  Methods for efficiently controlling dynamics propagated on networks are
usually based on identifying the most influential nodes. Knowledge of these
nodes can be used for the targeted control of dynamics such as epidemics, or
for modifying biochemical pathways relating to diseases. Similarly they are
valuable for identifying points of failure to increase network resilience in,
for example, social support networks and logistics networks. Many measures,
often termed `centrality', have been constructed to achieve these aims. Here we
consider Katz centrality and provide a new interpretation as a steady-state
solution to continuous-time dynamics. This enables us to implement a
sensitivity analysis which is similar to metabolic control analysis used in the
analysis of biochemical pathways. The results yield a centrality which
quantifies, for each node, the net impact of its absence from the network. It
also has the desirable property of requiring a node with a high centrality to
play a central role in propagating the dynamics of the system by having the
capacity to both receive flux from others and then to pass it on. This new
perspective on Katz centrality is important for a more comprehensive analysis
of directed networks.
