**Applications to Networks**

** Introduction**
A Network consists of branches and nodes. A typical example is the street
network where the branches are the streets and the nodes are the intersections.
Another example is an electrical network. Many network problems can be modeled
by systems of linear equations. The basic laws are explained below.

_{1)}_{ Electrical Networks}

In such networks, Ohm’s law and Kirchhoff’s laws govern current flow, as follows:

** **

**Ohm’s Law****: **The voltage drop across a resistor is the product of the
current and the resistance: *V=IR*

**Kirchhoff’s first Law:** The sum of the currents
flowing into a node is equal to the sum of the current flowing out.

**Kirchhoff’s second Law****: **The algebraic sum of the voltage drops
around a closed loop is equal to the total voltage in the loop.

** Example** Determine the currents I

Applying Kirchhoff’s first Law to either of the nodes B or
C, we find *I _{1}=I_{2}+I_{3}*

*I _{1}-I_{2}-I_{3
}=0.*

Applying Kirchhoff’s second Law to the loops BDCB and BCAB, we obtain the equations

*-10I _{1}+10I_{2}=10*.

*20I _{1}+10I_{2
}=5.*

This gives a linear system of three equations

_{}

The augmented matrix of the above system is

_{}

which can be reduced to

_{}

Therefore, the currents are:

_{}_{}

Since I_{3} is negative, the current flow is from C
to B rather than B to C, as tentatively assigned in the above diagram.

_{2)}_{ Traffic Networks}

The diagram below represents the traffic flow through a certain block of streets. (The numbers are the average flows into and out of the network at peak traffic hours)

By Kirchhoff’s first Law, the flow into an intersection is equal to the flow out. This gives the following system

_{}

_{} _{}_{}

The augmented matrix of the above system is

_{}_{}_{}_{}

Some calculations show that this matrix is row equivalent to

_{}

Therefore, the solution can be written as

_{}

For instance, if w=300 and t=1300 (in vehicles per hour),
then _{}

Suppose now that the streets from A to B and from B to C must be closed (for construction for instance), that is x=0 and y=0. How might the traffic be rerouted?

To answer this question, set x=y=0 in the above solution, we
get _{} _{}, _{}, _{} and _{}_{}. Of course, the negative value for _{}is not normal. In order to avoid negative flow, we must
reverse directions on the streets connecting C and D; this change makes _{} instead of _{}.