Improved Maximum Matching for Constrained Structural Controllability of Directed Networks

Although traditional structural controllability theory successfully determines the number and selection of driver nodes for complete network control in linear time-invariant systems, many real-world networks contain constrained nodes (e.g., buried pipeline junctions, undruggable proteins) that cannot function as driver nodes. To address this gap, this study develops a constrained structural controllability (CSC) framework to identify the minimal set of controllable nodes required for network control while systematically excluding constrained nodes. The key contribution involves an improved maximum matching algorithm that systematically accounts for node constraints in the driver selection process. Numerical simulations show that the CSC of directed networks with a higher average degree or clustering coefficient, or a more uneven degree distribution, typically require fewer driver nodes. However, assigning constraints to high-degree nodes significantly increases the number of driver nodes needed. Results also indicate that traditional strategies (e.g., naive elimination of constrained nodes or creation of artificial bridging connections) demonstrate fundamental limitations in addressing CSC challenges. For Erdős-Rényi and Watts-Strogatz small-world networks, the proposed algorithm outperforms existing greedy target control methods by requiring fewer driver nodes. These findings provide actionable insights for applications like optimal sensor placement in pipeline networks and related domains.