Modulus Consensus for Time-Varying Heterogeneous Opinion Dynamics on Multiple Interdependent Topics

ABSTRACTThis paper addresses the modulus consensus in a network of singularly perturbed systems, all of which own multi-time scales characteristics. The interactions among the agents are collaborative and antagonistic interactions, and it can been modelled by a signed directed graph. Also, the effect of time-varying delay has been taken into account when modelling the agent dynamics. Based on singularly perturbed decomposition method, Kronecker product technique and the comparison principle, several sufficient conditions are presented under which networked singularly perturbed time-varying delay systems with collaborative and antagonistic interactions achieve modulus consensus. Moreover, the upper bound of singular perturbation parameter is deduced explicitly. Two simulation examples are presented to illustrate the validity of the proposed results.

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.

Mobility-tolerant data forwarding over time-varying IoT networks is a challenging problem. Selection of a relay node at every instant in time is crucial and calls for robust and distributed methods to improve the quality of services in such networks. In this paper, an optimal relay node selection method is proposed for robust data forwarding in time-varying networks. The proposed method selects a relay node based on joint optimization of two network parameters, namely data latency and link reliability. The Pareto front of this multi-objective optimization problem is used to illustrate the tradeoff between low data latency and high link reliability. The proposed optimal relay node selection method first identifies all possible relay nodes at every time instant. The weights of the bi-objective problem are updated at every hop and an optimal relay node is selected among all the possible relay nodes, by solving the joint optimization problem. During performance evaluation, random spatial patterns of Internet of Things (IoT) devices are modeled using the homogeneous Poisson point processes. The connectivity duration information of all the IoT devices with their neighbors is updated continuously and a relay node found is in an online manner. Simulation results indicate that the proposed method significantly improves data latency and the packet loss risk for data forwarding over time-varying IoT networks, when compared to conventional single parameter optimization methods. The proposed data forwarding method indicates additional gains in terms of transmission range, mobility tolerance, packet replication cost, and connectivity parameters of the time-varying IoT network.

Bounded confidence opinion dynamics with opinion leaders and environmental noises

In the new media environment, the constantly emerging social platforms further expand the channels for the propagation of public opinion. Under the framework of complex network theory and faced the needs of management practice, modeling the spreading dynamics of public opinion in temporal group networks is of great significance for understanding its spreading law and improving the governance system of cyberspace and the development of network science. Through analyzing the changes of group networks topology and the spreading rules of public opinion, the spreading model of public opinion in temporal group networks was proposed by coupling the two dynamic processes, and the spreading thresholds of public opinion in static and temporal group networks were derived respectively. Then, the spreading characteristics of public opinion under different network topology, as well as the influence of important parameters on public opinion spreading process were discussed with the help of simulation experiments. The research results indicated that the propagation of public opinion in static and temporal group networks exhibits both similar trends and differentiated characteristics; compared with Spreader, the propagation of public opinion in temporal group networks is more sensitive to Ignorant’s behavior; both groups’ and netizens’ active probability have significant influence on public opinion propagation, but netizens’ affects more. Based on the relevant results, this paper proposed a series of countermeasures such as, grading social platforms, strengthening relationship management between them and introducing time management systems, so as to promote the formation of a good network ecosystem and the modernization of the national governance system.

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 this paper we introduce a novel model that can account for the spread of extreme opinions in a human population as a purely local, selforganising process. Our starting point is the well-known and influential Relative Agreement (RA) model of opinion dynamics introduced by Deffuant et al. (2002). The RA model explores the dynamics of opinions in populations that are initially seeded with some number of individuals, who hold opinions at the far ends of a continuous spectrum of opinions represented in the abstract RA model as a real value in the range [-1.0, +1.0]; but the majority of the individuals in the population are, at the outset, holding opinions closer to the central mid-range value of 0.0. Various researchers have demonstrated that the RA model generates opinion dynamics in which the influence of the extremists on the moderates leads, over time, to the distribution of opinion values in the population converging to attractor states that can be qualitatively characterised as one of either uni-polar and bi-polar extremes, or reversion to the centre (central convergence). However, a major weakness of the RA model is that it pre-supposes the existence of extremist individuals, and hence says nothing to answer the question of where do extremists come from? In this paper, we introduce the Relative Disagreement (RD) model, in which extremist individual arise spontaneously and can then exert influence over moderates, forming large groups of polar extremists, via an entirely internal, self-organisation process. We demonstrate that the RD model can readily exhibit the uni-polar, bi-polar, and central-convergence attractors that characterise the dynamics of the RA model, and hence this is the first paper to describe an opinion dynamic model in which extremist positions can spontaneously arise and spread in a population via a self-organising process opinion-influencing interactions between any two individuals are characterised not only by the extent to which they agree, but also by the extent to which they disagree.

