具有不完全复制的多状态选民模型
A multi-state voter model with imperfect copying
摘要：
具有多个状态的选民模型已经在人口遗传学、意见形成、物种竞争和语言动态等领域得到了应用。在动力学的单个步骤中，随机选择的个体复制了群体中随机邻居的状态。在这个基本公式中，假设复制是完美的，因此在每个时间步都会生成一个个体的精确副本。在这里，我们介绍和研究一个变种的多状态选民模型在平均场，在复制过程中，它包含一定程度的不完美或错误，这使得两个相互作用的个人的状态相似，但不完全相等。这种动态也可以被解释为一个完美的复制加上噪音;一个极简主义的模型群集。我们发现，这个多状态噪声选民模型的有序特性，测量参数$ψ$在[0，1]，依赖于复制错误或噪声的幅度$η$和人口规模N。在完全复制$η=0$的情况下，对于所有的N值，系统达到具有完全阶（$ψ=1$）的吸收构型。然而，对于任何程度的不完善$η>0$，我们表明，在稳态的平均值$ψ$减少与N为$\langle ψ\rangle \simeq 6/(π^2 η^2 N)$的$η\ll 1$和$η^2 N \gtrsim 1$，因此该系统成为完全无序的热力学极限$N \to \infty$。我们还表明，$\langle ψ\rangle \simeq 1-1.64 \, η^2 N$在消失的小误差限制$η\to 0$，这意味着完整的顺序是从来没有实现$η> 0$。这些结果得到了该模型的蒙特卡罗模拟的支持，该模拟也允许研究其他场景。
英文摘要：
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 multi-state 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 multi-state 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 $\langle ψ\rangle \simeq 6/(π^2 η^2 N)$ for $η\ll 1$ and $η^2 N \gtrsim 1$, and thus the system becomes totally disordered in the thermodynamic limit $N \to \infty$. We also show that $\langle ψ\rangle \simeq 1-1.64 \, η^2 N$ in the vanishing small error limit $η\to 0$, 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.
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