Continuous-Time Opinion Dynamics with Interdependent Topics

Abstract

This chapter considers a model in which a network of individuals discuss opinions on a set of logically interdependent topics simultaneously. Thus, the work is different to the EPO model (Part I), which assumes a single topic, and DeGroot–Friedkin model (Part II), which assumes multiple topics are discussed sequentially. Also, the model is in continuous-time, which is a departure from the previous discrete-time models considered. Key to this model is a “logic matrix” which describes the logical interdependence relations between the topics. Beginning with a network where no stubborn individuals exist, a necessary and sufficient condition is obtained for guaranteeing that the network reaches a consensus of opinions for each separate topic. The discrete-time counterpart has been studied to some degree by others, but to the author’s knowledge, this chapter is the first to study a continuous-time model for logically interdependent topic discussion. Surprisingly, different types of conclusions come from continuous time as opposed to discrete time. In fact, a key finding is that in continuous time, the condition for consensus couples the logic matrix with the network topology, whereas consensus in the discrete-time model is ensured if separate conditions on the logic matrix and network topology are satisfied. Last, stubborn individuals are considered, with some sufficient conditions obtained for the opinion to converge to a configuration of disagreement.