From Microscopic Heterogeneity to Macroscopic Complexity in the Contrarian Voter Model

An analytical treatment of a simple opinion model with contrarian behavior is presented. The focus is on the stationary dynamics of the model and in particular on the effect of inhomogeneities in the interaction topology on the stationary behavior. We start from a micro-level Markov chain description of the model. Markov chain aggregation is then used to derive a macro chain for the complete graph as well as a meso-level description for the two-community graph composed of two (weakly) coupled sub-communities. In both cases, a detailed understanding of the model behavior is possible using Markov chain tools. More importantly, however, this setting provides an analytical scenario to study the discrepancy between the homogeneous mixing case and the model on a slightly more complex topology. We show that memory effects are introduced at the macro level when we aggregate over agent attributes without sensitivity to the microscopic details and quantify these effects using concepts from information theory. In this way, the method facilitates the analysis of the relation between microscopic processes and a their aggregation to a macroscopic level of description and informs about the complexity of a system introduced by heterogeneous interaction relations.