Fig. 4.

Nodal importance rankings associated with nonlinear and linear control of a gene regulatory network. a, c Nonlinear and linear control importance rankings for the subnetwork of the giant component of size 60, respectively. In a, there are four nodes with R_NL = 1, because controlling any of these genes will make the system recover immediately from the tipping point collapse when the direction of the change in the bifurcation parameter is reversed. For linear control in c, the size of any minimum controller set is N_D = 4. b, d Nonlinear and linear nodal importance rankings for the subnetwork of 81 nodes with input connection. In b, there are several genes with R_NL = 0, as each gene in this group lacks the ability to restore the entire system even when its activity level is maintained at a high level through external control. For linear control in d, any minimum controller set has N_D = 17 nodes. In all panels, the nodal index along the abscissa is arranged in the descending order of the outgoing degrees of the genes