Analysis on critical nodes in controlling complex networks using dominating sets

We investigate the role of nodes when a minimum dominating set (MDS) approach is considered in controlling a network. We present an algorithmic procedure to compute the critical and redundant nodes using the MDS. The theoretical analysis for scale-free networks shows that (1) high-degree nodes tend to be critical nodes and (2) for scaling exponent gamma>2, such probability increases when gamma grows. These relationships are confirmed by computer simulations. Furthermore, our computational experiments also show that the MDS is unimodal when the average degree left grows. It means that most nodes are redundant for high-degree, therefore one can achieve control through a small set of nodes, highlighting a centralized control feature. The analysis of a set of real-world networks confirms the findings observed in simulation experiments. In particular, the U.S. airport network shows a more centralized control, whereas the Italy's nationwide power grid network exhibits a more distributed control. The human protein-protein interaction network exerts control with a configuration between both man-made networks.