Detection of minimal extended driver nodes in energetic costs reduction

Structures of complex networks are fundamental to system dynamics, where node state and connectivity patterns determine the cost of a control system, a key aspect in unraveling complexity. However, minimizing the energy required to control a system with the fewest input nodes remains an open problem. This study investigates the relationship between the structure of closed-connected function modules and control energy. We discovered that small structural adjustments, such as adding a few extended driver nodes, can significantly reduce control energy. Thus, we propose MInimal extended driver nodes in Energetic costs Reduction (MIER). Next, we transform the detection of MIER into a multi-objective optimization problem and choose an NSGA-II algorithm to solve it. Compared with the baseline methods, NSGA-II can approximate the optimal solution to the greatest extent. Through experiments using synthetic and real data, we validate that MIER can exponentially decrease control energy. Furthermore, random perturbation tests confirm the stability of MIER. Subsequently, we applied MIER to three representative scenarios: regulation of differential expression genes affected by cancer mutations in the human protein-protein interaction network, trade relations among developed countries in the world trade network, and regulation of body-wall muscle cells by motor neurons in Caenorhabditis elegans nervous network. The results reveal that the involvement of MIER significantly reduces control energy required for these original modules from a topological perspective. Additionally, MIER nodes enhance functionality, supplement key nodes, and uncover potential mechanisms. Overall, our work provides practical computational tools for understanding and presenting control strategies in biological, social, and neural systems.