Estimation Of Control Energy And Control Strategies For Complex Networks

The controlling of complex networks is one of the most challenging problems in modern network science. Accordingly, the required energy cost of control is a fundamental and significant problem. In this paper, we present the theoretical analysis and numerical simulations to study the controllability of complex networks from the energy perspective. First, by combining theoretical derivation and numerical simulations, we obtain lower bounds of the maximal and minimal energy costs for an arbitrary normal network, which are related to the eigenvalues of the state transition matrix. Second, we deduce that controlling unstable normal networks is easier than controlling stable normal networks with the same size. Third, we demonstrate a tradeoff between the control energy and the average degree (or the maximum degree) of an arbitrary undirected network. Fourth, numerical simulations show that the energy cost is negatively correlated with the degree of nodes. Moreover, the combinations of control nodes with the greater sum of degree need less energy to implement complete control. Finally, we propose a multi-objective optimization model to obtain the control strategy, which not only ensures the fewer control nodes but also guarantees the less energy cost of control.