Lecture 14 - Applications to Networks

Learning Objectives

• Set up and solve a system of linear equations to analyze flow within a network
• Analyze your network solution to find possible values for any free variable(s)

What is a "Network"?

A network is a set of points called nodes connected by branches:

The total flow into the network equals the total flow out of the network. The total flow into each node equals the total flow out of that node.

Networks can be used to model a variety of real world situations:

• Traffic flow
• Branches are one-way roads
• Nodes are intersections
• Total number of cars entering a neighborhood/city/etc. must equal the total number leaving
• Total number of cars entering an intersection must equal the total number leaving the intersection
• Water flow
• Branches are pipes
• Nodes are intersections of pipes
• Total amount of water entering the pipe system must equal the total flowing out
• Total amount of water flowing into a pipe intersection must equal the total amount flowing out
• Circuits
• Branches are wires
• Nodes are intersections of wires
• Total amount of current entering a circuit must equal the total amount leaving it
• Alternatively, for a closed circuit with a battery, the total rise in voltage (from the battery) equals the total drop (from resistors)
• Total amount of current entering a node must equal the total amount leaving it

An Extended Example

To understand how our linear algebra ideas can help us understand flow in a network, consider this network, representing streets and traffic flow in a neighborhood:

The branches labeled with numbers represent streets where we know the traffic flow. The variable branches are where the flow is unknown. We would like to use the information we have to draw conclusions about the possible flow rates on the variable branches.

The principle that "flow in equals flow out" allows us to set up equations involving these variables. We will have one equation for each node, plus one more equation for the network as a whole.

The equations are:

Now, we rewrite the equations in the standard format, moving the variables to the left-hand side and the numbers to the right-hand side: \[ \begin{eqnarray*} x_1 + x_4 & = & 125 \\ x_1 + x_2 & = & 200 \\ -x_2 + x_3 & = & 75 \\ -x_3 + x_5 & = & 50 \\ -x_4 + x_5 & = & 200 \end{eqnarray*} \]

Next, we construct and row-reduce the corresponding augmented matrix: \[ \begin{bmatrix} 1 & 0 & 0 & 1 & 0 & 125 \\ 1 & 1 & 0 & 0 & 0 & 200 \\ 0 & -1 & 1 & 0 & 0 & 75 \\ 0 & 0 & -1 & 0 & 1 & 50 \\ 0 & 0 & 0 & -1 & 1 & 200 \end{bmatrix} \longrightarrow \begin{bmatrix} 1 & 0 & 0 & 0 & 1 & 325 \\ 0 & 1 & 0 & 0 & -1 & -125 \\ 0 & 0 & 1 & 0 & -1 & -50 \\ 0 & 0 & 0 & 1 & -1 & -200 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \]

The general solution is:

• \( x_1 = -x_5 + 325 \)
• \( x_2 = x_5 - 125 \)
• \( x_3 = x_5 - 50 \)
• \( x_4 = x_5 - 200 \)
• \( x_5 \) is free

Since we have a free variable, there are infinitely many solutions. However, there are still limitations on what the solutions could be, based on the fact that the branches represent one-way streets. This means that the variables cannot be negative.

For example, we cannot have \( x_5 = 400 \), since this would make \( x_1 \) negative. Similarly, we cannot have \( x_5 = 100 \), since then \( x_2 \) would be negative.

The restriction \( x_1 \ge 0 \) gives \( x_5 \le 325 \). The restriction \( x_4 \ge 0 \) gives \( x_5 \ge 200 \). As long as \( 200 \le x_5 \le 325 \), all five variables will be nonnegative.

Summary

When using linear algebra to analyze network flow, you will have on variable for each unknown branch. You will also have one equation for each node, plus one more equation for the entire network.

Keep in mind that the system of equations may not be homogeneous, which means that there may not be any solutions! If the system is consistent, make sure to find solution(s) where all of the variables are nonnegative.