We propose a random adaptation variant of time-varying distributed averaging dynamics in discrete time. We show that this leads to novel interpretations of fundamental concepts in distributed averaging, opinion dynamics, and distributed learning. Namely, we show that the ergodicity of a stochastic chain is equivalent to the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">almost sure</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a.s.)</i> finite-time agreement attainment in the proposed random adaptation dynamics. Using this result, we provide a new interpretation for the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">absolute probability sequence</i> of an ergodic chain. We then modify the base-case dynamics into a time-reversed inhomogeneous Markov chain, and we show that in this case ergodicity is equivalent to the uniqueness of the limiting distributions of the Markov chain. Finally, we introduce and study a time-varying random adaptation version of the Friedkin-Johnsen model and a rank-one perturbation of the base-case dynamics.

Macroeconomic and stock market interactions with endogenous aggregate sentiment dynamics

We propose two different schemes of realizing a virtual walk corresponding to a kinetic exchange model of opinion dynamics. The walks are either Markovian or non-Markovian in nature. The opinion dynamics model is characterized by a parameter [Formula: see text] which drives an order disorder transition at a critical value [Formula: see text]. The distribution [Formula: see text] of the displacements [Formula: see text] from the origin of the walkers is computed at different times. Below [Formula: see text], two time scales associated with a crossover behaviour in time are detected, which diverge in a power law manner at criticality with different exponent values. [Formula: see text] also carries the signature of the phase transition as it changes its form at [Formula: see text]. The walks show the features of a biased random walk below [Formula: see text], and above [Formula: see text], the walks are like unbiased random walks. The bias vanishes in a power law manner at [Formula: see text] and the width of the resulting Gaussian function shows a discontinuity. Some of the features of the walks are argued to be comparable to the critical quantities associated with the mean-field Ising model, to which class the opinion dynamics model belongs. The results for the Markovian and non-Markovian walks are almost identical which is justified by considering the different fluxes. We compare the present results with some earlier similar studies. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.

Growing evidence links impairment of brain functions in Alzheimer's disease (AD) with disruptions of brain functional connectivity. However, whether the AD brain shows similar changes from a dynamic or cross-frequency view remains poorly explored. This paper provides an effective framework to investigate the properties of multiplex brain networks in AD considering inter-frequency and temporal dynamics. Using resting-state EEG signals, two types of multiplex networks were reconstructed separately considering the network interactions between different frequency bands or time points. We further applied multiplex network features to characterize functional integration and segregation of the cross-frequency or time-varying networks. Finally, machine learning methods were employed to evaluate the performance of multiplex-network-based indexes for detection of AD. Results revealed that the brain networks of AD patients are disrupted with reduced segregation particularly in the left occipital area for both cross-frequency and time-varying networks. However, the alteration of integration differs among brain regions and may show an increasing trend in the frontal area of AD brain. By combining the features of integration and segregation in time-varying networks, the best classification performance was achieved with an accuracy of 92.5%. These findings suggest that our multiplex framework can be applied to explore functional integration and segregation of brain networks and characterize the abnormalities of brain function. This may shed new light on the brain network analysis and extend our understanding of brain function in patients with neurological diseases.

Traditional models of opinion dynamics provide a simplified approach to understanding human behavior in basic social scenarios. However, when it comes to issues such as polarization and extremism, a more nuanced understanding of human biases and cognitive tendencies are required. This paper proposes an approach to modeling opinion dynamics by integrating mental models and assumptions of individuals agents using Bayesian-inspired methods. By exploring the relationship between human rationality and Bayesian theory, this paper demonstrates the usefulness of these methods in describing how opinions evolve. The analysis here builds upon the basic idea in the Continuous Opinions and Discrete Actions (CODA) model, by applying Bayesian-inspired rules to account for key human behaviors such as confirmation bias, motivated reasoning, and human reluctance to change opinions. Through this, This paper updates rules that are compatible with known human biases. The current work sheds light on the role of human biases in shaping opinion dynamics. I hope that by making the model more realistic this might lead to more accurate predictions of real-world scenarios.

Impact of informal networks on opinion dynamics in hierarchically formal organization

Analysis of opinion dynamics in social networks plays an important role in today's life. For applications such as predicting users' political preference, it is particularly important to be able to analyze the dynamics of competing opinions. While observing the evolution of polar opinions of a social network's users over time, can we tell when the network "behaved" abnormally? Furthermore, can we predict how the opinions of the users will change in the future? Do opinions evolve according to existing network opinion dynamics models? To answer such questions, it is not sufficient to study individual user behavior, since opinions can spread far beyond users' egonets. We need a method to analyze opinion dynamics of all network users simultaneously and capture the effect of individuals' behavior on the global evolution pattern of the social network. In this work, we introduce Social Network Distance (SND) - a distance measure that quantifies the "cost" of evolution of one snapshot of a social network into another snapshot under various models of polar opinion propagation. SND has a rich semantics of a transportation problem, yet, is computable in time linear in the number of users, which makes SND applicable to the analysis of large-scale online social networks. In our experiments with synthetic and real-world Twitter data, we demonstrate the utility of our distance measure for anomalous event detection. It achieves a true positive rate of 0.83, twice as high as that of alternatives. When employed for opinion prediction in Twitter, our method's accuracy is 75.63%, which is 7.5% higher than that of the next best method. Source Code: https://cs.ucsb.edu/~victor/pub/ucsb/dbl/snd/

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.