Network flow

What is a network flow?

A network consists of a set of points called junctions, all nodes, with lines or arcs called branches connecting some or all of the junctions. The direction of flow in each branch is indicated, and the flow amount or rate is either shown or denoted by a variable.

Systems of linear equations arise naturally when scientists engineers or economists study the flow of some quantity through a network. For instance, urban planners and traffic engineers monitor the pattern of traffic flow in a grid of City streets. Electrical engineers calculate current flow through electrical circuits. And economics analyzes the distribution of products and manufacturers to consumers through a network of wholesalers and retailers. For many networks, the systems of equations involve hundreds or even thousands of variables and equations.

The basic assumption of network flow is that the total flow into the network equals the total flow out of the network and that the total flow into a junction equals the total flow out of the junction for example 30 units flowing into a junction through one branch, with x₁ and x₂denoting the flows out of the junction through other branches. Since the floor is conserved at each junction, we must have x₁ + x₂= 30. in a similar fashion the floor at a junction is described by a linear equation. The problem of network analysis is to determine the floor in each branch when partial information such as the input to the network is known.

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