  This paper develops tools to quantify the importance of agent interactions
and its impact on global performance metrics for networks modeled as linear
time-invariant systems. We consider Gramian-based performance metrics and
propose a novel notion of edge centrality that encodes the first-order
variation in the metric with respect to the modification of the corresponding
edge weight, including for those edges not present in the network. The proposed
edge centrality matrix (ECM) is additive over the set of inputs, i.e., it
captures the specific contribution to each edge's centrality of the presence of
any given actuator. We provide a full characterization of the ECM structure for
the class of directed stem-bud networks, showing that non-zero entries are only
possible at specific sub/super-diagonals determined by the network size and the
length of its bud. We also provide bounds on the value of the trace, trace
inverse, and log-det of the Gramian before and after single-edge modifications,
and on the edge-modification weight to ensure the modified network retains
stability. Simulations show the utility of the proposed edge centrality notion
and validate our results.
