THE AXELROD MODEL FOR THE DISSEMINATION OF CULTURE REVISITED
THE AXELROD MODEL FOR THE DISSEMINATION OF CULTURE REVISITED
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DOI：
10.1214/11-aap790
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发表时间：
2012-04-01
影响因子：
1.8
通讯作者：
Lanchier, Nicolas
中科院分区：
文献类型：
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作者：
Lanchier, Nicolas
本文关注的是Axelrod模型，这是一个随机过程，与选民模型类似，包括社会影响，但与选民模型不同的是，它也考虑到同质性。交互网络的每个顶点由一组F个文化特征表征，每个文化特征可以假设q个状态。相邻的顶点对以与它们共享的特征数量成比例的速率相互作用，这导致相互作用的顶点对有一个共同的文化特征。在过去的十年中，Axelrod模型得到了广泛的研究，基于数值模拟和简单的平均场处理，而空间模型本身的分析结果却完全缺乏。一维系统的模拟结果使物理学家提出了以下猜想。当特征的数量F和状态的数量q都等于2时，或者当特征的数量超过状态的数量时，系统收敛于单一文化平衡，也就是说，随着人口趋于无穷大，由人口规模重新封闭的文化领域的数量收敛于零。相反，当状态的数量超过特征的数量时，系统就会冻结在一个高度碎片化的结构中，在这个结构中，文化领域的最终数量会像人口规模一样扩大。在本文中，我们解析地证明了当F = q = 2时，一维系统收敛到单文化平衡的聚类，以及当状态数量足够大于特征数量时，对高度碎片化配置的固定。我们的第一个结果也暗示了一维约束选民模型的聚类。
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 resealed 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.