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→Stochastic graph dynamical systems随机图动力系统
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马尔可夫链。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，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。-->.
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)对马尔可夫链的研究(其具体结构由 GDS 的组成部分控制) ，和(2)由此产生的马尔可夫链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 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。
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, 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 GDS using a synchronous update scheme and where the state space is K = {0, 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拓扑。
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>'''。
==Applications应用==
==Applications应用==