On a Modified DeGroot-Friedkin model of opinion dynamics

Abstract

Abstract-This paper studies the opinion dynamics that result when individuals consecutively discuss a sequence of issues. Specifically, we study how individuals' self-confidence levels evolve via a reflected appraisal mechanism. Motivated by the DeGroot-Friedkin model, we propose a Modified DeGrootFriedkin model which allows individuals to update their selfconfidence levels by only interacting with their neighbors and in particular, the modified model allows the update of selfconfidence levels to take place in finite time without waiting for the opinion process to reach a consensus on any particular issue. We study properties of this Modified DeGroot-Friedkin model and compare the associated equilibria and stability with those of the original DeGroot-Friedkin model. Specifically, for the case when the interaction matrix is doubly stochastic, we show that for the modified model, the vector of individuals' selfconfidence levels converges to a unique nontrivial equilibrium which for each individual is equal to 1 n , where n is the number of individuals. This implies that eventually individuals reach a democratic state. I. INTRODUCTIONOver the years, the advancement in information technology has enabled individuals to be more closely connected and the rapid expansion of online social networks has provided huge amount of data available for analysis regarding how individuals interact over networks. Consequently, much research attention has been drawn to understand how an individual's opinion evolves over time, in particular, how to model the underlying process of opinion formation [1], [2], [3]. There has been increasing interest in developing models of opinion dynamics to capture individuals' interaction, and some of these as relevant to the topic of this paper will be mentioned later. In the literature, two main approaches have been adopted on how each individual updates her opinion:In social sciences, quite a few models have been proposed for opinion dynamics. Notable among them are the three classical models, namely, the DeGroot model [1], the Friedkin-Johnsen model [2], and the Krause model [3]. In the DeGroot model, each individual has a fixed set of neighbors and the local interaction is captured by taking the convex combination of her own opinion and the opinions of her neighbors at each time step. The model can be extended naturally to the case in which the neighbor sets change over time. The Friedkin-Johnsen model is a variation of the DeGroot model in which each individual is assumed to adhere to her initial opinion to a certain degree, which brings

Stability Theory of Stochastic Models in Opinion Dynamics

“…). (23), γ = 4, and C given by (26), the unique fixed point p = (0.4173, 0.1537, 0.4298) T is unstable and there is an attractive 2-periodic orbit between p a and p b , verified by the time history (inset graph).…”

On convergence rate of a continuous-time distributed self-appraisal model with time-varying relative interaction matrices

“…Remark 1: System (1) with a fixed relative interaction (i.e., c ji (t) ≡ c ji for all time t) was proposed and studied in [20]. The system can be viewed as a continuous-time version of the modified DeGroot-Friedkin model studied in [10], [17].…”

“…Probably the most well-known model for opinion dynamics is the classical DeGroot model [1]. Various models have been proposed for opinion dynamics to understand how an individual's opinion evolves over time, including the Friedkin-Johnsen model [2], [3], the Hegselmann-Krause model [4]- [6], the DeGroot-Friedkin model [7]- [10], and the Altafini model [11]- [15]. However, there is few work concerning the self-confidence levels of the individuals in a social network.…”

“…Lately, a modified DeGroot-Friedkin model has been proposed in [10] which provides a time-efficient, distributed implementation of the original DeGroot-Friedkin model. The model has been studied in [10] with a fixed doubly stochastic relative interaction matrix, and in [17] with time-varying doubly stochastic relative interaction matrices, while the analysis of the modified DeGroot-Friedkin model with a fixed stochastic (not doubly stochastic) relative interaction matrix still remains open.…”

Evolution of social power for opinion dynamics networks

“…Based on DeGroot's model [1] combined with Friedkin's [22], in [25,23] they constructed the DeGroot-Friedkin model of opinion dynamics and influence networks. Moreover, [27] proposed the modified DeGroot-Friedkin model of opinion dynamics in order to allow the update of self-appraisal levels in finite time. Last works include the extension of Friedkin-Johnsen's [3] model to the evolution of social power [28] and the study of the coevolution of appraisal and influence networks [24].…”