Selection and optimization of drive nodes in drive-response networks

This paper considers the selection and optimization of drive nodes based on the controllability of multilayer networks. The intra-layer network topologies are arbitrary, and the node dynamics are linear time-invariant dynamical systems. The study focuses on the number and selection of drive nodes in a special class of drive-response networks. Several conclusions are drawn through the investigation: (1) All the drive nodes cannot be placed in the response layer but can be contained in the drive layer; (2) The minimum number of drive nodes placed in the drive layer is equal to the maximum geometric multiplicity of the system matrix of the drive layer; (3) The configuration of interlayer coupling weight significantly affects the number and distribution of drive nodes. Moreover, an optimization scheme is proposed based on the Gershgorin circle theorem, which aims to minimize the number of drive nodes in the entire network. This scheme ensures that regardless of the drive nodes originally needed, they can be reduced to the maximum geometric multiplicity of the system matrix of the drive layer. Numerical simulations on a general two-layer network as well as various synthetic networks are provided to validate the results.