Network Controllability — Minimum Driver Nodes (Liu et al.)

Liu et al. (2011) showed that structural controllability of a directed network — the ability to drive the system from any initial to any target state — requires a minimum set of driver nodes whose size is determined by the maximum matching of the network graph. Scale-free networks are surprisingly hard to control (many driver nodes needed), while random networks are easier. Driver nodes are shown in gold.

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Driver nodes n_D
Controllability n_D/N
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Max matching size