  We present an algorithm for the problem of linear distributed estimation of a
parameter in a network where a set of agents are successively taking
measurements. The approach considers a roaming token in a network that carries
the estimate, and jumps from one agent to another in its vicinity according to
the probabilities of a Markov chain. When the token is at an agent it records
the agent's local information. We analyze the proposed algorithm and show that
it is consistent and asymptotically optimal, in the sense that its
mean-square-error (MSE) rate of decay approaches the centralized one as the
number of iterations increases. We show these results for a scenario where the
network changes over time, and we consider two different set of assumptions on
the network instantiations: they are i.i.d. and connected on the average, or
they are deterministic and strongly connected for every finite time window of a
fixed size. Simulations show our algorithm is competitive with
consensus+innovations type algorithms, achieving a smaller MSE at each
iteration in all considered scenarios.
