Human opinion dynamics: An inspiration to solve complex optimization problems

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.

Bounded confidence opinion dynamics with opinion leaders and environmental noises

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.

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

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.

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.

Eulerian Opinion Dynamics with Bounded Confidence and Exogenous Inputs

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.

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.

Opinion formation and propagation are crucial phenomena in social networks and have been extensively studied across several disciplines. Traditionally, theoretical models of opinion dynamics have been proposed to describe the interactions between individuals (i.e., social interaction) and their impact on the evolution of collective opinions. Although these models can incorporate sociological and psychological knowledge on the mechanisms of social interaction, they demand extensive calibration with real data to make reliable predictions, requiring much time and effort. Recently, the widespread use of social media platforms provides new paradigms to learn deep learning models from a large volume of social media data. However, these methods ignore any scientific knowledge about the mechanism of social interaction. In this work, we present the first hybrid method called Sociologically-Informed Neural Network (SINN), which integrates theoretical models and social media data by transporting the concepts of physics-informed neural networks (PINNs) from natural science (i.e., physics) into social science (i.e., sociology and social psychology). In particular, we recast theoretical models as ordinary differential equations (ODEs). Then we train a neural network that simultaneously approximates the data and conforms to the ODEs that represent the social scientific knowledge. In addition, we extend PINNs by integrating matrix factorization and a language model to incorporate rich side information (e.g., user profiles) and structural knowledge (e.g., cluster structure of the social interaction network). Moreover, we develop an end-to-end training procedure for SINN, which involves Gumbel-Softmax approximation to include stochastic mechanisms of social interaction. Extensive experiments on real-world and synthetic datasets show SINN outperforms six baseline methods in predicting opinion dynamics.

In this era of fast and large-scale opinion formation, a mathematical understanding of opinion evolution, a.k.a. opinion dynamics, is especially important. Linear graph-based dynamics and bounded confidence dynamics are the two most popular models for opinion dynamics in social networks. Recently, stochastic bounded confidence opinion dynamics were proposed as a general framework that incorporates both these dynamics as special cases and also captures the inherent stochasticity and noise (errors) in real-life social exchanges. Although these dynamics are quite general and realistic, their analysis is particularly challenging compared to other opinion dynamics models. This is because these dynamics are nonlinear and stochastic, and belong to the class of Markov processes that have asymptotically zero drift and unbounded jumps. The asymptotic behavior of these dynamics was characterized in prior works. However, they do not shed light on their finite-time behavior, which is often of interest in practice. We take a stride in this direction by analyzing the finite time behavior of a two-agent system, which is fundamental to the understanding of multi-agent dynamics. In particular, we show that the opinion difference between the two agents is well concentrated around zero under the conditions that lead to asymptotic stability of the dynamics.

In recent years, many models have been introduced to study on that how those microcosmic individuals with different opinions tend to format macroscopic consensus in the opinion interaction, which have been described by evolutionary game theory. This paper introduce the game theory to model the formation of binary opinions in signed network: cooperative game is used to model the interaction rules of positive(+) relationship between two individuals; minority game is used to model the interaction rules of negative(-) relationship between two individuals. When we calculate those individuals cumulative payoffs, we are firstly induce the probability parameter ; which represent the impact of balance payoff base on structure balance theory to opinion dynamics. Meanwhile, in our work, the result also suggest that when individual resort to more neighbors' opinion as reference, the more difficulty for individual to decide which opinion should be selected, ultimately more consensus time (Tc) will spend on those debate process among individuals.

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.