Controllability of complex networks

This Supplementary Information is organized as follows. In Sec. II, we clarify the fundamental differences between our work and previous research on network controllability. In Sec. III, we give a short introduction to the structural control theory, which can be simply applied to directed networks. The reasons why we focus on linear dynamics are given in Sec. III A. Some simple examples in Sec. III B demonstrate the difference between controllability, structural controllability and strong structural controllability. The sufficient and necessary conditions for a linear system to be structurally controllable are given by Lin’s structural controllability theorem (SCT), which is discussed in Sec. III C. Based on SCT, we derived the minimum input theorem in Sec. III D, which gives the minimum number of inputs that we need to fully control a directed network. This theorem also enables us to find the driver nodes to which the external inputs should be injected, based on a deep relation between structural controllability and maximum matching. In Sec. IV, we analytically derived the average size and number of the maximum matchings for a random directed network ensemble with a prescribed degree distribution, using the cavity method developed in statistical physics. In Sec. V, we show the results on control robustness against node failure, compared to the results on link failure shown in the main text. The real-world networks analyzed in this work are listed and briefly described in Sec. VI.