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A network-compressive transmission protocol is developed in which correlated sensor observations belonging to a finite alphabet are linearly combined as they traverse the network on their way to a sink node. Statistical dependencies are modeled using factor graphs. The sum-product algorithm is run under different modeling assumptions to estimate the maximum a posteriori set of observations given the compressed measurements at the sink node. Error exponents are derived for cyclic and acyclic factor graphs using the method of types, showing that observations can be recovered with arbitrarily low probability of error as the network size grows. Simulated tests corroborate the theoretical claims.