Difficult control is related to instability in biologically inspired Boolean networks

Previous work in Boolean dynamical networks has suggested that the number of components that must be controlled to select an existing attractor is typically set by the number of attractors admitted by the dynamics, with no dependence on the size of the network. Here we study the rare cases of networks that defy this expectation, with attractors that require controlling most nodes. We find empirically that unstable fixed points are the primary recurring characteristic of networks that prove more difficult to control. We describe an efficient way to identify unstable fixed points and show that, in both existing biological models and ensembles of random dynamics, we can better explain the variance of control kernel sizes by incorporating the prevalence of unstable fixed points. In the end, the fact that these exceptions are associated with dynamics that are unstable to small perturbations hints that they are likely an artifact of using deterministic models. These exceptions are likely to be biologically irrelevant, supporting the generality of easy controllability in biological networks.