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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 '''<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。

例如，从应用程序的角度来看，思考 '''<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) ，以及所有实际应用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，因为它们的描述要么复杂要么笨拙，所以考虑概率近似是有意义的。

Every element of a graph dynamical system can be made stochastic in several ways. For example, in a sequential dynamical system the update sequence can be made stochastic. At each iteration step one may choose the update sequence w at random from a given distribution of update sequences with corresponding probabilities. The matching probability space of update sequences induces a probability space of SDS maps. A natural object to study in this regard is the Markov chain on state space induced by this collection of SDS maps. This case is referred to as update sequence stochastic GDS and is motivated by, e.g., processes where "events" occur at random according to certain rates (e.g. chemical reactions), synchronization in parallel computation/discrete event simulations, and in computational paradigms described later<!-- Make sure this cross ref stays/works. -->.

Every element of a graph dynamical system can be made stochastic in several ways. For example, in a sequential dynamical system the update sequence can be made stochastic. At each iteration step one may choose the update sequence w at random from a given distribution of update sequences with corresponding probabilities. The matching probability space of update sequences induces a probability space of SDS maps. A natural object to study in this regard is the Markov chain on state space induced by this collection of SDS maps. This case is referred to as update sequence stochastic GDS and is motivated by, e.g., processes where "events" occur at random according to certain rates (e.g. chemical reactions), synchronization in parallel computation/discrete event simulations, and in computational paradigms described later<!-- Make sure this cross ref stays/works. -->.

'''<font color="#ff8000"> 图动力系统graph dynamical system</font>'''的每个元素都可以通过几种方式随机化。例如，在顺序动力系统中，更新序列可以是随机的。在每个迭代步骤中，可以从给定的更新序列分布中随机选择具有相应概率的更新序列 w。更新序列的匹配概率空间引出 SDS 地图的概率空间。在这方面需要研究的一个自然对象是 SDS 映射集合在状态空间上产生的'''<font color="#ff8000"> 马尔可夫链Markov chain </font>'''。这种情况被称为更新序列随机 GDS，其动机是，例如，“事件”按照一定的速率随机发生的过程(例如:。化学反应) ，在并行计算 / 离散事件模拟中的同步，以及在后面描述的计算范例中的同步! ——确保这个交叉引用保持 / 工作。-->.

'''<font color="#ff8000"> 图动力系统graph dynamical system</font>'''的每个元素都可以通过几种方式随机化。例如，在顺序动力系统中，新序列可以是随机的。在每个迭代步骤中，可以从给定的新序列分布中随机选择具有相应概率的更新序列 w。新序列的匹配概率空间引出 SDS 地图的概率空间。在这方面需要研究的一个自然对象是 SDS 映射集合在状态空间上产生的'''<font color="#ff8000"> 马尔可夫链Markov chain </font>'''。这种情况被称为新序列随机 GDS，其目的如，“事件”按照一定的速率随机发生的过程(如化学反应)，在并行计算 / 离散事件模拟中的同步，以及在后面描述的计算范例中的同步! ——确保这个交叉引用保持 / 工作。-->.