Markov Chain Aggregation for Simple Agent-Based Models on Symmetric Networks: The Voter Model

For Agent Based Models, in particular the Voter Model (VM), a general framework of aggregation is developed which exploits the symmetries of the agent network $G$. Depending on the symmetry group $Aut_{\omega} (N)$ of the weighted agent network, certain ensembles of agent configurations can be interchanged without affecting the dynamical properties of the VM. These configurations can be aggregated into the same macro state and the dynamical process projected onto these states is, contrary to the general case, still a Markov chain. 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. In some cases the macro chain is solvable.