Controllability preservation in complex networks via minimal edge configurations

Recent advances in networked control systems have resulted in linear time-invariant (LTI) networks of increasing scale and complexity, thereby presenting significant challenges for controllability optimization. This study proposes a novel optimization framework to enhance control efficiency and reduce network complexity in large-scale complex networks, leveraging exact controllability theory and matrix similarity transformations for both directed and undirected topologies. Firstly, a method is introduced for identifying node correspondences before and after similarity transformations of the network's adjacency matrix. Subsequently, a minimal-edge connection framework is proposed to link Jordan blocks associated with distinct eigenvalues in the Jordan canonical form. Finally, building upon this method and framework, a specific optimization strategy is developed to minimize the number of inter-node connections while preserving network controllability. The proposed approach effectively simplifies the network structure and improves control efficiency.