Energy-Transfer Edge Centrality and Its Role in Enhancing Network Controllability

The ability to modify the structure of network systems offers great opportunities to enhance their operation, improve their efficiency, and increase their resilience against failures and attacks. This paper focuses on the edge modification problem, i.e., improving network controllability by adding and/or re-weighting interconnections while keeping the actuation structure fixed. We consider a network system following linear dynamics and propose a novel edge centrality measure that captures the extent to which an edge facilitates energy exchange across the network through its defining nodes. We analyze the effectiveness of the proposed measure by characterizing its relationship with the gradient (with respect to edge weights) of trace, log determinant, and inverse of the trace inverse of the Gramian. We show that the optimal solution of the edge modification problem lies on the boundary of the feasible search space when the objective is the trace of the Gramian or the network has a diagonal controllability Gramian and the objective is either log determinant or the inverse of the trace inverse of the Gramian. Finally, using the proposed edge centrality measure we design two network modification algorithms that restrict the search space to a smaller subset of all possible edges and numerically demonstrate their efficacy.