  This paper studies the problem of distributed weighted least-squares (WLS)
estimation for an interconnected linear measurement network with additive
noise. Two types of measurements are considered: self measurements for
individual nodes, and edge measurements for the connecting nodes. Each node in
the network carries out distributed estimation by using its own measurement and
information transmitted from its neighbours. We study two distributed
estimation algorithms: a recently proposed distributed WLS algorithm and the
so-called Gaussian Belief Propagation (BP) algorithm. We first establish the
equivalence of the two algorithms. We then prove a key result which shows that
the information matrix is always generalised diagonally dominant, under some
very mild condition. Using these two results and some known convergence
properties of the Gaussian BP algorithm, we show that the aforementioned
distributed WLS algorithm gives the globally optimal WLS estimate
asymptotically. A bound on its convergence rate is also presented.
