We consider the scenario of N distributed noisy sensors observing a single event. The sensors are distributed and can only exchange messages through a network. The sensor network is modelled by means of a graph, which captures the connectivity of different sensor nodes in the network. The task is to arrive at a consensus about the event after exchanging such messages. The focus of this paper is twofold: a) characterize conditions for reaching a consensus; b) derive conditions for when the consensus converges to the centralized MAP estimate. The novelty of the paper lies in applying belief propagation as a message passing strategy to solve a distributed hypothesis testing problem for a pre-specified network connectivity. We show that the message evolution can be re-formulated as the evolution of a linear dynamical system, which is primarily characterized by network connectivity. This leads to a fundamental understanding of as to which network topologies naturally lend themselves to consensus building and conflict avoidance.