Flow network
directed graph where each edge has a capacity and each edge receives a flow
In graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge. Often in operations research, a directed graph is called a network, the vertices are called nodes and the edges are called arcs. A flow must satisfy the restriction that the amount of flow into a node equals the amount of flow out of it, unless it is a source, which has only outgoing flow, or sink, which has only incoming flow. A flow network can be used to model traffic in a computer network, circulation with demands, fluids in pipes, currents in an electrical circuit, or anything similar in which something travels through a network of nodes. As such, efficient algorithms for solving network flows can also be applied to solve problems that can be reduced to a flow network, including survey design, airline scheduling, image segmentation, and the matching problem.
Definition
A network is a directed graph G = (V, E) with a non-negative capacity function c for each edge, and without multiple arcs (i.e. edges with the same source and target nodes). If (u, v) ∈ E, then (v, u) ∉ E. By convention, when (v, u) ∉ E, we say that c(v, u) = 0.
If two nodes in G are distinguished – one as the source s and the other as the sink t – then (G, c, s, t) is called a flow network.
The public source identifies “Flow network” as directed graph where each edge has a capacity and each edge receives a flow. This brief keeps that definition visible, then builds a research path around Flow, network and directed.
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This entry incorporates text from “Flow network” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.