Strongly uncontrollable network topologies

TL;DR

This paper identifies a class of network topologies with Laplacian consensus dynamics that are strongly uncontrollable under broadcast control signals, revealing fundamental limitations in controllability for certain network structures.

Contribution

It introduces a sufficient condition for strong uncontrollability in network topologies and analyzes how network perturbations affect this property.

Findings

Networks can be uncontrollable regardless of control input subset

Controllability may require controlling nearly half the nodes

Certain topologies are invariant under specific perturbations

Abstract

In this paper, we present a class of network topologies under which the Laplacian consensus dynamics exhibits 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 non-trivial network topologies that require the control of approximately half the nodes in the network as a necessary condition for controllability.