Homology on semimodule-valued sheaves naturally generalizes network flows from the setting of numerical capacity constraints to other sorts of constraints (e.g. stochastic, multicommodity). In this talk, we present new work relating the algebraic structure of flows with local network properties and algebraic properties of the ground semiring. For example, a sheaf-valued flow admits an interpretation as an equalizer if the in-degree or out-degree of each vertex is no more than 1 or the stalks are invertible or the stalks are flat - applications include calculational methods. For another example, a constant semiring-valued flow decomposes into loops under certain factorization properties of the semiring. For yet another example, semimodule-valued sheaves under which homology satisfies an exactness property exhibit rigid constraints.