  We attempt to answer the question what data brings adaptive diffusion
algorithms converging to true parameters. The discussion begins with the
diffusion recursive least squares (RLS). When unknown parameters are scalar,
the necessary and sufficient condition of the convergence for the diffusion RLS
is established, in terms of the strong consistency and mean-square convergence
both. However, for the general high dimensional parameter case, our results
suggest that the diffusion RLS in a connected network might cause a diverging
error, even if local data at every node could guarantee the individual RLS
tending to true parameters. Due to the possible failure of the diffusion RLS,
we prove that the diffusion Robbins-Monro (RM) algorithm could achieve the
strong consistency and mean-square convergence simultaneously, under some
cooperative information conditions. The convergence rates of the diffusion RM
are derived explicitly.
