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Networks of classifiers are capturing the attention of system and algorithmic researchers because they offer improved accuracy over single model classifiers, can be distributed over a network of servers for improved scalability, and can be adapted to available system resources. This work provides a principled approach for the optimized allocation of system resources across a networked chain of classifiers. We begin with an illustrative example of how complex classification tasks can be decomposed into a network of binary classifiers. We formally define a global performance metric by recursively collapsing the chain of classifiers into one combined classifier. The performance metric trades off the end-to-end probabilities of detection and false alarm, both of which depend on the resources allocated to each individual classifier. We formulate the optimization problem and present optimal resource allocation results for both simulated and state-of-the-art classifier chains operating on telephony data.