Network control theory can be used to model how one should steer the brain between different states by driving a specific region with an input. The needed energy to control a network is often used to quantify its controllability, and controlling brain networks requires diverse energy depending on the selected input region. We use the theory of how input node placement affects the longest control chain (LCC) in the controllability of brain networks to study the role of the architecture of white matter fibers in the required control energy. We show that the energy needed to control human brain networks is related to the LCC, i.e., the longest distance between the input region and other regions in the network. We indicate that regions that control brain networks with lower energy have small LCCs. These regions align with areas that can steer the brain around the state space smoothly. By contrast, regions that need higher energy to move the brain toward different target states have larger LCCs. We also investigate the role of the number of paths between regions in the control energy. Our results show that the more paths between regions, the lower cost needed to control brain networks. We evaluate the number of paths by counting specific motifs in brain networks since determining all paths in graphs is a difficult problem.

翻译：网络控制理论可用于建模如何通过输入驱动特定区域来引导大脑在不同状态间切换。控制网络所需的能量常被用于量化其可控性，而控制大脑网络所需的能量因输入区域的选择而异。我们利用输入节点位置如何影响大脑网络可控性中最长控制链（LCC）的理论，研究白质纤维结构在控制能量需求中的作用。研究表明，控制人脑网络所需的能量与LCC相关，即输入区域与网络中其他区域之间的最长距离。我们指出，以较低能量控制大脑网络的区域具有较小的LCC，这些区域能够平滑地引导大脑在状态空间中运动。相反，需要更高能量将大脑导向不同目标状态的区域具有较大的LCC。我们还探究了区域间路径数量对控制能量的影响。结果显示，区域间路径越多，控制大脑网络所需的能量越低。由于确定图中的所有路径是困难问题，我们通过计数大脑网络中的特定模体来评估路径数量。