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Mathematical models of cellular physiological mechanisms often involve randomwalks on graphs representing transitions within networks of functional states.Schmandt and Gal\'{a}n recently introduced a novel stochastic shieldingapproximation as a fast, accurate method for generating approximate samplepaths from a finite state Markov process in which only a subset of states areobservable. For example, in ion channel models, such as the Hodgkin-Huxley orother conductance based neural models, a nerve cell has a population of ionchannels whose states comprise the nodes of a graph, only some of which allow atransmembrane current to pass. The stochastic shielding approximation consistsof neglecting fluctuations in the dynamics associated with edges in the graphnot directly affecting the observable states. We consider the problem offinding the optimal complexity reducing mapping from a stochastic process on agraph to an approximate process on a smaller sample space, as determined by thechoice of a particular linear measurement functional on the graph. Thepartitioning of ion channel states into conducting versus nonconducting statesprovides a case in point. In addition to establishing that Schmandt andGal\'{a}n's approximation is in fact optimal in a specific sense, we use recentresults from random matrix theory to provide heuristic error estimates for theaccuracy of the stochastic shielding approximation for an ensemble of randomgraphs. Moreover, we provide a novel quantitative measure of the contributionof individual transitions within the reaction graph to the accuracy of theapproximate process.

Measuring edge importance: a quantitative analysis of the stochastic shielding approximation for random processes on graphs