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Connectivity Matrices Graphs are very useful for picturing relationships between objects. But graphs do not have to be represented by pictures. They can also be represented by lists or by matrices. For example, this graph B

A

C

D

can be represented by this matrix: To A B C D A 0 1 0 1 B 1 0 1 0 From C 0 1 0 1 D 1 0 1 0

The vertex set or node set is [A B C D] The edge set is [AB, BC, CD, DA]

Representing a graph by a matrix allows the information stored in the graph to be manipulated by a graphic calculator or computer. When graphs are converted to matrices to show how nodes/vertices are connected, they become known as connectivity matrices, which are labelled C. To construct a connectivity matrix there are four points to remember:

• Size of the connectivity matrix – involves a number of rows and elements equivalent to the number of nodes in • • •

the network. The example above had four nodes, therefore the connectivity matrix would be 4 by 4. Connection – each element represents a connection between two nodes and receives a value of one. Non-connection – each element that does not represent a direct connection receives a value of zero. If all connections in the network are bi-directional (movement in both directions from two nodes), the connectivity matrix is transposable.

From connectivity matrices it is possible to determine the degree of a node. This is achieved by adding up a row or a column. What the degree of a node shows is the most connected compared to all other nodes. However, this many not hold true on a more complex network because of a larger number of indirect paths which are not considered in the connectivity matrix. It is also possible to calculate two-stage, three-stage, etc. routes. Multi-stage routes focus on the number of paths possible from one node to another node. Let us consider our original network diagram. B

A

B

C

A

C

Note: Edges without arrows are bi-directional. D

D

If I wanted to calculate the number of paths from node A to node C the possibilities are A → B → C or A → D → C. As there are two possible paths, we call these 2nd stage routes.

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MATRICES

PHYSICS PHYSICS ESSENTIALS ESSENTIALS STAGE STAGE 22 ESSENTIALS STAGE 2 MATHEMATICAL APPLICATIONS

To B C 1 0 0 1 1 0 0 1

D 1 0 1 0

1 0 1 0

0 1 0 1

1 0 1 0

To B C 0 2 2 0 0 2 2 0

D 0 2 0 2

A 0 1 0 1

A B ∴C = From C D then

C2 =

0 1 0 1

1 0 1 0

A B = From C D

0 1 0 1

A 2 0 2 0

Note where “From” and “To” are located on the matrix. The order could be remembered as “From left” “To right”.

0 1 0 1

1 0 1 0

There are 2 two-stage routes from A to C A → B → C and A → D → C

Note: C2 represents two-stage routes. C3 represents three-stage routes. For example, from B to C, B → A → D → C and B → C → D → C routes. → Cn represents n-stage routes. The Matrices C + C2 and C + C2 + C3 What is the significance of C + C2 and C + C2 + C3 within networks? So far, we have understood that C2, C3… Cn represents the two-stage, three-stage and … n-stage routes of a network. We have also understood that if C is a connectivity matrix, then a zero in the matrix indicates no connection between the nodes where as a 1 indicates that the two nodes are connected. It can also be observed that C2 and C3 operate in the same way. That is, zero indicates no connection with nodes in twostage and three-stage routes, whereas a one indicates that two nodes are connected via a third node in C2 and in C3 nodes are connected via two other nodes. B

A

B

C

C2 1 route from A to C i.e. A → B → C (note: 2 connections)

C

A

D

C3 1 route from A to D i.e. A → B → C → D (note: 3 connections)

So what does C + C2 and C + C2 + C3 tell us? It shows potential problems or inefficiencies that can occur in the system, where zero indicates no connection and a 1 tells us there is a connection. Other numbers in the matrix, such as 2 or 3 for example, tell us of an oversupply of possible pathways. The greater the value the more paths and oversupply. It is also important to understand that a high connectivity can have varying meanings. For example, if our network focused on a road network, then oversupply of possible routes would be silly and a waste of resources and money, whereas in a computer network, an oversupply of pathways could be a good thing if one or more connections fail.

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EXAMPLE 1 From a connectivity network, a connectivity matrix is constructed. To A 0 1 0 1

A C= B From C D

B 0 0 1 1

A

C 1 1 0 1

D 1 0 1 0

B D C

The degree of the nodes can be found by adding up the row or column for each node. This is a ONE stage matrix. A 0 1 0 1

A C= B C D

B 0 0 1 1

C 1 1 0 1

D 1 0 1 0

2 2 2 3

Node D is the most connected with a degree of 3. A one-stage matrix shows the number of single routes from one node to another. The route does NOT pass through any other nodes. A→D

B→A

C→B

D→A

A→C

B→C

C→D

D→B

D→C A two stage matrix passes through one node. That is, there are two routes used to move from one node to another.

