  We consider general, steady, balanced flows of a commodity over a network
where an instance of the network flow is characterized by edge flows and nodal
potentials. Edge flows in and out of a node are assumed to be conserved, thus
representing standard network flow relations. The remaining freedom in the flow
distribution over the network is constrained by potentials so that the
difference of potentials at the head and the tail of an edge is expressed as a
nonlinear function of the edge flow. We consider networks with nodes divided
into three categories: sources that inject flows into the network for a certain
cost, terminals which buy the flow at a fixed price and "internal" customers
each withdrawing an uncertain amount of flow, which has a priority and thus it
is not priced. Our aim is to operate the network such that the profit, i.e.
amount of flow sold to terminals minus cost of injection, is maximized, while
maintaining the potentials within prescribed bounds. We also require that the
operating point is robust with respect to the uncertainty of customers'
withdrawals. In this setting we prove that potentials are monotonic functions
of the withdrawals. This observation enables us to replace in the maximum
profit optimization infinitely many nodal constraints, each representing a
particular value of withdrawal uncertainty, by only two constraints
representing the cases where all nodes with uncertainty consume their minimum
and maximum amounts respectively. We illustrate this general result on example
of the natural gas transmission network. In this enabling example gas
withdrawals by consumers are assumed uncertain, the potentials are gas
pressures squared, the potential drop functions are bilinear in the flow and
its intensity with an added tunable factor representing compression.
