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 $F$ cultural features, each of which can assume
$q$ 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
during 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 itself. Simulation results for the one-dimensional system led physicists
to formulate the following conjectures. When the number of features $F$ and the
number of states $q$ both equal two, or when the number of features exceeds the
number of states, the system converges to a monocultural equilibrium in the
sense that the number of cultural domains rescaled by the population size
converges to zero as the population goes to infinity. In contrast, when the
number of states exceeds the number of features, the system freezes in a highly
fragmented configuration in which the ultimate number of cultural domains
scales like the population size. In this article, we prove analytically for the
one-dimensional system convergence to a monocultural equilibrium in terms of
clustering when $F=q=2$, as well as fixation to a highly fragmented
configuration when the number of states is sufficiently larger than the number
of features. Our first result also implies clustering of the one-dimensional
constrained voter model.

本文研究了 Axelrod 模型，证明了当文化特征数和状态数为 2 时，该系统会收敛于单一文化均衡；当状态数大于特征数时，该系统会被冻结于高度碎片化的配置。