Schmandt and Galán (Phys. Rev. Lett., 2012) introduced the Stochastic Shielding Approximation (SSA) as a fast, accurate simplification of randomly gated ion channel models. Viewing the channel as a discrete process on a directed graph, driven by an independent noise source for each edge, the SSA accurately represents the process using independent noise sources for only a small subset of the edges. Schmidt and Thomas (J. Math. Neurosci., 2014) showed the variance of the channel conductance decomposes into a sum of contributions from each directed edge, providing a metric for ranking the relative importance of each edge. Moreover they showed that SSA is equivalent to a dimension-reducing projection acting on the sample space, rather than on the state space. The SSA preserves the mean field behavior while selectively incorporating only the independent underlying noise sources that contribute the most significantly to observable system behavior. Thus the stochastic shielding heuristic provides an analytically tractable example of incorporating fluctuations "beyond the mean field" in a manner relevant to the network's physiological function.