  We derive conditions for monotonicity properties that characterize general
flows of a commodity over a network, where the flow is described by potential
and flow dynamics on the edges, as well as potential continuity and
Kirchhoff-Neumann mass balance requirements at nodes. The transported commodity
may be injected or withdrawn at any of the network nodes, and its movement
throughout the network is controlled by nodal actuators. For a class of
dissipative nonlinear parabolic partial differential equation (PDE) systems on
networks, we derive conditions for monotonicity properties in steady-state
flow, as well as for propagation of monotone ordering of states with respect to
time-varying boundary condition parameters. In the latter case, initial
conditions, as well as time-varying parameters in the coupling conditions at
vertices, provide an initial boundary value problem (IBVP). We prove that
ordering properties of the solution to the IBVP are preserved when the initial
conditions and the parameters of the time-varying coupling law are
appropriately ordered. Then, we prove that when monotone ordering is not
preserved, the first crossing of solutions occurs at a network node. We
consider the implications for robust optimization and optimal control
formulations and real-time monitoring of uncertain dynamic flows on networks,
and discuss application to subsonic compressible fluid flow with energy
dissipation on physical networks. The main result and monitoring policy are
demonstrated for gas pipeline test networks and a case study using data
corresponding to a real working system. We propose applications of this general
result to the control and monitoring of natural gas transmission networks.
