Flow networks: some definitions
Flow network: A flow network is an undirected graph through whose vertices certain
input currents are injected, and those currents are distrbuted through the
edges in the form of flows according to a flow model.
Definition:
Let a flow network be defined as a tuple \((G, \vec{I}, f)\), where :
- \(G\) is a graph with vertex set \(V\) and edge set \(E\).
- \(I \in R^{|V|}\) is a vector specifying input currents at each vertex \(v \in V\)
- \(f\) is a flow model specifying flows \(F_{uv}\) from node \(v\) to node \(u\)
across each edge \((u,v)\in E\):
\[f: E \to R^{|E|}\]\[f: (u,v) \mapsto F_{uv}\]
The flows must obey:
- the continuity equation at each node:
\[\sum_{(u,v)\in E} I_u + F_{uv} = 0\]
- the reflexivity of flows across each edge:
\[F_{uv} = -F_{vu}\]
