Voter models on deterministic and random graphs

After a brief introduction to the structure of the voter model (in several incarnations), we discuss its duality with a suitable coalescing random walk (following the work of Aldous). The classic voter model on the complete graph is then studied in detail: an explicit expression for the expected consensus time is proved in the case where initial opinions are all different; in the case with only two initial opinions, the probability that each ends up as the winner is computed. We then look at the more general case of a voter model on an initially sampled inhomogeneous random graph, as treated in a recent article by Fernley and Ortgiese. In particular, after introducing the SNR random graph model together with some related structural results, asymptotic expressions for the expected consensus time of the voter model on inhomogeneous random graphs are proved.