更改

From, e.g., the point of view of applications it is interesting to consider the case where one or more of the components of a GDS contains stochastic elements. Motivating applications could include processes that are not fully understood (e.g. dynamics within a cell) and where certain aspects for all practical purposes seem to behave according to some probability distribution. There are also applications governed by deterministic principles whose description is so complex or unwieldy that it makes sense to consider probabilistic approximations.

From, e.g., the point of view of applications it is interesting to consider the case where one or more of the components of a GDS contains stochastic elements. Motivating applications could include processes that are not fully understood (e.g. dynamics within a cell) and where certain aspects for all practical purposes seem to behave according to some probability distribution. There are also applications governed by deterministic principles whose description is so complex or unwieldy that it makes sense to consider probabilistic approximations.

例如，从应用程序的角度来看，考虑 GDS 的一个或多个组件包含随机元素的情况是有趣的。激励应用程序可以包括不完全理解的过程(例如:。细胞内部的动力学) ，以及所有实际用途的某些方面似乎都符合某些概率分布。还有一些由确定性原理控制的应用程序，它们的描述是如此复杂或笨拙，以至于考虑概率近似是有意义的。

例如，从应用程序的角度来看，考虑 '''<font color="#ff8000"> GDS</font>'''的一个或多个组件包含随机元素的情况是有趣的。激励应用程序可以包括不完全理解的过程(例如:。细胞内部的动力学) ，以及所有实际用途的某些方面似乎都符合某些概率分布。还有一些由确定性原理控制的应用程序，它们的描述是如此复杂或笨拙，以至于考虑概率近似是有意义的。

This specific example with stochastic update sequence illustrates two general facts for such systems: when passing to a stochastic graph dynamical system one is generally led to (1) a study of Markov chains (with specific structure governed by the constituents of the GDS), and (2) the resulting Markov chains tend to be large having an exponential number of states. A central goal in the study of stochastic GDS is to be able to derive reduced models.

This specific example with stochastic update sequence illustrates two general facts for such systems: when passing to a stochastic graph dynamical system one is generally led to (1) a study of Markov chains (with specific structure governed by the constituents of the GDS), and (2) the resulting Markov chains tend to be large having an exponential number of states. A central goal in the study of stochastic GDS is to be able to derive reduced models.

这个带有随机更新序列的具体例子说明了这类系统的两个一般事实: 当传递到一个'''<font color="#ff8000"> 随机图动力系统Stochastic graph dynamical system</font>'''时，一般会导致: (1)对'''<font color="#ff8000"> 马尔可夫链Markov chain </font>'''的研究(其具体结构由 GDS 的组成部分控制) ，和(2)由此产生的'''<font color="#ff8000"> 马尔可夫链Markov chain </font>'''趋向于具有指数数量的状态。随机 GDS 研究的一个中心目标是能够推导出简化模型。

这个带有随机更新序列的具体例子说明了这类系统的两个一般事实: 当传递到一个'''<font color="#ff8000"> 随机图动力系统Stochastic graph dynamical system</font>'''时，一般会导致: (1)对'''<font color="#ff8000"> 马尔可夫链Markov chain </font>'''的研究(其具体结构由 '''<font color="#ff8000"> GDS</font>''' 的组成部分控制) ，和(2)由此产生的'''<font color="#ff8000"> 马尔可夫链Markov chain </font>'''趋向于具有指数数量的状态。随机 '''<font color="#ff8000"> GDS</font>'''  研究的一个中心目标是能够推导出简化模型。

One may also consider the case where the vertex functions are stochastic, i.e., ''function stochastic GDS''. For example, Random [[Boolean network]]s are examples of function stochastic GDS using a synchronous update scheme and where the state space is ''K'' = {0,&nbsp;1}. Finite [[probabilistic cellular automata]] (PCA) is another example of function stochastic GDS. In principle the class of Interacting particle systems (IPS) covers finite and infinite [[probabilistic cellular automata|PCA]], but in practice the work on IPS is largely concerned with the infinite case since this allows one to introduce more interesting topologies on state space.

One may also consider the case where the vertex functions are stochastic, i.e., ''function stochastic GDS''. For example, Random [[Boolean network]]s are examples of function stochastic GDS using a synchronous update scheme and where the state space is ''K'' = {0,&nbsp;1}. Finite [[probabilistic cellular automata]] (PCA) is another example of function stochastic GDS. In principle the class of Interacting particle systems (IPS) covers finite and infinite [[probabilistic cellular automata|PCA]], but in practice the work on IPS is largely concerned with the infinite case since this allows one to introduce more interesting topologies on state space.

One may also consider the case where the vertex functions are stochastic, i.e., function stochastic GDS. For example, Random Boolean networks are examples of function stochastic GDS using a synchronous update scheme and where the state space is K = {0,&nbsp;1}. Finite probabilistic cellular automata (PCA) is another example of function stochastic GDS. In principle the class of Interacting particle systems (IPS) covers finite and infinite PCA, but in practice the work on IPS is largely concerned with the infinite case since this allows one to introduce more interesting topologies on state space.

One may also consider the case where the vertex functions are stochastic, i.e., function stochastic GDS. For example, Random Boolean networks are examples of function stochastic '''<font color="#ff8000"> GDS</font>'''  using a synchronous update scheme and where the state space is K = {0,&nbsp;1}. Finite probabilistic cellular automata (PCA) is another example of function stochastic GDS. In principle the class of Interacting particle systems (IPS) covers finite and infinite PCA, but in practice the work on IPS is largely concerned with the infinite case since this allows one to introduce more interesting topologies on state space.

我们也可以考虑顶点函数是随机的情况，即函数是随机的 GDS。例如，随机布尔网络是函数随机 GDS 使用同步更新方案的例子，其中状态空间为 k {0,1}。有限概率'''<font color="#ff8000"> 细胞自动机（PCA)</font>'''是功能随机 GDS 的另一个例子。原则上，相互作用粒子系统(IPS)包括有限和无限的 PCA，但实际上，IPS 的工作主要是关注无限的情况，因为这允许在状态空间中引入更多有趣的'''<font color="#ff8000">拓扑Topologies </font>'''。

我们也可以考虑顶点函数是随机的情况，即函数是随机的 GDS。例如，随机布尔网络是函数随机 GDS 使用同步更新方案的例子，其中状态空间为 k {0,1}。有限概率'''<font color="#ff8000"> 细胞自动机（PCA)</font>'''是功能随机 GDS 的另一个例子。原则上，相互作用粒子系统(IPS)包括有限和无限的 PCA，但实际上，IPS 的工作主要是关注无限的情况，因为这允许在状态空间中引入更多有趣的'''<font color="#ff8000">拓扑Topologies </font>'''。