Strongly Uncontrollable Network Topologies

In this article, we present a class of network topologies under which the Laplacian consensus dynamics exhibit undesirable controllability properties under a broadcast control signal. Specifically, the networks we characterize are uncontrollable for any subset of the nodes chosen as control inputs and that emit a common control signal. We provide a sufficient condition for a network to contain this strong uncontrollability property and describe network perturbations that leave the uncontrollability property invariant. As a by-product, we identify nontrivial network topologies that require the control of approximately half the nodes in the network as a necessary condition for controllability.