Effect of correlations on controllability transition in network control

The network control problem has recently attracted an increasing amount of attention, owing to concerns including the avoidance of cascading failures of power-grids and the management of ecological networks. It has been proven that numerical control can be achieved if the number of control inputs exceeds a certain transition point. In the present study, we investigate the effect of degree correlation on the numerical controllability in networks whose topological structures are reconstructed from both real and modeling systems, and we find that the transition point of the number of control inputs depends strongly on the degree correlation in both undirected and directed networks with moderately sparse links. More interestingly, the effect of the degree correlation on the transition point cannot be observed in dense networks for numerical controllability, which contrasts with the corresponding result for structural controllability. In particular, for directed random networks and scale-free networks, the influence of the degree correlation is determined by the types of correlations. Our approach provides an understanding of control problems in complex sparse networks.