复杂网络的结构是系统动力学的基础，其中节点状态和连接模式决定控制系统的成本，这是解决复杂性的一个关键方面。然而，最小化控制具有最少输入节点的系统所需的能量仍然是一个悬而未决的问题。本研究研究了闭连接功能模块的结构与控制能量之间的关系。我们发现小的结构调整，例如添加一些扩展驱动节点，可以显着降低控制能量。因此，我们在能源成本降低（MIER）中提出了 MInimal 扩展驱动节点。接下来，我们将MIER的检测转化为多目标优化问题，并选择NSGA-II算法来解决它。与基线方法相比，NSGA-II能够最大程度地逼近最优解。通过使用合成数据和真实数据的实验，我们验证了 MIER 可以指数级降低控制能量。此外，随机扰动测试证实了 MIER 的稳定性。随后，我们将MIER应用于三个代表性场景：人类蛋白质-蛋白质相互作用网络中受癌症突变影响的差异表达基因的调节、世界贸易网络中发达国家之间的贸易关系以及运动神经元对体壁肌肉细胞的调节秀丽隐杆线虫神经网络。结果表明，从拓扑角度来看，MIER 的参与显着降低了这些原始模块所需的控制能量。此外，MIER 节点增强功能、补充关键节点并揭示潜在机制。总的来说，我们的工作为理解和呈现生物、社会和神经系统中的控制策略提供了实用的计算工具。© 2024 作者。由 AIP Publishing 独家许可出版。

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.© 2024 Author(s). Published under an exclusive license by AIP Publishing.