Edge Centrality Matrix: Impact of Network Modification on Gramian Controllability Metrics

Due to recent technological advances, performance enhancement of complex networked control systems by edge modification done according to their importance in the network is becoming increasingly feasible. Unlike the nodal case, edge characterization with respect to a given performance metric is a rather unexplored research area. In this work, we seek to address this problem by proposing a novel Gramian-based edge centrality matrix which characterizes all the possible edges in the network with respect to physically realizable energy-based performance metrics. We rigorously prove the relationship of the various edge centrality matrix for different performance metrics with the gradient of the controllability Gramian with respect to edge weights. Notable feature of our proposed edge characterization is that it exhibits the contribution of individual inputs. We then analyze the edge centrality matrix for directed ring and line networks. Finally, through numerical examples, we validate a structural property of proposed edge centrality matrix and demonstrate its utility in network edge modification.