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Consensus+innovations detection: Phase transition under communication noise

We consider the tradeoffs between sensing and communication in a consensus+innovations distributed detection problem when the local communications among agents are noisy. Intuitively, we can expect that the error performance of the distributed detector is affected by both the sensing noise and the noise corrupting the communication among agents in the network. Too little communication (cooperation) and the distributed detector error performance will be dominated by the sensing noise. Too much communication and the detector error performance is dominated by the communication noise. We make this tradeoff precise through a large deviations analysis, i.e., by studying the exponential decay rate of the probability of error of the consensus+innovations distributed detector at each agent. Under a mild assumption of network connectedness, we show: 1) the weight sequences affecting the consensus and innovations potentials in the distributed detector need to be carefully designed for the error probability at every agent detector to decay exponentially fast; 2) the network exhibits a phase transition with respect to the communication noise power. Below a threshold on the communication noise power, cooperation (communication) among agents improves the error detection performance; above threshold, inter-agent communication does not enhance the error detection performance.