In this talk we consider distributed detection systems (sensor networks) where the sensors are subject to stringent communication constraints, mostly stringent power or bandwidth constraints.
We present the optimal Stein-error exponents for different distributed detection systems and explain the communication and testing schemes that achieve these optimal exponents. Our results reveal interesting dichotomies of the Stein-exponent with respect to the connectivity of the channel: If all inputs can lead to all outputs, then the Stein-exponent under stringent (probability 1) power constraints equals the exponent of a local test at the decision center only. I.e., The sensors and their observations cannot improve this exponent. In contrast, if the channel is degenerate the Stein-exponent is the same as when communication is over a noiseless channel. Under slightly less stringent power constraints (on expectation instead of probability 1) the Stein-exponent will depend on the exact channel law if the channel is fully connected and also on whether the power constraint is imposed under all hypotheses or not. We completely characterize the Stein-exponent also when stringent power constraints are imposed on expectation.