Network Controllability

Minimum Driver Nodes — Liu, Slotine & Barabási (2011)
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Structural Controllability (Lin 1974 + Liu et al. 2011)

ẋ = Ax + Bu    (linear dynamics)

n_D = max(1, N - |maximum matching|)

A system is controllable if any state can be reached from any initial condition. The minimum number of driver nodes equals N minus the size of a maximum matching on the network. Scale-free networks (hubs) surprisingly require more driver nodes than random — hubs create bottlenecks. Highlighted nodes (red) are driver nodes; blue arrows show matched edges.