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We study, by large deviations analysis, the asymptotic performance of Gaussian running consensus distributed detection over random networks; in other words, we determine the exponential decay rate of the detection error probability. With running consensus, at each time step, each sensor averages its decision variable with the neighbors' decision variables and accounts on-the-fly for its new observation. We show that: 1) when the rate of network information flow (the speed of averaging) is above a threshold, then Gaussian running consensus is asymptotically equivalent to the optimal centralized detector, i.e., the exponential decay rate of the error probability for running consensus equals the Chernoff information; and 2) when the rate of information flow is below a threshold, running consensus achieves only a fraction of the Chernoff information rate. We quantify this achievable rate as a function of the network rate of information flow. Simulation examples demonstrate our theoretical findings on the behavior of running consensus detection over random networks.