Continuous Opinion Dynamics Under Bounded Confidence on a Complete Network

Models of continuous opinion dynamics under bounded confidence show a sharp transition between a consensus and a polarization phase at a critical global bound of confidence. In this paper, heterogeneous bounds of confidence are studied. The surprising result is that a society of agents with two different bounds of confidence (open‐ and closed‐minded agents) can find consensus even when both bounds of confidence are significantly below the critical bound of confidence of a homogeneous society. The phenomenon is shown by examples of agent‐based simulation and by numerical computation of the time evolution of the agents density. The result holds for the bounded confidence model of Deffuant, Weisbuch, and others (Weisbuch et al., Complexity 2002, 7, 55–63), as well as for the model of Hegselmann and Krause (Hegselmann and Krause, Journal of Artificial Societies and Social Simulation 2002, 5, 2). Thus, given an average level of confidence, diversity of bounds of confidence enhances the chances for consensus. The drawback of this enhancement is that opinion dynamics becomes suspect to severe drifts of clusters, where open‐minded agents can pull closed‐minded agents towards another cluster of closed‐minded agents. A final consensus might thus not lie in the center of the opinion interval as it happens for uniform initial opinion distributions under homogeneous bounds of confidence. It can be located at extremal locations. This is demonstrated by example, which also show that the extension to heterogeneous bounds of confidence enriches the complexity of the dynamics tremendously. © 2009 Wiley Periodicals, Inc. Complexity, 2010

We study a tractable opinion dynamics model that generates long-run disagreements and persistent opinion fluctuations. Our model involves an inhomogeneous stochastic gossip process of continuous opinion dynamics in a society consisting of two types of agents: (1) regular agents who update their beliefs according to information that they receive from their social neighbors and (2) stubborn agents who never update their opinions and might represent leaders, political parties, or media sources attempting to influence the beliefs in the rest of the society. When the society contains stubborn agents with different opinions, the belief dynamics never lead to a consensus (among the regular agents). Instead, beliefs in the society fail to converge almost surely, the belief profile keeps on fluctuating in an ergodic fashion, and it converges in law to a nondegenerate random vector. The structure of the graph describing the social network and the location of the stubborn agents within it shape the opinion dynamics. The expected belief vector is proved to evolve according to an ordinary differential equation coinciding with the Kolmogorov backward equation of a continuous-time Markov chain on the graph with absorbing states corresponding to the stubborn agents, and hence to converge to a harmonic vector, with every regular agent's value being the weighted average of its neighbors' values, and boundary conditions corresponding to the stubborn agents' beliefs. Expected cross products of the agents' beliefs allow for a similar characterization in terms of coupled Markov chains on the graph describing the social network. We prove that, in large-scale societies, which are highly fluid, meaning that the product of the mixing time of the Markov chain on the graph describing the social network and the relative size of the linkages to stubborn agents vanishes as the population size grows large, a condition of homogeneous influence emerges, whereby the stationary beliefs' marginal distributions of most of the regular agents have approximately equal first and second moments.

The formation of opinions in a large population is governed by endogenous (interactions with peers) and exogenous (influence of media) factors. In the analysis of opinion evolution in a large population, decision making rules are often approximated with non-Bayesian “rule of thumb” methods. Adopting a non-Bayesian averaging rule, this paper focuses on an Eulerian bounded-confidence model of opinion dynamics and studies the information assimilation process resulting from exogenous inputs. In this model, a population is distributed over an opinion set, and each individual updates its opinion via (i) opinions of the population inside the individual's confidence range and (ii) the information from an exogenous input in that range. First, we establish various mathematical properties of this system's dynamics with time-varying inputs. Second, for the case of no exogenous input, we prove the convergence of the population's distribution to a sum of Dirac delta distributions. We further derive a simple sufficient condition for opinion consensus under the influence of a time-varying input. Third, regarding information assimilation, we define the attracted population of a constant input. For a weighted Dirac delta input and for uniformly distributed initial population, we establish an upper bound on the attracted population valid under some technical assumptions. This upper bound is an increasing function of the population's confidence bound and a decreasing function of the input's measure (i.e., the integral of input's distribution over the opinion space). Fourth, for a normally distributed input with truncated support, we conjecture that the attracted population is approximately an increasing affine function of the population's confidence bound and of the input's standard deviation; we illustrate this conjecture numerically.

