Dominance Matrices

2. Learning Intention and Success
Criteria
 Learning Intention: Students will understand the
meaning of the word dominance with regards to
matrices, and will be able to apply their knowledge of
graphs to convert to matrices
 Success criteria: Students can apply their knowledge
of dominance to produce a ranking of teams in a
round-robin tournament

3. Graphs to Matrices
 The graph to the left
represents the
relationship between
four points, called
vertices (one vertex)
 Each line connecting the
vertices is called an edge
 Directed Graph: Each
edge has a direction to it
(from A to B or from B to
A, but not both)

4. More Graph Vocabulary
 Degree: The number of
edges a vertex has attached
to it
 Ex. The degree of A is 4
 In-degree: The number of
directed edges that go into
the vertex
 Ex. The in-degree of D is 1
 Out-degree: The number
of directed edges that go
out from the vertex
 Ex. The out-degree of C is
2

5. One-stage pathways
 One-stage pathway: a
path from one vertex to
another that goes through
exactly one edge
 Ex. There are two one-
stage pathways from B to
C
 A matrix showing the
number one-stage
pathways is a matrix
showing edges from one
vertex to another

6. One-Stage Pathways
𝑇𝑜
𝐴 𝐵 𝐶
𝐴
𝐹𝑟𝑜𝑚 𝐵
𝐶

7. One-Stage Pathways
𝑇𝑜
𝐴 𝐵 𝐶
𝐴
𝐹𝑟𝑜𝑚 𝐵
𝐶
0

8. One-Stage Pathways
𝑇𝑜
𝐴 𝐵 𝐶
𝐴
𝐹𝑟𝑜𝑚 𝐵
𝐶
0 1

9. One-Stage Pathways
𝑇𝑜
𝐴 𝐵 𝐶
𝐴
𝐹𝑟𝑜𝑚 𝐵
𝐶
0 1 1

10. One-Stage Pathways
𝑇𝑜
𝐴 𝐵 𝐶
𝐴
𝐹𝑟𝑜𝑚 𝐵
𝐶
0 1 1
1 0 2

11. One-Stage Pathways
𝑇𝑜
𝐴 𝐵 𝐶
𝐴
𝐹𝑟𝑜𝑚 𝐵
𝐶
0 1 1
1 0 2
1 0 0
The rows of a matrix when
looking at pathways always
represent the “from”, and
the columns represent “to”

12. Two-Stage Pathways
 Two-stage pathway: a
path from one vertex to
another that goes through
exactly two edges
 Ex. There is one two-stage
pathway from B to C
(through A)
 Create a matrix showing
the number of two-stage
pathways from each vertex
to another

13. Two-Stage Pathways
𝑇𝑜
𝐴 𝐵 𝐶
𝐴
𝐹𝑟𝑜𝑚 𝐵
𝐶

14. Two-Stage Pathways
𝑇𝑜
𝐴 𝐵 𝐶
𝐴
𝐹𝑟𝑜𝑚 𝐵
𝐶
2
We cannot go from 𝐴 to 𝐵 in two
steps (it would have to go
through 𝐶)

15. Two-Stage Pathways
𝑇𝑜
𝐴 𝐵 𝐶
𝐴
𝐹𝑟𝑜𝑚 𝐵
𝐶
2 0
The pathways from 𝐴 to 𝐴 are:
• 𝐴 to 𝐵 and back to 𝐴
• 𝐴 to 𝐶 and back to 𝐴

16. Two-Stage Pathways
𝑇𝑜
𝐴 𝐵 𝐶
𝐴
𝐹𝑟𝑜𝑚 𝐵
𝐶
2 0 2

17. Two-Stage Pathways
𝑇𝑜
𝐴 𝐵 𝐶
𝐴
𝐹𝑟𝑜𝑚 𝐵
𝐶
2 0 2
2 1 1

18. Two-Stage Pathways
𝑇𝑜
𝐴 𝐵 𝐶
𝐴
𝐹𝑟𝑜𝑚 𝐵
𝐶
2 0 2
2 1 1
0 1 1

19. One and Two Stage Pathways
𝑇𝑜
𝐴 𝐵 𝐶
𝐴
𝐹𝑟𝑜𝑚 𝐵
𝐶
0 1 1
1 0 2
1 0 0
One-stage pathways
𝑇𝑜
𝐴 𝐵 𝐶
𝐴
𝐹𝑟𝑜𝑚 𝐵
𝐶
2 0 2
2 1 1
0 1 1
Two-stage pathways
What is the relationship between the two matrices we have
created?

