Controllability of Multi-Relational Networks with Heterogeneous Dynamical Nodes

This paper studies the controllability of networked systems, in which the nodes are heterogeneous high-dimensional dynamical systems, and the links between nodes are multi-relational. Our aim is to find controllability criteria for heterogeneous networks with multi-relational links beyond those only applicable to networks with single-relational links. It is found a network with multi-relational links can be controllable even if each single-relational network topology is uncontrollable, and vice versa. Some sufficient and necessary conditions are derived for the controllability of multi-relational networks with heterogeneous dynamical nodes. For two typical multi-relational networks with star-chain topology and star-circle topology, some easily verified conditions are presented. For illustration and verification, several examples are presented. These findings provide practical insights for the analysis and control of multi-relational complex systems.