  We show that for a certain class of dynamics at the nodes the response of a
network of any topology to arbitrary inputs is defined in a simple way by its
response to a monotone input. The nodes may have either a discrete or
continuous set of states and there is no limit on the complexity of the
network. The results provide both an efficient numerical method and the
potential for accurate analytic approximation of the dynamics on such networks.
As illustrative applications, we introduce a quasistatic mechanical model with
objects interacting via frictional forces, and a financial market model with
avalanches and critical behavior that are generated by momentum trading
strategies.
