  This work considers the robustness of uncertain consensus networks. The first
set of results studies the stability properties of consensus networks with
negative edge weights. We show that if either the negative weight edges form a
cut in the graph, or any single negative edge weight has magnitude less than
the inverse of the effective resistance between the two incident nodes, then
the resulting network is unstable. These results are then applied to analyze
the robustness properties of the consensus network with additive but bounded
perturbations of the edge weights. It is shown that the small-gain condition is
related again to cuts in the graph and effective resistance. For the single
edge case, the small-gain condition is also shown to be exact. The results are
then extended to consensus networks with non-linear couplings.
