We consider steady states of dynamics that have an underlying network structure. We study how a steady state responds to small perturbations in the network parameters and how this sensitivity is connected to the network structure. We introduce a prototypical linear response equation and determine its sensitivity. This abstract result is applied to study the sensitivity of steady states in two common dynamics on networks: continuous-time Markov chains and deterministically modelled chemical reaction networks. For continuous-time Markov chains, we are able to efficiently compute the signs of the response in terms of the underlying network structure. The study of chemical reaction networks extends the sensitivity analysis to open systems with more complex network structures.