Before revision:   As illustrative applications, we propose and discuss a quasistatic mechanical model with objects interacting via friction forces, and a financial market model with avalanches and critical behavior induced by momentum trading strategies.
After revision:  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 propose and discuss a quasistatic mechanical model with objects interacting via friction forces, and a financial market model with avalanches and critical behavior induced by momentum trading strategies.