Combinatorial Average Energy Controllability (CAEC) for Analyzing Interaction of Functional Brain Networks

ABSTRACT Understanding how different functional brain networks interact is crucial for revealing the complexity of brain function and behavior. This study addresses this gap by investigating how brain transitions occur between functional brain networks, focusing on the controllability of brain structural subsets. Previous studies on brain controllability have primarily focused on whole‐brain connectivity networks, which do not adequately capture the transition abilities of weakly connected regions. To address this issue, we introduce a new metric—combinatorial average energy controllability (CAEC)—which assesses the influence of functional networks based on their ability to modulate other networks using low‐energy control inputs. By employing manifold learning and geodesic distance calculations, we aggregate influence vectors to provide a comprehensive view of energy propagation capacities in less connected functional networks, complementing conventional average controllability measures. Our findings demonstrate that even regions with weak connections can propagate input energy, while some moderately connected ones do not, and strong connections preserve their distribution abilities. Additionally, we utilize optimal control cost calculations to compare with CAEC results, revealing how the brain's structure and connections affect its function. This study offers new insights into how increased activity in different functional networks influences brain activity, with implications for understanding cognitive processes and addressing neurological disorders.