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 association of these outliers with dynamics that are unstable to small perturbations reveals them as artifacts of deterministic models, making them less biologically relevant and reinforcing the generality of easy controllability in biological networks.Author summary: What sets how easily a living system can be controlled? Such a question can be operationalized in terms of the number of system components that need to be forced to guarantee a desired outcome. Previous results in Boolean networks have suggested that control does not intrinsically become more difficult in larger networks, but instead depends mostly on the number of distinct long-term dynamics exhibited by the system. Yet there are exceptions to this rule, cases in which most or all nodes in a network must be controlled to achieve certain end states. Here, we study these cases in detail and show that they are related to instability. We view our results as encouraging for the hypothesis that even large biological networks may be typically easy to control.