Finite-time convergence of opinion dynamics in homogeneous asymmetric bounded confidence models☆
Introduction
Opinion dynamics has been widely used to analyze the temporal evolution of the behaviors and attitudes of groups of agents in social networks [18]. The agents are modeled as nodes of a graph, and the state of each agent represents her opinion which can be interpreted as the intensity of a cognitive orientation (attitude) toward a particular goal or issue. In this class of models, each agent updates her opinion by weighting the opinions of the agents who are selected as her neighbors [1], [9], i.e., her adjacent nodes in the graph. The dependence on the opinion values for the neighbors’ selection allows one to model the homophily concept, which is defined as the tendency of individuals to bond with their similar [14].
In the Hegselmann–Krause (HK) model, at each time-step the neighbors of each agent are those who have an opinion close to her own up to some confidence thresholds [10], [20]. This rule implies a dynamic topology of the graph. The characteristics of the confidence thresholds allow a classification for different types of the HK model: the model is said to be homogeneous when the same confidence thresholds characterize all agents, and heterogeneous otherwise; the model is said to be symmetric if the same confidence thresholds are used to select the neighbors with lower and upper opinions, and asymmetric otherwise [10]. In this paper, we consider discrete-time homogeneous asymmetric HK models.
Complex behaviors are exhibited by HK models and several issues arise in their formal analysis [15]. A relevant problem analyzed in opinion dynamics focuses on the convergence of the opinions to a steady state, which can be a consensus, if a common value is reached, or a clustering, if the opinions split into groups achieving consensus within a group but with different values for different groups. The fact that HK models typically exhibit a clustering behavior reflects well the homophily principle, according to which individuals tend to get closer to other individuals that have similar opinions, and to get away from those with opinion that contradicts their own [1].
For the symmetric case, the convergence to a steady state of a homogeneous HK model has been theoretically analyzed in many studies, see among others [5], [8], [11], [12]. For the same class of models, an upper bound on the convergence time, which has a polynomial dependence on the number of agents, has been provided in [4], [16], and numerical studies have shown that the number of clusters depends on the confidence intervals [5], [13]. For the asymmetric case, numerical analyses have shown that even the homogeneous HK model can exhibit behaviors not directly deducible from the symmetric class [2], [10]. On the other hand, the properties of homogeneous HK models cannot be easily derived by particularizing the (few) theoretical results available for the heterogeneous case. The convergence to a consensus has been analyzed for heterogeneous HK models in the presence of group pressure [7] and with confidence thresholds fixed for each pair of nodes [6], but the results presented therein are valid only for the symmetric case. The model considered in [17] assumes uniform weights but restricts the ability of certain agents to communicate with each other, unlike [6]. The theoretical findings in [3], [21] depend on the specific rules chosen for the adaptation of the confidence thresholds. In [4] a particular structure of opinion dynamics, which includes the class of homogeneous asymmetric HK models considered in this paper, was presented. In the model of [4], the agents have the same upper confidence thresholds but different lower confidence thresholds (or viceversa).
For the asymmetric homogeneous HK model, in this paper we present conditions for the convergence of the opinions to a constant steady state, which can be a consensus or a clustering, and a corresponding upper bound which improves the one obtained in [4]. By assuming equal confidence thresholds, the results presented in this paper can be directly applied to symmetric HK models too.
The rest of the paper is organized as follows. Section 2 presents the discrete-time HK model, useful definitions and some preliminary propositions. Section 3 contains new results on the convergence to a steady state for the asymmetric homogeneous HK models. Section 4 concludes the paper by tracing a direction for future research.
Section snippets
Bounded confidence opinion dynamics
The HK model considered in this paper belongs to a more general class of discrete-time opinion dynamics.
Consider a set of agents whose opinions are represented through scalar state variables , , where is the set of positive natural numbers and is the set of real numbers. A typical opinion dynamics assumes the update of the opinion of each agent obtained as the average of the opinions of her neighbors; this class of models can be described by the following set of
Convergence results
In order to prove some convergence properties of the asymmetric HK model, it is useful to define the minimum and the maximum of the confidence thresholds, i.e., and , respectively. In the following we exclude the trivial case .
It easy to verify that the asymmetric HK model satisfies some properties typical of the symmetric case: the range of opinions is nonincreasing, the opinion at the next time-step is bounded by the opinions of the neighbors, the order of the
Conclusions
For the discrete-time homogeneous asymmetric HK models considered in this paper, three main results, novel for the asymmetric case, have been obtained: i) the convergence to a constant steady state is always achieved and in finite time if the upper and lower confidence thresholds are both nonzero; ii) the agents reach a consensus if and only if the connectivity of the graph is maintained over time; iii) the convergence time has an upper bound which depends on the number of agents, on the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper
References (22)
Mathematical models in social psychology
Advances in Experimental Social Psychology
(1967)- et al.
A mixed logical dynamical model of the Hegselmann–Krause opinion dynamics
IFAC-PapersOnLine
(2020) Consensus formation under bounded confidence
Nonlinear Analysis: Theory, Methods & Applications
(2001)The problem of social control and coordination of complex systems in sociology
IEEE Control Systems Magazine
(2015)Consensus strikes back in the Hegselmann–Krause model of continuous opinion dynamics under bounded confidence
Journal of Artificial Societies and Social Simulation
(2006)- et al.
Consensus stability in the Hegselmann–Krause model with coopetition and cooperosity
IFAC-PapersOnLine
(2017) - et al.
Practical consensus in bounded confidence opinion dynamics
Automatica
(2021) - et al.
Knowledge sharing, heterophily, and social network dynamics
The Journal of Mathematical Sociology
(2020) - et al.
Heterogeneous opinion dynamics with confidence thresholds adaptation
IEEE Transactions on Control of Network Systems
(2021) - et al.
On the convergence of the Hegselmann–Krause system
Proceedings of the 4th Conference on Innovations in Theoretical Computer Science
(2013)
On Krause’s multi-agent consensus model with state-dependent connectivity
IEEE Transactions on Automatic Control
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