Instituto de Cálculo, FCEN, Universidad de Buenos Aires and CONICET, Buenos Aires, Argentina.
Instituto de Física de Líquidos y Sistemas Biológicos (IFLYSIB), UNLP, CCT La Plata-CONICET, Calle 59 no. 789, B1900BTE La Plata, Argentina.
Phys Rev E. 2019 Oct;100(4-1):042301. doi: 10.1103/PhysRevE.100.042301.
The voter model with multiple states has found applications in areas as diverse as population genetics, opinion formation, species competition, and language dynamics, among others. In a single step of the dynamics, an individual chosen at random copies the state of a random neighbor in the population. In this basic formulation, it is assumed that the copying is perfect, and thus an exact copy of an individual is generated at each time step. Here, we introduce and study a variant of the multistate voter model in mean field that incorporates a degree of imperfection or error in the copying process, which leaves the states of the two interacting individuals similar but not exactly equal. This dynamics can also be interpreted as a perfect copying with the addition of noise: a minimalistic model for flocking. We found that the ordering properties of this multistate noisy voter model, measured by a parameter ψ in [0,1], depend on the amplitude η of the copying error or noise and the population size N. In the case of perfect copying η=0, the system reaches an absorbing configuration with complete order (ψ=1) for all values of N. However, for any degree of imperfection η>0, we show that the average value of ψ at the stationary state decreases with N as 〈ψ〉≃6/(π^{2}η^{2}N) for η≪1 and η^{2}N≳1, and thus the system becomes totally disordered in the thermodynamic limit N→∞. We also show that 〈ψ〉≃1-π^{2}/6η^{2}N in the vanishing small error limit η→0, which implies that complete order is never achieved for η>0. These results are supported by Monte Carlo simulations of the model, which allow to study other scenarios as well.
多态投票模型在人口遗传学、观点形成、物种竞争和语言动态等多个领域都有应用。在动力学的单个步骤中，随机选择的个体复制群体中随机邻居的状态。在这种基本形式中，假设复制是完美的，因此在每个时间步中都会生成个体的精确副本。在这里，我们在平均场中引入并研究了多态投票模型的一个变体，该变体在复制过程中存在一定程度的不完美或错误，使得两个相互作用的个体的状态相似但不完全相同。这种动力学也可以被解释为带有噪声的完美复制：一种用于群体行为的简约模型。我们发现，通过参数 ψ 在 [0,1] 内测量，这种多态噪声投票模型的排序性质取决于复制错误或噪声的幅度 η 和群体大小 N。在完美复制的情况下 η=0，系统对于所有 N 值都达到具有完全秩序的吸收配置（ψ=1）。然而，对于任何程度的不完美 η>0，我们表明，在稳定状态下 ψ 的平均值随着 N 的减小而减小，对于 η≪1 和 η^{2}N≳1，有〈ψ〉≃6/(π^{2}η^{2}N)，因此系统在热力学极限 N→∞时变得完全无序。我们还表明，在 η→0 的微小误差极限中，〈ψ〉≃1-π^{2}/6η^{2}N，这意味着对于 η>0，永远无法达到完全秩序。这些结果得到了模型的蒙特卡罗模拟的支持，这些模拟也允许研究其他场景。
