Flows with Negative Values?

Define a "non-standard" flow to be a flow where the quantity flowing through an edge may be negative.

Formally, given a directed graph $G$, and two designated and distinct vertices $s$ and $t$ (such that no edges are leaving $t$, and no edges are entering $s$), a non-standard flow $f : G_E \to \mathbb{R}$ is a valuation on the edges such that

For each $v \in G_V\setminus \{s, t\}$, the sum of the values of $f$ on the edges entering $v$ is equal to the sum of the values of $f$ on the edges leaving $v$.

Assuming all the capacities are positive, does the Max-Flow Min-Cut theorem still hold for non-standard flows?

Furthermore, is there a network with positive edge capacities, such that the maximum non-standard flow through the network is larger than the maximum (standard) flow through the network?

The standard proof for Max-Flow Min-Cut theorem uses the non-negativity of the flow to claim that each flow is smaller than each cut. I would guess that this tells us that the counterexample (if any) would have a cut that has negative flow into its incoming edges.

• I think you could transform your negative-flow graph into a normal flow graph by simply adding reversed copies of all the edges. Move the negative flows to the new edges (with negated values) and you're set. In a more specific (non-abstract) problem, you might need to consider if the capacities of the reversed edges should be always identical to the capacities of the corresponding original edges, but I suspect if that wasn't the case you wouldn't have been able to have negative flow values in the first place. – Blckknght Nov 21 '17 at 1:21
• @Blckknght But that is not the answer I am looking for. I want to know if there is a graph where the usual max flow is smaller than the nonstandard max flow on the same graph. – Agnishom Chattopadhyay Nov 21 '17 at 3:10