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：网络控制理论可建模如何通过向特定脑区施加输入来引导大脑在不同状态间转换。控制网络所需的能量常被用于量化其可控性，而控制脑网络的能量需求会因所选输入区域的不同而产生差异。我们利用输入节点布局如何影响脑网络可控性中最长控制链（LCC）的理论，研究白质纤维架构在调控能量需求中的作用。研究表明，控制人脑网络所需能量与LCC（即输入区域到网络中其他区域的最长距离）相关。能够以较低能量控制脑网络的脑区具有较小的LCC值，这些区域与可平滑引导大脑在状态空间中移动的脑区相吻合。相反，需要更高能量驱动大脑趋向不同目标状态的脑区则具有较大的LCC值。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，控制脑网络的成本越低。由于求解图中所有路径属于困难问题，我们通过统计脑网络中的特定模体来评估路径数量。