The Axelrod model for the dissemination of culture revisited

Abstract

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.Comment: Published in at http://dx.doi.org/10.1214/11-AAP790 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Cultural diversity dynamics

“…Equilibrium is reached when adjacent cultural groups have no traits in common and therefore interaction is not possible anymore. However, as has been put forward by other authors (for instance [9]) and we have verified experimentally ourselves, this result is a combined side-effect of the model assumptions of 4-connectivity and fewer features than traits per feature. If instead, more features are used with fewer traits each, all systems eventually lead to monoculture.…”

Ecosystems perspective on financial networks: Diagnostic tools

“…The changes in the investment return expectations in RexðtÞ of Investors is modeled through a modified version of the Axelrod model for the dissemination of culture [17,18]. This is applied in the model using a stochastic process in which every Investor earning returns below the benchmark has a probability of increasing in RexðtÞ toward the benchmark.…”

Constraints and effects of partner selection on the emergence of structures in social environments

“…We are using as basis an Axelrod model of cultural imitation [3][4][5] [6], where an agent is characterized by a number of cultural features, each one of which may take one of several values, called traits, and where an initially fully heterogeneous population (with random traits) may lead eventually to compact cultural groups that are homogeneous internally but with large differences between them, provided that a simple imitation process takes place: an agent may imitate (copy) a trait of a neighbor probabilistically based on the affinity with him (the proportion of traits that coincide). We have modified and extended this model with a Moore neighborhood, heterogeneous sets of cultural features per agent and a number of psychologically realistic, basic and more advanced, conceptual models of cultural affinity perception and imitation [7].…”

“…Our models have been inspired by the observation (also put forward by other authors, for instance [6]) that the original Axelrod result, where cultural grouping and polarization emerges in an initially diverse society, is a combined sideeffect of the model assumptions of 4-connectivity and fewer features than traits per feature. If instead, more features are used with fewer traits each, all systems eventually lead to monoculture.…”