Impact of informal networks on opinion dynamics in hierarchically formal organization

Eulerian Opinion Dynamics with Bounded Confidence and Exogenous Inputs

We explore the possibilities of enforcing and preventing consensus in continuous opinion dynamics that result from modifications in the communication rules. We refer to the model of Weisbuch and Deffuant, where n agents adjust their continuous opinions as a result of random pairwise encounters whenever their opinions differ not more than a given bound of confidence ε. A high ε leads to consensus, while a lower ε leads to a fragmentation into several opinion clusters. We drop the random encounter assumption and ask: How small may ε be such that consensus is still possible with a certain communication plan for the entire group? Mathematical analysis shows that ε may be significantly smaller than in the random pairwise case. On the other hand, we ask: How large may ε be such that preventing consensus is still possible? In answering this question, we prove Fortunato's simulation result that consensus cannot be prevented for ε > 0.5 for large groups. Next, we consider opinion dynamics under different individual strategies and examine their power to increase the chances of consensus. One result is that balancing agents increase chances of consensus, especially if the agents are cautious in adapting their opinions. However, curious agents increase chances of consensus only if those agents are not cautious in adapting their opinions.

Dynamics of an agent-based opinion model with complete social connectivity network

In this paper, we propose a new model for continuous time opinion dynamics on an evolving network. As opposed to existing models, in which the network typically evolves by discretely adding or removing edges, we instead propose a model for opinion formation which is coupled to a network evolving through a system of ordinary differential equations for the edge weights. We interpret each edge weight as the strength of the relationship between a pair of individuals, with edges increasing in weight if pairs continually listen to each other’s opinions and decreasing if not. We investigate the impact of various edge dynamics at different timescales on the opinion dynamics itself. This is done partly through analytic results and partly through extensive numerical simulations of two case studies: one using bounded confidence interaction dynamics in the opinion formation process (as in the classical Hegselmann–Krause model) and one using an exponentially decaying interaction function. We find that the dynamic edge weights can have a significant impact on the opinion formation process, since they may result in consensus formation but can also reinforce polarisation. Overall, the proposed modelling approach allows us to quantify and investigate how the network and opinion dynamics influence each other.

In order to characterize asset price and wealth dynamics arising from the interaction of heterogeneous agents with CRRA utility, a discrete-time stationary model in terms of return and wealth proportions (among different types of agents) is established. When fundamentalists and chartists are the main heterogeneous agents in the model, it is found that in the presence of heterogeneous agents the stationary model can have multiple steady states. The steady state is unstable when the chartists extrapolate strongly and (locally) stable when they extrapolate weakly. The convergence to the steady state follows an optimal selection principle - the return and wealth proportions tend to the steady state which has relatively higher return. More importantly, heterogeneity can generate instability which, under the stochastic processes of the dividend yield and extrapolation rates, results in switching of the return among different states, such as steady-state, periodic and aperiodic cycles from time to time. The model that is finally developed displays the essential characteristics of the standard asset price dynamics model assumed in continuous-time finance, in that the asset price is fluctuating around a geometrically growing trend. The model also displays the volatility clustering that is an essential feature of empirically observed asset returns.

Two phenomena that are central to simulation research on opinion dynamics are opinion divergence—the result that individuals interacting in a group do not always collapse to a single viewpoint, and group polarization—the result that average group opinions can become more extreme after discussions than they were to begin with. Standard approaches to modeling these dynamics have typically assumed that agents have an influence bound, such that individuals ignore opinions that differ from theirs by more than some threshold, and thus converge to distinct groups that remain uninfluenced by other distinct beliefs. Additionally, models have attempted to account for group polarization either by assuming the existence of recalcitrant extremists, who draw others to their view without being influenced by them, or negative reaction—movement in opinion space away from those they disagree with. Yet these assumptions are not well supported by existing social/cognitive theory and data, and insofar as there are data, it is often mixed. Moreover, an alternative cognitive assumption is able to produce both of these phenomena: the need for consistency within a set of related beliefs. Via simulation, we show that assumptions about knowledge or belief spaces and conceptual coherence naturally produce both convergence to distinct groups and group polarization, providing an alternative cognitively grounded mechanism for these phenomena.

