The Roles of Input Matrix and Nodal Dynamics in Network Controllability

Recently, there has been a growing interest in network controllability for determining the number and placement of controllers. In the framework of linear dynamics, this paper regarding the network controllability studies both the formulation of the control input matrix and the influence of the linear nodal dynamics, with precise guidelines derived on how to design the input matrix. This design strategy takes the column vectors of the input matrix as the basis to maximize the controllable subspace. In the one-dimensional case, it is found that nodal dynamics with high degrees of heterogeneity dominate the network topology, but identical nodal dynamics have no effect on the network controllability. In the higher dimensional case, it reveals why controllable network topology and controllable nodal dynamics together are still insufficient to ensure the controllability of the whole network, with a feasible solution to the problem presented based on a suitably designed input matrix. All characteristics of the network topology and nodal dynamics are integrated into the input matrix formulation, which plays an essential role in determining the network controllability.