(

**a**) Exact controllability measure

*n*_{D} as a function of connecting probability

*p* in directed ER random networks with identical weights (DER) for different network sizes. (

**b**) Exact controllability and structural controllability

*n*_{D} versus

*p* in directed ER random networks with random weights (RW DER) in the absence of bidirectional links. The inset of

**b**, exact controllability and structural controllability

*n*_{D} versus

*p* in unweighted, directed ER random networks (UW DER) in the absence of bidirectional links. Here MMT stands for the exact controllability determined by the maximum geometric multiplicity , SoD stands for the exact controllability of sparse or dense networks from and , respectively, S denotes the exact controllability of sparse networks from and, LSB denotes the structural controllability from the maximum matching algorithm. In

**a**,

and

are in good agreement with each other for different network sizes. In

**b**,

,

and

coincide exactly without any difference in the network described by the structural matrix. In the inset of

**b**, the assumption of structural controllability is weakly violated, ascribed to the unweighted property of the network. The structural controllability

is still consistent with the exact controllability

but with negligible difference. For example, at

*p*=8 × 10

^{−4},

=0.21584, whereas

=0.21576. At

*p*=10

^{−3},

=0.1352, whereas

=0.1350. For the networks in

**a**, the connecting probabilities of two directions between two nodes are both

*p*, but the corresponding random variables are independent of each other. For the networks in

**b**, based on the structure of an undirected ER random network, we randomly assign a direction to each undirected link. All the results are from 20 independent realizations and the error bars denote the s.d. The network size in

**b** is 5,000.

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