The discrete-time Altafini model is an opinion dynamics model in which the interactions among a group of agents are described by a time-varying signed digraph. This paper first uses graph theoretic constructions to study modified versions of the Altafini model in which there are communication delays or quantized communication. The condition under which consensus in absolute value or bipartite consensus is achieved proves to be the same as the condition in the delay-free case. The paper also analyzes the performance of the model where the information exchanged between neighboring agents is subject to a certain type of deterministic uniform quantization. We show that in finite time and depending on initial conditions, the model on any static, connected, undirected signed graph will either cause all agents to reach a quantized consensus in absolute value, or will lead all variables to oscillate in a small neighborhood around the absolute value.

This article investigates the Multiple Equilibria Regulation (MER) model, i.e., an agent-based simulation model, to represent opinion dynamics in social networks. It relies on a small set of micro-prerequisites (intra-individual balance and confidence bound), leading to emergence of (non)stationary macro-outcomes. These outcomes may refer to consensus, polarization or fragmentation of opinions about taxation (e.g., congestion pricing) or other policy measures, according to the way communication is structured. In contrast with other models of opinion dynamics, it allows for the impact of both the regulation of intra-personal discrepancy and the interpersonal variability of opinions on social learning and network dynamics. Several simulation experiments are presented to demonstrate, through the MER model, the role of different network structures (complete, star, cellular automata, small-world and random graphs) on opinion formation dynamics and the overall evolution of the system. The findings can help to identify specific topological characteristics, such as density, number of neighbourhoods and critical nodes-agents, that affect the stability and system dynamics. This knowledge can be used to better organize the information diffusion and learning in the community, enhance the predictability of outcomes and manage possible conflicts. It is shown that a small-world organization, which depicts more realistic aspects of real-life and virtual social systems, provides increased predictability and stability towards a less fragmented and more manageable grouping of opinions, compared to random networks. Such macro-level organizations may be enhanced with use of web-based technologies to increase the density of communication and public acceptability of policy measures.

Individuals of modern societies share ideas and participate in collective processes within a pervasive, variable, and mostly hidden ecosystem of content filtering technologies that determine what information we see online. Despite the impact of these algorithms on daily life and society, little is known about their effect on information transfer and opinion formation. It is thus unclear to what extent algorithmic bias has a harmful influence on collective decision-making, such as a tendency to polarize debate. Here we introduce a general theoretical framework to systematically link models of opinion dynamics, social network structure, and content filtering. We showcase the flexibility of our framework by exploring a family of binary-state opinion dynamics models where information exchange lies in a spectrum from pairwise to group interactions. All models show an opinion polarization regime driven by algorithmic bias and modular network structure. The role of content filtering is, however, surprisingly nuanced; for pairwise interactions it leads to polarization, while for group interactions it promotes coexistence of opinions. This allows us to pinpoint which social interactions are robust against algorithmic bias, and which ones are susceptible to bias-enhanced opinion polarization. Our framework gives theoretical ground for the development of heuristics to tackle harmful effects of online bias, such as information bottlenecks, echo chambers, and opinion radicalization.

Bounded confidence opinion dynamics with opinion leaders and environmental noises

Formal models of opinion formation commonly represent an individual’s opinion by a value on a fixed opinion interval. We propose an alternative modeling method wherein interpretation is only provided to the relative positions of opinions vis-à-vis each other. This method is then considered in a similar setting as the discrete-time Altafini model (an extension of the well-known DeGroot model), but with more general influence weights. Even in a linear framework, the model can describe, in the long run, polarization, dynamics with a periodic pattern, and (modulus) consensus formation. In addition, in our alternative approach key characteristics of the opinion dynamic can be derived from real-valued square matrices of influence weights, which immediately allows one to transfer matrix theory insights to the field of opinion formation dynamics under more relaxed conditions than in the DeGroot or discrete-time Altafini models. A few specific themes are covered: (i) We demonstrate how stable patterns in relative opinion dynamics are identified which are hidden when opinions are considered in an absolute opinion framework. (ii) For the two-agent case, we provide an exhaustive closed-form description of the relative opinion model’s dynamic in the long run. (iii) We explore group dynamics analytically, in particular providing a non-trivial condition under which a subgroup’s asymptotic behavior carries over to the entire population.