C2 =

0 1 0 1

A C2 = B C D

0 0 1 1 A 1 0 2 1

1 1 0 1

1 0 1 0 B 2 1 1 1

×

C 1 1 2 2

0 1 0 1 D 1 2 0 2

0 0 1 1

1 1 0 1

1 0 1 0

=

1 0 2 1

2 1 1 1

1 1 2 2

1 2 0 2 A

5 4 5 6

There are now 2 two-stage routes from A to B.

B D C

For example, looking at the two-stage routes from A, the network has A→D→A

1 two-stage route from A to A

A→D→B A→C→B

2 two-stage routes from A to B

A→D→C

1 two-stage route from A to C

A→C→D

1 two-stage route from A to D

When C and C2 are added (C + C2), the resulting matrix will be the total number of one- and two-stage routes from one node to another.

C + C2 =

0 1 0 1

0 0 1 1

1 1 0 1

1 0 1 0

+

1 0 2 1

2 1 1 1

1 1 2 2

1 2 0 2

=

1 1 2 2

2 1 2 2

2 2 2 3

2 2 1 2

There are 3 one- and two-stage routes from D to C.

The 3 one- and two-stage routes from D to C are D → C, D → B → C and D → A → C.

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MATRICES

PHYSICS PHYSICS ESSENTIALS ESSENTIALS STAGE STAGE 22 ESSENTIALS STAGE 2 MATHEMATICAL APPLICATIONS

EXAMPLE 2 A transport company provides the following pickups and deliveries between five major cities on Mondays, Wednesdays and Fridays: Adelaide (A), Darwin (D), Melbourne (M), Sydney (S) and Perth (P). The network below represents the available transport routes:

D S P A

M

(a) Construct a connectivity matrix, C, as a model to describe the transport network above.

(b) Calculate matrix C 2 and explain what information it contains.

(c) Calculate the matrix C + C2 + C3. What information does this contain?

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(d) Describe the possible transport routes from Sydney to Perth. State the transport route(s).

(e) (i) From which city is a driver unable to leave and return without passing through one or two cities?

(ii) What single alteration or addition do you think would improve the system? Give a reason for your answer.

(f ) Describe one limitation of using the matrix model for transport between the five major cities.

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PHYSICS PHYSICS ESSENTIALS ESSENTIALS STAGE STAGE 22 ESSENTIALS STAGE 2 MATHEMATICAL APPLICATIONS

Solution (a) Construct a connectivity matrix, C, as a model to describe the transport network.

D S C=

P M

A

A

A D M 0 0 1

S 0

P 1

D

1

0

0

0

0

M

1

0

0

1

0

S

1

1

1

0

0

P

1

0

0

1

0

(b) Calculate matrix C 2 and explain what information it contains.

C2 =

0 1 1 1 1

0 0 0 1 0

1 0 0 1 0

0 0 1 0 1

1 0 0 0 0

0 1 1 1 1

×

0 0 0 1 0

1 0 0 1 0

0 0 1 0 1

1 0 0 0 0

2 0 1 2 1

=

0 0 1 0 1

0 1 2 1 2

2 0 0 1 0

0 1 1 1 1

C2 represents the number of two-stage routes between two cities. (c) Calculate the matrix C + C2 + C3. What information does this contain?

C3 =

0 1 1 1 1

0 0 0 1 0

1 0 0 1 0

C + C2 + C3 =

0 0 1 0 1

0 1 1 1 1

3

1 0 0 0 0

0 0 0 1 0

1 0 0 1 0

=

2 2 4 3 4

0 0 1 0 1

1 0 0 0 0

2 0 0 1 0

4 0 1 3 1

0 2 3 2 3

2 0 1 2 1

+

2 0 1 2 1

0 0 1 0 1

0 1 2 1 2

2 0 0 1 0

0 1 1 1 1

+

2 2 4 3 4

2 0 0 1 0

4 0 1 3 1

0 2 3 2 3

2 0 1 2 1

A D = M S P

A 4 3 6 6 6

D 2 0 1 2 1

M 5 1 3 5 3

S 2 2 4 3 4

P 3 1 2 3 2

This matrix describes the number of one-stage, two-stage and three-stage transport routes between cities. (d) Describe the possible transport routes from Sydney to Perth. State the transport route(s). There are three transport routes from Sydney to Perth passing through the minimum number of cities, 1 two-stage route and 2 three-stage routes. The routes are S → A → P and S → D → A → P and S → M → A → P. (e) (i) From which city is a driver unable to leave and return without passing through one or two cities? A driver cannot leave and return to Darwin without passing through one or two cities. (ii) What single alteration or addition do you think would improve the system? Give a reason for your answer. Add the transport route Darwin → Sydney. This would allow a return trip to Darwin through Sydney and also significantly increase the access to other cities via Darwin. (f ) Describe one limitation of using the matrix model for transport between the five major cities. The model assumes that all routes are open at all times (not closed due to accident, road works, bushfires etc.).

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