Binary consensus with Gaussian communication noise: A probabilistic approach

In this paper we consider the impact of Gaussian communication noise on a network that is trying to reach consensus on the occurrence of an event. We take a probabilistic approach and formulate the consensus problem using Markov chains. We show that the steady state behavior in the presence of any amount of non-zero communication noise is unfavorable as the network loses the memory of the initial state. However, we show that the network can still reach and stay in accurate consensus for a long period of time. In order to characterize this, we derive a close approximation for the second largest eigenvalue of the network and show how it is related to the size of the network and communication noise variance.