Average Controllability of Complex Networks With Laplacian Dynamics

The trace of the controllability Gramian quantifies the average controllability in all directions in the system state space. In this paper, we investigate the average controllability of a semistable networked system with Laplacian dynamics and derive upper and lower bounds on the trace of its pseudo-controllability Gramian matrix. We show that these bounds are solely determined by the network topology, which can be obtained without computing any higher-dimensional matrix. We find that a sparse or a scale-free network is easy to control in terms of the average controllability. We then investigate the effect of the edges with negative weights on the average controllability for a signed network with Laplacian dynamics. We find that a small number of negatively-weighted edges can significantly affect the average controllability of the signed network. We finally demonstrate that many real-world networks are easy to control via manipulating negatively-weighted edges.