Circulation in a flow network (${\sum }_j{f}_{ij}={\sum }_j{f}_{ji}\text{ at every }i$)

A network circulation is a flow with zero net supply at every vertex. A feasible residual network circulation changes one feasible flow into another with the same prescribed supplies. It decomposes into directed cycles, giving the negative-cycle optimality test for minimum-cost flow.

A flow on a directed graph satisfying capacity constraints and a specified net inflow $d_v$ at each vertex. Positive $d_v$ denotes consumption and negative $d_v$ supply. Summing all balances cancels each edge once as inflow and once as outflow, so ${\sum }_v{d}_v=0$ is necessary. With balanced demands and no lower bounds, add a source edge of flow network edge capacity $-d_v$ into every supply vertex and an edge of flow network edge capacity $d_v$ from each demand vertex to a new sink. A maximum flow saturating all auxiliary edges exists exactly when the demands are feasible.