Mathematics – Probability
Scientific paper
2010-04-02
Mathematics
Probability
17 pages, 3 figures, 1 table
Scientific paper
This article is concerned with the Axelrod model, a stochastic process which similarly to the voter model includes social influence, but unlike the voter model also accounts for homophily. Each vertex of the network of interactions is characterized by a set of cultural features, each of which can assume a fixed number of states. Pairs of adjacent vertices interact at a rate proportional to the number of features they share, which results in the interacting pair having one more cultural feature in common. The Axelrod model has been extensively studied the past ten years based on numerical simulations and simple mean-field treatments while there is a total lack of analytical results for the spatial model. Numerical results emerging from spatial simulations of the one-dimensional system are difficult to interpret but suggest that (i) when the number of cultural features and the number of states per feature both equal two or (ii) when the number of features exceeds the number of states per feature, the system converges to a monocultural equilibrium in the sense that a single culture ultimately occupies a macroscopically large fraction of the graph while (iii) when the number of states per feature exceeds the number of features, the system freezes in a highly fragmented configuration. In this article, we prove analytically conjecture (i) in terms of a clustering of the infinite system, and part of conjecture (iii). Our first result also applies to the constrained voter model.
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