed path to a fixed agent, then even if different agents follow different fixed agents, all agents obtain their information reliably from anchors at fixed distances from each other in opinion space. Therefore, they will not wander off to arbitrary distances. To accommodate this concept, introduce an auxiliary zero-th agent , representing a vacuous agent. We create a directed edge from to in the leadership graph if agent is fixed in their opinion. If every agent in the society is stably connected to a fixed agent, then falls in the leadership core. Hypothesis 5.3. Let be the Markov process which includes all opinion differences in our society. If the relationship graph is connected, and furthermore the leadership core of the society is non-empty, then admits an invariant measure. 6. Discussion and conclusion In this article, we have outlined a flexible model framework of noisy, non-strategic social interactions based on relaxing the bounded confidence assumption as encountered in the Deffuant–Weisbuch (Weisbuch et al., Citation2003) and the Hegselmann–Krause (Hegselmann & Krause, Citation2002). Our model allows for continuous opinions of any dimension and heterogeneity at the level of individual agents (i.e. agent-specific latent factors). We have provided the reader with a threefold of applications of the model environment, namely the smoothed bounded confidence model, decaying confidence model, and beta model. Our work provides an extensive basis for modelers in opinion dynamics to work with. We have also provided results on stability of these systems that significantly generalize the type of analysis performed by Baccelli et al. (Citation2017) for their stochastic bounded confidence model (what we call the decaying confidence model), as we allow for real-valued, multidimensional opinions and provide both necessary and sufficient conditions for stability in this more general framework. We moreover give results on asymmetric interactions and provide a hypothesis which would generalize our results to arbitrary asymmetric societies. This could be the starting point for future research. Research focused on stability of social systems can provide great insight into the formation and evolution of cultural norms and beliefs. Fully generalizing the stability theorems to asymmetric social networks can provide an insightful interplay between the parameters of the system (i.e. the bond strengths, interaction probabilities, and static compatibilities) and the formation of clusters and collective patterns. It can also identify main clusters and components in social systems and highlight groups that are at risk for alienation. As we had discussed in Section 4.1, we stress that a stable society (in terms of opinion differences) does not inherently imply that there is consensus among the agents. Nor does an unstable society signal polarization, as it could also imply fragmentation of the social system. The context in which the model is placed will determine what combination of social networks and behavioral parameters will imply consensus, polarization, and fragmentation. Although our model and the technical results are developed for broad applicability, we propose any empirical or real-world application to start out as simple as possible. This would mean that the personal characteristics of agents represent available (anonymous) data of agents: age, profession, nationality, social status, hobbies. A good starting might be a reciprocal-polynomial penalty function, where the new opinion of an individual depends on their initial opinion , the opinion of the other as for a noise term. The personal characteristics of the agents then act as moderators in the relationship via the -exponent . Relationships of this type could be empirically or experimentally evaluated by choosing a specific set of covariates of interest as motivated by literature. The increments of the time-series can subsequently be evaluated by experimental methods (letting individuals debate pairwise in controlled environments) or via data available on social networks. Once more research is done on the paired interactions, the model can be extended to approximate or analyze social systems using data from social networks. The contact probabilities of individuals coming into contact are then best approximated using the data available on the social platform. Our stability results can be adapted to remain applicable to opinion spaces that are countably infinite, or even finite after a transformation. Instead of considering to be the opinion space, one could consider this a latent opinion space that determines how people interact. The actual opinions or resulting actions are obtained via a map from to a compact or finite space. We also note that there, currently, is no dependency on initial opinions. As the opinion processes are irreducible, any point in opinion space is attainable and will be assigned some positive probability under the invariant measure (if it exists). For a promising direction of future research, we suggest to consider the static components of the system (interaction probabilities and compatibilities) as co-evolving, albeit on long timescales. Thus, the system moves to stability (or instability) in opinion space on short timescales. After a short-timescale burn-in period, the static parameters of the model are updated to represent evolving social bonds in terms of trust, fit, and the social infrastructure. Then, the opinion processes burn-in again, and so forth. This induces feedback loop between the social network and the opinion evolution. As individuals more often repulse each other, their compatibility might decrease; if they more often find agreement in their interaction, their compatibility might increase. As algorithms on social networks learn patterns, they alter the contact probabilities of the individuals to match personalities. All this could be fluently modeled using our setup and would be very valuable to further explore. Supplemental material Response to Reviewer 1.pdf Download PDF (97 KB)Response to Reviewer 1.pdfResponse to Reviewer 2.pdf Download PDF (106.1 KB)Response to Reviewer 2.pdfAcknowledgments The authors would like to thank the reviewers for their time and effort in providing feedback for this paper. This has undoubtedly improved the quality of the final article. Disclosure statement No potential conflict of interest was reported by the author(s). Supplementary material Supplemental data for this article can be accessed online at https://doi.org/10.1080/0022250X.2025.2529179 Additional information Funding Notes 1 Quite relevant here is the famous statement by Shizuo Kakutani: a drunk man will find his way home eventually, but a drunk bird may as well be lost forever. 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