Control capacity and bimodality in target control

Abstract

Controlling large networks is a fundamental problem and a great challenge in network science. Typically, full control is not necessary and infeasible. In many cases, only a preselected subset of nodes is required to be controlled, which is the target control problem. Each node does not participate in controlling the target set with equal probability, prompting us to quantify their contributions for target control. Here we develop a random sampling method to estimate the likelihood of each node participating as a driver node in target control configurations and demonstrate the unbiasedness of sampling. Each node is assigned with a role of critical, intermittent or redundant as it appears in all, some and none of the minimum driver node sets accordingly. We apply the method to Erdős-Rényi (ER) and scale-free (SF) networks and find that the fractions of critical and intermittent nodes increase as the scale of the target set increases. Furthermore, when the size of target node is fixed in SF networks, the fraction of redundant nodes may show a bimodal behavior as the networks become denser, leading to two control modes: centralized control and distributed control. The findings help understand the dynamics of control and offer tools for target control in complex systems.