Network Matrices

Today in class we talked about NETWORK MATRICES. These matrices are used in the real world. Whether it is cables between computers, wries between telephones, railway networks, airplane routes between major centers, there is always a connection between places and things.

Connections may be achieved by "one-hop", "two-hop", "three-hop", "four-hop", etc.

I'll use airplanes as an example.

A "one-hop" is a direct connection between the two cities.

ex: one airplane trip (one hop) from Atlanta to Boston

A "two-hop" is the 2 different ways from one city to another with 2 different airplanes.

ex: one airplane trip (one hop) from Atlanta to Boston then another airplane trip (2 hop) either back to Atlanta, or Charlotte.

A "three-hop" is the 3 different ways from one city to another with 3 different airplanes.

ex: one airplane trip (one hope) from Atlanta to Boston, then another airplane trip (2 hop) to Charlotte, then another airplane trip (3 hop) back to Atlanta.

And so on...

For example:

We can create a "connectivity matrix" showing the direct routes of the network on the left side of this page.

We can create a matrix that represents the connection between A, B, and C. In this matrix we will need 3 rows and 3 columns.

When there is one direct route, you place a "1" in any cell which indicates a direct connection between two places. For example there is one route from A to B, so a "1" could go into the cell that connects A to B. When there is an indirect route, you place a "0" in any cell which indicates no direct connection. In this case, there is no route from A to C, so therefore in the cell that connects A to C, you would place a "0"

The matrix for the image above should be:

We also learned how to indicate where there is a "2 hop" route between places. In our example, is there a 2 hop route between A and A? YES! Because you are able to take a plane to Boston, and take another plane back to Atlanta. So in a matrix that indicates "2 hops", you'd place a "1" showing that there is a "2 hop" route between A and A. In our case, there is no 2 hop route from A to B. Because there is a plane that takes you from A to B, but the next plane will take you somewhere else besides B. So you'd place a "0" in the cell connecting A and B for "2 hops"

The matrix indicating "2 hop" routes should look like this:

Whatever number a connectivity matrix is raised to, will tell you how many different routes there are.

For example, if the connectivity matrix is raised to the power of 2, the result of that would tell us the number of ways from one place to another with 2 airplanes.

If the connectivity matrix is raised to power of 3, the result of that would tell us the number of ways from one place to another with 3 airplanes. ETC.

And tomorrow's scribe is....... JHAY-AR!!

## Thursday, February 8, 2007

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Jen, you've done a good job of summarizing what we learned today very succinctly. A number of your classmates today seemed to be struggling with understanding what the exponent on the first connectivity matrix tells us ... you did a really good job explaining here.

ReplyDeleteWay to go!

clap! clap!

ReplyDeleteJENNIFER!! =) good job on your FIRST scribe post.

ReplyDeleteLOVE MATH,

Lindsay