  Distributed parameter estimation for large-scale systems is an active
research problem. The goal is to derive a distributed algorithm in which each
agent obtains a local estimate of its own subset of the global parameter
vector, based on local measurements as well as information received from its
neighbours. A recent algorithm has been proposed, which yields the optimal
solution (i.e., the one that would be obtained using a centralized method) in
finite time, provided the communication network forms an acyclic graph. If
instead, the graph is cyclic, the only available alternative algorithm, which
is based on iterative matrix inversion, achieving the optimal solution, does so
asymptotically. However, it is also known that, in the cyclic case, the
algorithm designed for acyclic graphs produces a solution which, although non
optimal, is highly accurate. In this paper we do a theoretical study of the
accuracy of this algorithm, in communication networks forming cyclic graphs. To
this end, we provide bounds for the sub-optimality of the estimation error and
the estimation error covariance, for a class of systems whose topological
sparsity and signal-to-noise ratio satisfy certain condition. Our results show
that, at each node, the accuracy improves exponentially with the so-called
loop-free depth. Also, although the algorithm no longer converges in finite
time in the case of cyclic graphs, simulation results show that the convergence
is significantly faster than that of methods based on iterative matrix
inversion. Our results suggest that, depending on the loop-free depth, the
studied algorithm may be the preferred option even in applications with cyclic
communication graphs.
