Detection of the Important Control Nodes in Network Controllability Processes on Temporal Networks Using Temporal Graph Transformer Neural Networks

Controllability in temporal networks refers to the ability to steer the state of a time-evolving system toward any desired configuration through external inputs while explicitly respecting the temporal ordering and availability of interactions. Identifying control-important nodes is a fundamental challenge in temporal network controllability, as effective control depends not only on network topology but also on the timing and sequence of connections, making the detection of critical control nodes essential for reducing control cost and improving robustness. In this paper, we propose a novel learning-based framework that formulates the identification of control-important nodes as a node classification problem on temporal networks and addresses it using a temporal graph transformer neural network. The proposed method integrates spatial attention to capture structural dependencies within each temporal snapshot and temporal attention to model the evolution of node influence across time, followed by a spatiotemporal feature fusion mechanism that generates expressive node representations. Based on these representations, nodes are classified into critical, ordinary, and redundant categories without explicitly enumerating all possible minimum driver node sets or temporal matchings. Experimental evaluations on both synthetic and real-world temporal networks demonstrate that the proposed approach achieves high accuracy, robustness, and scalability compared with existing graph-based and temporal learning methods. The results confirm that the model effectively captures time-respecting control patterns and provides an efficient and interpretable solution for identifying control-important nodes, thereby bridging temporal network controllability theory with modern transformer-based graph learning techniques.