A relative approach to opinion formation

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.

Opinion dynamics in dynamical hypergraphs, mirroring the dynamical nature of high-order social interactions, are attracting increasing attention. Opinion dynamics on high-order interactions lead to non-trivial dynamical patterns compared with that on pairwise networks. How should we systematically understand the intrinsic differences between opinion dynamics in hypergraphs and that in networks? We establish a voter model in a dynamical hypergraph. We find that both opinion formation and transient topology are captured by a single multi-player game, provided that the hypergraphs evolve sufficiently fast. The Nash equilibrium analysis facilitates us to reveal the intrinsic differences between high-order interactions and pairwise interactions in the empirical system. Furthermore, we perform simulations to show how robust our theoretical results are beyond fast rewiring. Our work provides a game route for opinion dynamics and sheds a hidden connection between opinion formation on dynamical high-order interactions and multi-player games.

To investigate modulus consensus on multiple interdependent topics with heterogeneous logical relationships, a comprehensive opinion dynamics model is proposed in a time-varying influence network. There exist cooperative and antagonistic interpersonal interactions in the time-varying influence network. A logic matrix is designed to describe the heterogeneous logical relationships within multiple interdependent topics. Based on graph theory, both convergence and modulus consensus are shown for the heterogeneous time-varying opinion dynamics. Simulation results are given to illustrate effectiveness on modulus consensus of the heterogeneous time-varying opinion dynamics.

Abstractsocial networks Online provide a globally available, large-scale infrastructure for people to exchange information and ideas. A topic of great interest in internet research is how to model this information exchange and, in particular, how to model and analyze the effects of interpersonal influence on processes such as information diffusion, influence propagation, and opinion formation. Recent empirical studies indicate that, in order to accurately model communication in online social networks, it is important to consider not just relationships between individuals, but also the frequency with which these individuals interact. We study a model of opinion formation in social networks proposed by De Groot and Lehrer and show how this model can be extended to include interaction frequency. We prove that, for the purposes of analysis and design, the opinion formation process with probabilistic interactions can be accurately approximated by a deterministic system where edge weights are adjusted for the probability of interaction. We also present simulations that illustrate the effects of different interaction frequencies on the opinion dynamics using real-world social network graphs.KeywordsOpinion DynamicsSocial InfluenceComplex NetworkAgent-based Modeling

Bounded confidence opinion dynamics with opinion leaders and environmental noises

We extend a classical model of continuous opinion formation to explicitly include an age-structured population. We begin by considering a stochastic differential equation model which incorporates ageing dynamics and birth/death processes, in a bounded confidence type opinion formation model. We then derive and analyse the corresponding mean field partial differential equation and compare the complex dynamics on the microscopic and macroscopic levels using numerical simulations. We rigorously prove the existence of stationary states in the mean field model, but also demonstrate that these stationary states are not necessarily unique. Finally, we establish connections between this and other existing models in various scenarios.

This article estimates a simple univariate model of expectation or opinion formation in continuous time adapting a ‘canonical’ stochastic model of collective opinion dynamics (Weidlich and Haag, 1983; Lux, 1995, 2009a). This framework is applied to a selected data set on survey-based expectations from the rich EU business and consumer survey database for 12 European countries. The model parameters are estimated through Maximum Likelihood (ML) and numerical solution of the transient probability density functions for the resulting stochastic process. The model's success is assessed with respect to its out-of-sample forecasting performance relative to univariate Time Series (TS) models of the Autoregressive Moving Average model, ARMA(p, q) and Autoregressive Fractionally Integrated Moving Average, ARFIMA(p, d, q) varieties. These tests speak for a slight superiority of the canonical opinion dynamics model over the alternatives in the majority of cases.

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.

Opinion dynamics under double thresholds

Researches on opinion dynamics aim at modeling the formation and evolution of public opinions from individual level. This paper reviews basic models and results in the field of opinion dynamics, focusing on the starting points and the modeling principles. The influence of network structure on the dynamical result is discussed. The comparison of models and results with empirical data on the Internet is emphasized. By employing the particularities of the Internet, we propose several future research issues in the area.

[formula omitted]-player game formulation of the majority-vote model of opinion dynamics

We study the effect of adaptivity on a social model of opinion dynamics and consensus formation. We analyse how the adaptivity of the network of contacts between agents to the underlying social dynamics affects the size and topological properties of groups and the convergence time to the stable final state. We find that, while on static networks these properties are determined by percolation phenomena, on adaptive networks the rewiring process leads to different behaviors: adaptive rewiring fosters group formation by enhancing communication between agents of similar opinion, though it also makes possible the division of clusters. We show how the convergence time is determined by the characteristic time of link rearrangement. We finally investigate how the adaptivity yields nontrivial correlations between the internal topology and the size of the groups of agreeing agents.

Social media platforms have amplified opinion polarization, complicating policymaking and public sentiment management. Existing models primarily address either consensus formation or binary polarization, failing to account for the diverse, multi-polar opinion distributions observed in real-world social interactions. This work introduces a dynamic opinion model integrating individual stubbornness and opinion sensitivity, enabling the simulation of multi-opinion coexistence within social networks. It is formally proven that, under low-impact events, opinions converge to a neutral consensus under both synchronous and asynchronous updates. For high-impact events, where theoretical analysis is challenging, this work employs simulations under asynchronous updating, which better reflect real conditions. The results indicate that mixed groups may exhibit multi-state coexistence, with homogenized social relationships leading to weak opinion cohesion and social heterogeneity fostering community formation. To regulate diverse opinions in social networks, this work introduces two governance strategies designed for events with different impacts. Notably, for events with limited impact, soft opinion guidance shapes group preferences more effectively than forceful interventions. These findings offer insights for refining policy communication strategies and advancing the broader understanding of opinion dynamics in diverse societies.

Human interactions give rise to the formation of different kinds of opinions in a society. The study of formations and dynamics of opinions has been one of the most important areas in social physics. The opinion dynamics and associated social structure leads to decision making or so called opinion consensus. Opinion formation is a process of collective intelligence evolving from the integrative tendencies of social influence with the disintegrative effects of individualisation, and therefore could be exploited for developing search strategies. Here, we demonstrate that human opinion dynamics can be utilised to solve complex mathematical optimization problems. The results have been compared with a standard algorithm inspired from bird flocking behaviour and the comparison proves the efficacy of the proposed approach in general. Our investigation may open new avenues towards understanding the collective decision making.

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.