  An information theoretic formulation of the distributed averaging problem
previously studied in computer science and control is presented. We assume a
network with m nodes each observing a WGN source. The nodes communicate and
perform local processing with the goal of computing the average of the sources
to within a prescribed mean squared error distortion. The network rate
distortion function R^*(D) for a 2-node network with correlated Gaussian
sources is established. A general cutset lower bound on R^*(D) is established
and shown to be achievable to within a factor of 2 via a centralized protocol
over a star network. A lower bound on the network rate distortion function for
distributed weighted-sum protocols, which is larger in order than the cutset
bound by a factor of log m is established. An upper bound on the network rate
distortion function for gossip-base weighted-sum protocols, which is only log
log m larger in order than the lower bound for a complete graph network, is
established. The results suggest that using distributed protocols results in a
factor of log m increase in order relative to centralized protocols.
