Extended Data Figure 2.

Illustration of structural controllability, the construction of the linking graph, and the derivation of the controllability criterion. (a,b) The blue nodes receive external signals and the pink nodes are those we aim to control. Thus, S = 1 and M = 2 for both networks. (c) The construction of the linking graph for the network in (a). (d) The calculation of the linking size can be mapped into a multi-source-multi-sink max-flow problem, with the constraint that the capacity of each node and each edge is one. The red edges show the two disjoint paths that achieve the maximum flow. (e,f) Schematic picture for the derivation of the lower bound z_*.