Mathematical models of cellular physiological mechanisms often involve random walks on graphs representing transitions within networks of functional states. Schmandt and Galán recently introduced a novel stochastic shielding approximation as a fast, accurate method for generating approximate sample paths from a finite state Markov process in which only a subset of states are observable. For example, in ion-channel models, such as the Hodgkin-Huxley or other conductance-based neural models, a nerve cell has a population of ion channels whose states comprise the nodes of a graph, only some of which allow a transmembrane current to pass. The stochastic shielding approximation consists of neglecting fluctuations in the dynamics associated with edges in the graph not directly affecting the observable states. We consider the problem of finding the optimal complexity reducing mapping from a stochastic process on a graph to an approximate process on a smaller sample space, as determined by the choice of a particular linear measurement functional on the graph. The partitioning of ion-channel states into conducting versus nonconducting states provides a case in point. In addition to establishing that Schmandt and Galán's approximation is in fact optimal in a specific sense, we use recent results from random matrix theory to provide heuristic error estimates for the accuracy of the stochastic shielding approximation for an ensemble of random graphs. Moreover, we provide a novel quantitative measure of the contribution of individual transitions within the reaction graph to the accuracy of the approximate process.

中文翻译：

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测量边缘重要性：图上随机过程的随机屏蔽近似的定量分析。

细胞生理机制的数学模型通常涉及在表示功能状态网络内转换的图上的随机游走。Schmandt 和 Galán 最近引入了一种新颖的随机屏蔽近似，作为一种快速、准确的方法，用于从有限状态马尔可夫过程中生成近似样本路径，其中只有一部分状态是可观察的。例如，在离子通道模型中，例如 Hodgkin-Huxley 或其他基于电导的神经模型，神经细胞具有一群离子通道，其状态包括图的节点，其中只有一些允许跨膜电流经过。随机屏蔽近似包括忽略与图中边缘相关的动力学波动，不直接影响可观察状态。我们考虑找到最佳复杂度的问题，以减少从图上的随机过程到较小样本空间上的近似过程的映射，这取决于图上特定线性测量函数的选择。将离子通道状态划分为导电状态和非导电状态就是一个很好的例子。除了确定 Schmandt 和 Galán 的近似实际上在特定意义上是最佳的之外，我们还使用随机矩阵理论的最新结果为随机图集合的随机屏蔽近似的准确性提供启发式误差估计。此外，我们提供了一种新颖的定量测量方法，用于衡量反应图中各个转变对近似过程准确性的贡献。