Asymptotic Analysis of Opinion Dynamic Models

Abstract

Opinion dynamics is the study of the evolution of opinions through interactions among a group of people referred to as agents. With the development of technology and growth of social networking among individuals, the quantitative study of social dynamics has attracted many researchers from diverse disciplines. Study of social psychology suggests that the opinion of an agent is a function of environmental influences and his/her personality. In this thesis, we analyze opinion dynamic models for the long term behavior of opinions in both deterministic and stochastic settings. Deterministic opinion dynamic models are based on the interaction policies between agents. These interaction policies depend on the opinions of interacting agents and their confidence bounds. Agent's opinions can be considered as d-dimensional vectors and confidence bounds as scalar values. Considering real life examples of interpersonal relations leads to the observation that not everyone trusts everyone else. This brings the idea of bounded confidence (BC) into the modeling of opinion dynamics. The BC models suggest that an agent will only be influenced by those whose opinions are closer to his/her own. We study a well known BC model, the Hegselmann-Krause (HK) model, for continuous time and multidimensional opinions where agents have heterogeneous but symmetric confidence bounds. We prove that all trajectories approach an equilibrium as time t \to \infty. We study the model and investigate two forms of stability of equilibria. We prove the Lyapunov stability of all equilibria in the relative interior of the set of equilibria. Considering a particular interaction function, we provide a necessary condition and a sufficient condition for a form of structural stability of the equilibria. This structural stability is a notion of stability where a new agent with an arbitrarily small weight is introduced to a system in equilibrium. We also study a stochastic binary opinion dynamics model where agents are considered to hold an opinion of "yes" or "no" at each moment. In this model the number of agents with opinion "yes" at time t is considered to be a birth-death process, X^N(t), and the configuration of the opinions at each moment is determined by a probability distribution. We consider that the rate an agent changes his/her opinion depends on the coefficients of self-motivation, conformity and a rate function. We give asymptotic (in the sense of large population size) analysis of the stationary probabilities for continuously differentiable (C^1) rate functions. We present computational results using various examples of the rate functions. We also analyze the stationary probabilities for a particular discontinuous rate function. We use Laplace's method and obtain approximations to the stationary probabilities when the number of agents is very large. We present computational results supporting the obtained asymptotic approximations.