We consider the problem of classifying among a set of M hypothesis with N distributed noisy sensors. The N sensors can collaborate over a finite link-capacity network. The task is to arrive at a consensus about the event after exchanging such messages. In contrast to the conventional decentralized detection approach, wherein the bit rates for each link is explicitly constrained, our approach is based on high-rate limit perspective. We apply a variant of belief propagation as a strategy for collaboration to arrive at a solution to the distributed classification problem. We show that the message evolution can be reformulated as the evolution of a linear dynamical system, which is primarily characterized by network connectivity. It turns out that consensus is almost always reached by the sensors for any arbitrary network. We then derive conditions under which the consensus is the centralized MAP estimate and show that this is achieved with O(M log/sub 2/ N) bits.