Convergence Analysis of Distributed Inference with Vector-Valued Gaussian Belief Propagation

In networks such as the smart grid, communication networks, and social networks, local measurements/observations are scattered over a wide geographical area. Centralized inference algorithm are based on gathering all the observations at a central processing unit. However, with data explosion and ever-increasing network sizes, centralized inference suffers from large communication overhead, heavy computation burden at the center, and susceptibility to central node failure. This paper considers inference over networks using factor graphs and a distributed inference algorithm based on Gaussian belief propagation. The distributed inference involves only local computation of the information matrix and of the mean vector and message passing between neighbors. We discover and show analytically that the message information matrix converges exponentially fast to a unique positive definite limit matrix for arbitrary positive semidefinite initialization. We provide the necessary and sufficient convergence condition for the belief mean vector to converge to the optimal centralized estimator. An easily verifiable sufficient convergence condition on the topology of a factor graph is further provided.