Opinion Dynamics and the Evolution of Social Power in Social Networks

Abstract

A fundamental aspect of society is the exchange and discussion of opinions between individuals, occurring in mediums and situations as varied as company boardrooms, elementary school classrooms and online social media. This thesis studies several mathematical models of how an individual’s opinion(s) evolves via interaction with others in a social network, developed to reflect and capture different socio-psychological processes that occur during the interactions.

In the first part, and inspired by Solomon E. Asch’s seminal experiments on conformity, a novel discrete-time model of opinion dynamics is proposed, with each individual having both an expressed and a private opinion on the same topic. Crucially, an individual’s expressed opinion is altered from the individual’s private opinion due to pressures to conform to the majority opinion of the social network. Exponential convergence of the opinion dynamical system to a unique configuration is established for general networks. Several conclusions are established, including how differences between an individual’s expressed and private opinions arise, and how to estimate disagreement among the private opinions at equilibrium. Asch’s experiments are revisited and re-examined, and then it is shown that a few extremists can create “pluralistic ignorance”, where people believe there is majority support for a position but in fact the position is privately rejected by the majority of individuals!

The second part builds on the recently proposed discrete-time DeGroot–Friedkin model, which describes the evolution of an individual’s self-confidence (termed social power) in his/her opinion over the discussion of a sequence of issues. Using nonlinear contraction analysis, exponential convergence to a unique equilibrium is established for networks with constant topology. Networks with issue-varying topology (which remain constant for any given issue) are then studied; exponential convergence to a unique limiting trajectory is established. In a social context, this means that each individual forgets his/her initial social power exponentially fast; in the limit, his/her social power for a given issue depends only on the previously occurring sequence of dynamic topology. Two further related works are considered; a network modification problem, and a different convergence proof based on Lefschetz Fixed Point Theory.

In the final part, a continuous-time model is proposed to capture simultaneous discussion of logically interdependent topics; the interdependence is captured by a “logic matrix”. When no individual remains attached to his/her initial opinion, a necessary and sufficient condition for the network to reach a consensus of opinions is provided. This condition depends on the interplay between the network interactions and the logic matrix; if the network interactions are too strong when compared to the logical couplings, instability can result. Last, when some individuals remain attached to their initial opinions, sufficient conditions are given for opinions to converge to a state of persistent disagreement. Show more