On evolving network models and their influence on opinion formation

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

One of the most studied problems among researchers in recent years is how individuals form their opinions. This problem has become more urgent with the advent of social networks, which can easily influence a huge number of followers and have become increasingly pervasive over time. The produced effect is the rise of polarized opinions among different groups of people. Understanding polarization is of great relevance across various application domains, such as economics and politics. Opinion dynamics has often been studied by exploiting the popular Friedkin–Johnsen model. In this paper, we propose a different modeling approach based on the Markovian agents paradigm for deriving metrics characterizing polarized opinions. The main goal of this work is to demonstrate the potential of the Markovian agent modeling paradigm for the analysis of opinion dynamics. The main advantages of Markovian agents are the ease of setting a large number of behavioral parameters, spatial distribution of agents, scalability, and numerical tractability. We extend our previous work, in which we analyzed a peer assembly and validated it against other commonly used modeling approaches. In our opinion, the Markovian agent approach offers an effective modeling framework due to its scalability and flexibility in handling parameters that describe the behavior of individuals in the opinion formation process. The context we will discuss is inspired by election rallies, where an assembly attends a speech by a political candidate. The crowd consists of individuals with diverse initial political opinions, and the candidate seeks to polarize them toward his/her own political stance.

The opinion dynamics literature argues that the way people perceive social influence depends not only on the opinions of interacting individuals, but also on the individuals’ non-opinion characteristics, such as age, education, gender, or place of residence. The current paper advances this line of research by studying longitudinal data that describe the opinion dynamics of a large sample (~30,000) of online social network users, all citizens of one city. Using these data, we systematically investigate the effects of users’ demographic (age, gender) and structural (degree centrality, the number of common friends) properties on opinion formation processes. We revealed that females are less easily influenced than males. Next, we found that individuals that are characterized by similar ages have more chances to reach a consensus. Additionally, we report that individuals who have many common peers find an agreement more often. We also demonstrated that the impacts of these effects are virtually the same, and despite being statistically significant, are far less strong than that of opinion-related features: knowing the current opinion of an individual and, what is even more important, the distance in opinions between this individual and the person that attempts to influence the individual is much more valuable. Next, after conducting a series of simulations with an agent-based model, we revealed that accounting for non-opinion characteristics may lead to not very sound but statistically significant changes in the macroscopic predictions of the populations of opinion camps, primarily among the agents with radical opinions (≈3% of all votes). In turn, predictions for the populations of neutral individuals are virtually the same. In addition, we demonstrated that the accumulative effect of non-opinion features on opinion dynamics is seriously moderated by whether the underlying social network correlates with the agents’ characteristics. After applying the procedure of random shuffling (in which the agents and their characteristics were randomly scattered over the network), the macroscopic predictions have changed by ≈9% of all votes. What is interesting is that the population of neutral agents was again not affected by this intervention.

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.

<p>pinion dynamics models study how the interaction among people influences the opinion formation process. In most opinion dynamics models, only one opinion can exist in the steady state, which is different from the real-life opinion formation process. In 2009, Shao et at. introduced a Non-Consensus Opinion (NCO) model, which allows different opinions to coexist in the steady state. This paper extends the NCO model by introducing a special type of nodes, namely Byzantine nodes, to play the role of dishonest people. We perform simulations on three different network models: small-scale graphs, Erdős-Rényi random graphs and scale-free networks. We find a new steady state for the NCO model: the cyclic steady state. The cyclic behavior of the NCO and Byzantine NCO model is discussed, including a method to generate networks with extremely long cycle lengths. Other properties of the Byzantine NCO model, such as the probability of cyclic behavior and the final opinion distribution, are also studied. We find that the introduction of Byzantine nodes generally steers towards a more balanced steady state and increases the probability of cyclic behavior. The latter is particularly problematic in communication systems, where the large cycle lengths may cause a very slow consensus process and thus stalling future communications.</p>

It is well known that a model of any electrical circuit can be translated into a system of ordinary differential equations. Usually, the simulation is performed in the time-domain using classical discrete approaches convenient for both linear and non-linear systems. An other method consists in the quantization on the variable magnitude which is based on discrete events. The discrete event system specification formalism is adapted to simulate any electrical circuit described with a system of ordinary differential equations using the system state quantification. In this paper, a solution to make more efficient the magnitude quantification for a system of linear ordinary differential equations is given. The proposed technique is used to simulate a simple three phase coupled power systems and it is compared to other solutions given by the explicit integration using formal language and the time-domain discretization using fixed-step fifth order Runge-Kutta integration method.

We consider boundary value problems with periodic boundary conditions for first-order linear systems of impulsive ordinary differential equations with unknown right-hand sides and jumps of solutions at the impulse points entering into the statement of these problems which are assumed to be subjected to some quadratic restrictions. From indirect noisy observations of their solutions on a finite system of intervals, we obtain the optimal, in certain sense, estimates of images of their right-hand sides under linear continuous operators. Under the condition that the unknown correlation functions of noises in observations belong to some special sets, it is established that such estimates and estimation errors are expressed explicitly via solutions of special periodic boundary value problems for linear impulsive systems of ordinary differential equations.

Impact of informal networks on opinion dynamics in hierarchically formal organization

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.

Bounded confidence opinion dynamics with opinion leaders and environmental noises

The non-standard Hamiltonian system, also referred to as a partial Hamiltonian system in the literature, of the form q ˙ i = ∂ H ∂ p i , p ˙ i = − ∂ H ∂ q i + Γ i ( t , q i , p i ) ${\dot q^i} = \frac{{\partial H}}{{\partial {p_i}}},{\text{ }}{\dot p^i} = - \frac{{\partial H}}{{\partial {q_i}}} + {\Gamma ^i}(t,{\text{ }}{q^i},{\text{ }}{p_i})$ appears widely in economics, physics, mechanics, and other fields. The non-standard (partial) Hamiltonian systems arise from physical Hamiltonian structures as well as from artificial Hamiltonian structures. We introduce the term ‘artificial Hamiltonian’ for the Hamiltonian of a model having no physical structure. We provide here explicitly the notion of an artificial Hamiltonian for dynamical systems of ordinary differential equations (ODEs). Also, we show that every system of second-order ODEs can be expressed as a non-standard (partial) Hamiltonian system of first-order ODEs by introducing an artificial Hamiltonian. This notion of an artificial Hamiltonian gives a new way to solve dynamical systems of first-order ODEs and systems of second-order ODEs that can be expressed as a non-standard (partial) Hamiltonian system by using the known techniques applicable to the non-standard Hamiltonian systems. We employ the proposed notion to solve dynamical systems of first-order ODEs arising in epidemics.

Rational forecasts or social opinion dynamics? Identification of interaction effects in a business climate survey

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.

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.

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.