Applied 40S February 9, 2009

More About Networks & Connectivity Matrices

The FlickrVerse by flickr user GustavoG

Write a matrix to summarize each network.

A B

C D

W

X Y

Draw each network.

(a) On the soap opera The Young and the Mathematical, Ann is in love with Buck and Dash. Dash is in love with Ann, Charlotte and Elvira. Charlotte, who just broke up with her fiancée after learning that he was dating the University of Winnipeg women’s basketball team, has fallen in love with Frank, not realizing that he is actually her step-brother who has returned to Winnipeg with amnesia and major plastic surgery after the tragic explosion of a graphing calculator at a secret CSIS cryptography lab. Frank is in love with Elvira. Buck is in love only with himself. Write a connection matrix detailing the relationships among the six named characters. Include labels or other explanations to make your work clear.

(b) Suppose that you made a connection matrix describing the relationships of all 23 characters on the soap opera. How big would the matrix be?

Matrix Multiplication

CD =

DC =

Matrix Multiplication

DI =

ID =

Solve for x and y.

One hop routes ...

rabbit

Two hop routes ...

Two headed bunny rabbit

=

=

Three hop routes ...

Rabbits

One hop routes ...

A more realistic example ...

Two hop routes ...

Three hop routes ...

Four hop routes ...

Campfire CamaraderieBear, Squirrel, Rabbit, Deer, and Owl are seating themselves around a campfire to roast marshmallows. However, they have to be careful about who sits where!

Question: In what order are they sitting around the campfire?

• Owl is allergic to chestnuts so he cannot be next to Squirrel because she smells like them.

• Squirrel's arms are really short so Bear is next to her to help with the marshmallows.

• Every time Owl hoots Deer gets nervous so they do not sit next to each other.

• Bear's favorite snack is rabbit, therefore they cannot sit together.

A matrix describing the connections between five websites (call them 1, 2, 3, 4, 5) is given below.

(a) How many links are there on website 1?

(c) Suppose that you are on website 1. Describe a path that would take you to website 3.

(b) How many sites have links to website 5?

HOMEWORK

Google maintains a connection matrix that tracks links between all the websites that are indexed by Google. Does the matrix contain mostly 1’s or mostly 0’s? Explain. HOMEWORK

HOMEWORK

The diagram shows a network that connects computer terminals in 5 offices. The arrows indicate the directions in which messages can be sent. A B

C D E

(a) Write the matrix, C, for the communication network.

(b) Find the matrix taht represents the number of ways a message can be sent from one office to another, using exactly one relay.

(c) Find the matrix that represents the number of ways a message can be sent from one office to another, using at most one relay.

Air freight is transported among several cities in Western Canada, with Calgary as the hub. This means all air freight either originates in Calgary or is transported through Calgary. HOMEWORK

(a) Write a matrix, A, to represent the network.

Vancouver Calgary

EdmontonWhitehorse

(b) Calculate the matrix A . What do the elements in this matrix represent?

2

(c) Explain why there is one element equal to 3, and why there are nine elements equal to 1, and six elements equal to 0 in the matrix A .2