Flow balance (${\sum }_j{f}_{ij}-{\sum }_j{f}_{ji}=b_i$)

A flow balance prescribes net outgoing minus incoming flow at a vertex. Positive $b_i$ is supply and negative $b_i$ is demand; zero is ordinary flow conservation. Summing all balances requires total supply to equal total demand. In a directed graph, these equations are represented by the oriented incidence matrix.