Memory effects play a crucial role in social interactions and decision-making processes. This paper proposes a novel fractional-order bounded confidence opinion dynamics model to characterize the memory effects in system states. Building upon the Hegselmann-Krause framework and fractional-order difference, a comprehensive model is established that captures the persistent influence of historical information. Through rigorous theoretical analysis, the fundamental properties including convergence and consensus is investigated. The results demonstrate that the proposed model not only maintains favorable convergence and consensus characteristics compared to classical opinion dynamics, but also addresses limitations such as the monotonicity of bounded opinions. This enables a more realistic representation of opinion evolution in real-world scenarios. The findings of this study provide new insights and methodological approaches for understanding opinion formation and evolution, offering both theoretical significance and practical applications.
We derive a differential-integral equation akin to the Hegselmann-Krause model of opinion dynamics, and propose a particle method for solving the equation. Numerical experiments demonstrate second-order convergence of the method in a weak sense. We also show that our differential-integral equation can equivalently be stated as a system of differential equations. An integration-by-parts argument that would typically yield an energy dissipation inequality in physical problems then yields a concentration inequality, showing that a natural measure of concentration increases monotonically.
The Hegselmann-Krause (HK) model allows one to characterize the continuous change of agents' opinions with the bounded confidence threshold $\varepsilon$. To consider the heterogeneity of agents in characteristics, we study the HK model on homogeneous and heterogeneous networks by introducing a kind of smart agent. Different from the averaging rule in opinion update of HK model, smart agents will consider, in updating their opinions, the environmental influence following the fact that an agent's behavior is often coupled with environmental changes. The environment is characterized by a paramet…
The original Hegselmann-Krause (HK) model consists of a set of~$n$ agents that are characterized by their opinion, a number in~$[0, 1]$. Each agent, say agent~$i$, updates its opinion~$x_i$ by taking the average opinion of all its neighbors, the agents whose opinion differs from~$x_i$ by at most~$ε$. There are two types of~HK models: the synchronous~HK model and the asynchronous~HK model. For the synchronous model, all the agents update their opinion simultaneously at each time step, whereas for the asynchronous~HK model, only one agent chosen uniformly at random updates its opinion at each ti…
We study the dynamics of opinion formation in the situation where changing opinion involves a cost for the agents. To do so we couple the dynamics of a heterogeneous bounded confidence Hegselmann-Krause model with that of the resources that the agents invest on each opinion change. The outcomes of the dynamics are non-trivial and strongly depend on the different regions of the confidence parameter space. In particular, a second order phase transition, for which we determine the corresponding critical exponents, is found in the region where a re-entrant consensus phase is observed in the hetero…
In this work, we investigate a heterogeneous population in the modified Hegselmann-Krause opinion model on complex networks. We introduce the Shannon information entropy about all relative opinion clusters to characterize the cluster profile in the final configuration. Independent of network structures, there exists the optimal stubbornness of one subpopulation for the largest number of clusters and the highest entropy. Besides, there is the optimal bounded confidence (or subpopulation ratio) of one subpopulation for the smallest number of clusters and the lowest entropy. However, network stru…
We introduce a geospatial bounded confidence model with mega-influencers, inspired by Hegselmann and Krause. The inclusion of geography gives rise to large-scale geospatial patterns evolving out of random initial data; that is, spatial clusters of like-minded agents emerge regardless of initialization. Mega-influencers and stochasticity amplify this effect, and soften local consensus. As an application, we consider national views on Covid-19 vaccines. For a certain set of parameters, our model yields results comparable to real survey results on vaccine hesitancy from late 2020.
Probabilistic neighbors' selection competes with confirmation bias in a bounded confidence model
CNRS, Complex Systems Institute of Paris Île-de-France (ISC-PIF)（法国国家科学研究中心，巴黎法兰西岛复杂系统研究所）；Laboratoire de Physique Théorique et Modélisation, CNRS-CY Cergy Paris Université（理论与建模物理实验室，CNRS-CY塞尔吉巴黎大学）；EHESS, Centre d’Analyse et de Mathématique Sociales (CAMS)（高等社会科学学院，社会分析与数学中心）；Cergy Paris Université（塞尔吉巴黎大学）
We perform a detailed study of the Hegselmann-Krause bounded confidence opinion dynamics model with heterogeneous confidence $\varepsilon_i$ drawn from uniform distributions in different intervals $[\varepsilon_l, \varepsilon_u]$. The phase diagram reveals a highly complex and nonmonotonous behaviour, with a re-entrant consensus phase in the region where fragmentation into multiple distinct opinions is expected for the homogeneous case. A careful exploration of the phase diagram, along with an extensive finite-size analysis, allows us to identify the mechanism leading to this counter-intuitive…
Often in the setting of human-based interactions, the existence of a temporal hierarchy of information plays an important role in diffusion and opinion dynamics within communities. For example at the individual agent level, more recently acquired information may exert greater influence during decision-making processes. To facilitate further exploration of this effect, we introduce an efficient method for modelling temporally asynchronous opinion updates, where the timing of updates depends on the timing of incoming opinion states received from neighbours. The framework enables the introduction…
Opinion dynamics models have been developed to study and predict the evolution of public opinion. Intensive research has been carried out on these models, especially exploring the different rules and topologies, which can be considered two degrees of freedom of these models. In this paper we introduce what can be considered a third degree of freedom. Since it is not possible to directly access someone's opinions without measuring them, we always need to choose a way to transform real world opinions (e.g. being anti-Trump) into numbers. However, the properties of this transformation are usually…
In this work we present novel results to the problem of the Hegselmann-Krause dynamics in networks obtained by an extensive study of the behavior of the standard order parameter sensitive to the onset of consensus: the normalized size of the giant cluster. This order parameter reveals the non trivial effect of the network topology on the steady states of the dynamics, overlooked by previous works, which concentrated on the onset of unanimity, and allows to detect regions of polarization between the fragmented and the consensus phases. While the previous results on unanimity are confirmed, the …
