On the Robustness of Uncertain Consensus Networks

This work considers the robustness of uncertain consensus networks. The stability properties of consensus networks with negative edge weights are also examined. We show that the network is unstable if either the negative weight edges form a cut in the graph or any single negative edge weight has a magnitude less than the inverse of the effective resistance between the two incident nodes. These results are then used to analyze the robustness 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 nonlinear couplings.