对称群体与文化传播的阿克塞尔罗德模型
Symmetric group and the Axelrod model for dissemination of cultures
摘要：
我们认为阿克塞尔罗德提出的模型传播文化的2维平方格。我们从分析的角度来审视这个模型。我们定义$\left\langle s(t)\right\rangle$来量化社会中时间$t$的可能的文化配置。该模型的典型的初始文化配置的特点。推导了以$\left\langle s(t)\right\rangle$表示的运动方程。研究了该系统向培养构型平衡发展的曲线图。一般来说，我们观察到该模型经历了三个发展阶段。我们对这三个不同的发展阶段给出了定量的解释。与Axelrod模型保持一致，我们刻画了它在平衡点处的构型空间，其中$\left\langle s(t_{\text{eq}})\right\rangle = 1$。这个空间被称为单一文化空间。理解这一空间，就相当于把文化形态的空间限制在模型中的一个个体。该个体文化空间被标识为直到同构的空间$V_N^{\otimes F}$。考虑置换群$S_N$在空间$V_{N}^{\otimes F}$上的作用.在这种作用下，可观测的$\left\langle s(t)\right\rangle$是Axelrod系统的不变量。我们探索这种对称性，并对构成单一文化空间的不同的不对等的文化形态进行分类。为了实现这一点，我们考虑的情况下$N\geq F$。我们提出了来自群表示理论的技术来执行这种分类。培养物构型的不等价类由与贝尔数$B_F$相关的迪考图索引。一个具体的例子与$F=4$和$N\geq 4$被认为是一个完整的说明我们的分析。
英文摘要：
We consider the model proposed by Axelrod for dissemination of cultures on a 2-dimensional squared lattice. We review this model from an analytic point of view. We define $\left\langle s(t)\right\rangle$ to quantify possible culture configurations at time $t$ in a society. Typical initial culture configurations of this model are characterised. Equation of motion in terms of $\left\langle s(t)\right\rangle$ is derived. We study the graph of development of this Axelrod system toward to its culture configurations equilibrium. Generically, we observe that this model undergoes three phases of development. We give a quantitative explanation about these three different phases of development. Keeping up with this Axelrod model, we characterize its culture configurations space at equilibrium point where $\left\langle s(t_{\text{eq}})\right\rangle = 1$. This space is called monoculture space. Understanding this space is equivalent to restrict to the space of culture configurations from one individual in the model. This individual culture space is identified to the space $V_N^{\otimes F}$ up to isomorphisms. Action of the permutation group $S_N$ on the space $V_{N}^{\otimes F}$ is considered. Under this action, the observable $\left\langle s(t)\right\rangle$ is an invariant of the Axelrod system. We explore this symmetry and classify the different inequivalent classes of culture configurations composing the monoculture space. To achieve this, we consider the case $N\geq F$. We propose techniques from group representation theory to perform this classification. The inequivalent classes of culture configurations are indexed by the Dickau diagrams which are associated to the Bell number $B_F$. A concrete example with $F=4$ and $N\geq 4$ is considered for a full illustration of our analysis.
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