The Axelrod model is a spatial stochastic model for the dynamics of cultures
which includes two important social factors: social influence, the tendency of
individuals to become more similar when they interact, and homophily, the
tendency of individuals to interact more frequently with individuals who are
more similar. Each vertex of the interaction network is characterized by its
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. In this article, we continue the
analysis of the Axelrod model initiated by the first author by proving that the
one-dimensional system fixates when $F\leq cq$ where the slope satisfies the
equation $e^{-c}=c$. In addition, we show that the two-feature model with at
least three states fixates. This last result is sharp since it is known from
previous works that the one-dimensional two-feature two-state Axelrod model
clusters.Comment: Published in at http://dx.doi.org/10.1214/12-AAP910 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
This article is concerned with a general class of stochastic spatial models for the dynamics of opinions. Like in the voter model, individuals are located on the vertex set of a connected graph and update their opinion at a constant rate based on the opinion of their neighbors. However, unlike in the voter model, the set of opinions is represented by the set of vertices of another connected graph that we call the opinion graph: when an individual interacts with a neighbor, she imitates this neighbor if and only if the distance between their opinions, defined as the graph distance induced by the opinion graph, does not exceed a certain confidence threshold. When the confidence threshold is at least equal to the radius of the opinion graph, we prove that the one-dimensional process fluctuates and clusters and give a universal lower bound for the probability of consensus of the process on finite connected graphs. We also establish a general sufficient condition for fixation of the infinite system based on the structure of the opinion graph, which we then significantly improve for opinion graphs which are distance-regular. Our general results are used to understand the dynamics of the system for various examples of opinion graphs: paths and stars, which are not distance-regular, and cycles, hypercubes and the five Platonic solids, which are distance-regular.
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