Figure 1. Control topologies in networks with linear, non-dissipative dynamics.

(a) A simple directed network. We omit cycles from this example because in this framework they are inherently self-regulatory and their control follows immediately once the remainder of the network has been controlled²³. (b) In a control topology, every node is either directly controlled (colored nodes) or indirectly controlled (white nodes with colored outlines). Indirect control is achieved by placing nodes on a path originating at a directly controlled node (white edges with colored outlines). Importantly, in this framework every node can control at most one of its downstream neighbors and every pair of such paths are necessarily node-disjoint. (c) A control topology is minimal if it minimizes the number of controls. In this example node A must be directly controlled (it has no upstream nodes through which a control path may be routed) and either node B or node C must be directly controlled because node A can control at most one of its downstream neighbors.