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Identifying the nodes that have the potential to influence the state of anetwork is a relevant question for many complex systems. In many applicationsit is often essential to test the ability of an individual node to control aspecific target subset of the network. In biological networks, this mightprovide precious information on how single genes regulate the expression ofspecific groups of molecules in the cell. Taking into account theseconstraints, we propose an optimized heuristic based on the Kalman rankcondition to quantify the centrality of a node as the number of target nodes itcan control. By introducing a hierarchy among the nodes in the target set, andperforming a step-wise research, we ensure for sparse and directed networks theidentification of a controllable driver-target configuration in a significantlyreduced space and time complexity. We show how the method works for simplenetwork configurations, then we use it to characterize the inflammatorypathways in molecular gene networks associated with macrophage dysfunction inpatients with multiple sclerosis. Results indicate that the targeted secretedmolecules can in general be controlled by a large number of driver nodes (51%)involved in different cell functions, i.e. sensing, signaling andtranscription. However, during the inflammatory response only a moderatefraction 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 wefind that this is related to the presence of dysregulated genes along thecontrollable walks. Our method, that we name step-wise target controllability,represents a practical solution to identify controllable driver-targetconfigurations in directed complex networks and test their relevance from afunctional perspective.

Step-wise target controllability identifies dysregulated pathways of macrophage networks in multiple sclerosis