  We consider the problem of robustness in large consensus networks that occur
in many areas such as distributed optimization. Robustness, in this context, is
the scaling of performance measures, e.g. H2-norm, as a function of network
dimension. We provide a formal framework to quantify the relation between such
performance scaling and the convergence speed of the network. Specifically, we
provide upper and lower bounds for the convergence speed in terms of robustness
and discuss how these bounds scale with the network topology. The main
contribution of this work is that we obtain tight bounds, that hold regardless
of network topology. The work here also encompasses some results in convergence
time analysis in previous literature.
