  Identifying the nodes that have the potential to influence the state of a
network is a relevant question for many complex systems. In many applications
it is often essential to test the ability of an individual node to control a
specific target subset of the network. In biological networks, this might
provide precious information on how single genes regulate the expression of
specific groups of molecules in the cell. Taking into account these
constraints, we propose an optimized heuristic based on the Kalman rank
condition to quantify the centrality of a node as the number of target nodes it
can control. By introducing a hierarchy among the nodes in the target set, and
performing a step-wise research, we ensure for sparse and directed networks the
identification of a controllable driver-target configuration in a significantly
reduced space and time complexity. We show how the method works for simple
network configurations, then we use it to characterize the inflammatory
pathways in molecular gene networks associated with macrophage dysfunction in
patients with multiple sclerosis. Results indicate that the targeted secreted
molecules can in general be controlled by a large number of driver nodes (51%)
involved in different cell functions, i.e. sensing, signaling and
transcription. However, during the inflammatory response only a moderate
fraction of all the possible driver-target pairs are significantly coactivated,
as measured by gene expression data obtained from human blood samples. Notably,
they differ between multiple sclerosis patients and healthy controls, and we
find that this is related to the presence of dysregulated genes along the
controllable walks. Our method, that we name step-wise target controllability,
represents a practical solution to identify controllable driver-target
configurations in directed complex networks and test their relevance from a
functional perspective.
