Flux and fixation for the voter model and the Axelrod model

Nicolas Lanchier (June 24, 2012)

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As a warming up, we will start with a brief overview of the main results about the voter model: clustering versus coexistence, cluster size and occupation time. The voter model is an example of interacting particle system - individual-based model - that models social influence, the tendency of individuals to become more similar when they interact. Each vertex of the lattice is characterized by one of two possible competing opinions and updates its state at rate one by mimicking one of its neighbors chosen uniformly at random. We will conclude with recent results about the one-dimensional Axelrod model which, like the voter model includes social influence, but unlike the voter model also accounts for homophily, the tendency of individuals to interact more frequently with individuals who are more similar. In the Axelrod model, each vertex of the lattice is now characterized by a culture, a vector of F cultural features that can each assumes q different states. Pairs of neighbors interact at a rate proportional to the number of cultural features they have in common, which results in the interacting pair having one more cultural feature in common.