Opinion Dynamics on Discourse Sheaves

We introduce a novel class of network structures called discourse sheaves, together with generalizations of graph Laplacians and diffusion dynamics, in order to model dynamics on networks with application to opinion dynamics on social networks. Such sheaves are algebraic structures tethered to a network that can represent various modes of communication, including selective opinion expression and lying. After introducing the discourse sheaf model for communication in a social network, we develop a sheaf Laplacian in this context and show how to evolve both opinions and communications with diffusion dynamics over the network. Issues of controllability, reachability, bounded confidence, and harmonic extension are addressed using this framework, providing a rich class of analyzable examples.