Table 2.

The number of unrestricted or unilateral control inputs required for controllability of some specific network structures.

Network type	N u	N r	η
Circular network, positive cycle	1	2	2
Circular network, negative cycle	1	1	1
Hub with n − 1 outgoing edges	n − 1	n + 1	η → 1 when n → ∞
Hub with n − 1 incoming edges	n − 1	2(n − 1)	2
Rooted directed tree, outgoing edges from the root	n leaf	n leaf + r + 1	(n leaf + r + 1)/n leaf
Full network, simple eigenvalues	1	2	2
Full network, skew-symmetric	1	1	1

For the RDT, the number of leaves coincides with the dimension of the null space, i.e. n _leaf = μ ₀, and r is the number of nodes with outdegree ≥2.