Dominating communities for hierarchical control of complex networks

Well-selected nodes driven by external signals in complex networks are of great importance for a dynamic system's structural controllability, which can also be achieved by selecting nodes in the minimum dominating set (MDS) of the networks as driver nodes. However, hierarchical structures such as communities are widely observed in complex networks, which motivates us to develop the hierarchical control of complex networks by dominating communities, and the control over whole networks is transformed into that within communities, i.e., a non-driver node is effectively controlled by its driver node if they belong to the same community. For the hierarchical control, we propose the structure-driven minimum dominating set (SD-MDS), which assumes that a driver node can control a non-driver node if the link between them belongs to a cyclic structure of the networks. We further investigate the impact of community dynamics on our framework. The results show that it is easier to control the networks with stronger communities and more heterogeneous community sizes, and the SD-MDS achieves more effective control for non-driver nodes than the MDS. This work also indicates that the number of driver nodes for the SD-MDS is associated with the degree distribution of the network.