  We derive conditions for the propagation of monotone ordering properties for
a class of nonlinear parabolic partial differential equation (PDE) systems on
metric graphs. For such systems, PDE equations with a general nonlinear
dissipation term define evolution on each edge, and balance laws create
Kirchhoff-Neumann boundary conditions at the vertices. Initial conditions, as
well as time-varying parameters in the coupling conditions at vertices, provide
an initial value problem (IVP). We first prove that ordering properties of the
solution to the IVP are preserved when the initial conditions and time-varying
coupling law parameters at vertices are appropriately ordered. In addition, we
prove that when monotone ordering is not preserved, the first crossing of
solutions occurs at a graph vertex. We consider the implications for robust
optimal control formulations and real-time monitoring involving uncertain
dynamic flows on networks, and discuss application to subsonic compressible
fluid flow with energy dissipation on physical networks.