20. 2 0 2
2 1 1
0 1 1
=
0 1 1
1 0 2
1 0 0
2
In general:
If a matrix of one-stage pathways is called 𝐴:
 Note that the sum of the columns of 𝐴 represents the in-
degrees of the vertices, and the sum of the rows represents
the out-degrees
 The matrix showing the number of two stage pathways
matrix is 𝐴2
 The matrix showing the number of n-stage pathways is 𝐴 𝑛

21. Which Team is Best?
Winner Loser
Croatia Spain
Croatia Lithuania
Argentina Croatia
Croatia Brazil
Nigeria Croatia
Spain Lithuania
Spain Argentina
Brazil Spain
Spain Nigeria
Lithuania Argentina
Lithuania Brazil
Lithuania Nigeria
Argentina Brazil
Argentina Nigeria
Brazil Nigeria
Country Abbreviatio
n
Argentina A
Brazil B
Croatia C
Lithuania L
Nigeria N
Spain S
*Actual data from the Rio Olympics Men’s Basketball tournament

22. Wins and Losses
Country Number of Wins Number of Losses Rank
Argentina 3 2
Brazil 2 3
Croatia 3 2
Lithuania 3 2
Nigeria 1 4
Spain 3 2
?

23. Dominance
 If there are more edges going from A to B than from B
to A, then we say the A is more dominant than B.
 In general, how do we determine the most dominant
vertex in a matrix
 Generally, a vertex with a large out-degree will be more
dominant than one with a large in-degree
 We can use the one-stage pathways matrix to try and
determine the dominance of a network, by adding up
the rows
 The one-stage pathways matrix is called a one-step
dominance matrix

24. Create a Directed Graph and a Matrix to Represent This Situation
Winner Loser
Croatia Spain
Croatia Lithuania
Argentina Croatia
Croatia Brazil
Nigeria Croatia
Spain Lithuania
Spain Argentina
Brazil Spain
Spain Nigeria
Lithuania Argentina
Lithuania Brazil
Lithuania Nigeria
Argentina Brazil
Argentina Nigeria
Brazil Nigeria
Country Abbreviatio
n
Argentina A
Brazil B
Croatia C
Lithuania L
Nigeria N
Spain S
*Actual data from the Rio Olympics Men’s Basketball tournament

25. Example
𝐿𝑜𝑠𝑒𝑟𝑠
𝐴 𝐵 𝐶 𝐿 𝑁 𝑆
𝑊 𝐴
𝑖 𝐵
𝑛 𝐶
𝑛 𝐿
𝑒 𝑁
𝑟 𝑆

26. Example
𝐿𝑜𝑠𝑒𝑟𝑠
𝐴 𝐵 𝐶 𝐿 𝑁 𝑆
𝑊 𝐴
𝑖 𝐵
𝑛 𝐶
𝑛 𝐿
𝑒 𝑁
𝑟 𝑆
0 1 1 0 1 0

27. Example
𝐿𝑜𝑠𝑒𝑟𝑠
𝐴 𝐵 𝐶 𝐿 𝑁 𝑆
𝑊 𝐴
𝑖 𝐵
𝑛 𝐶
𝑛 𝐿
𝑒 𝑁
𝑟 𝑆
0 1 1 0 1 0
0 0 0 0 1 1
0 1 0 1 0 1
1 1 0 0 1 0
0 0 1 0 0 0
1 0 0 1 1 0

28. Example
One-stage Dominance
Matrix:
 The sum of each row
represents the number of
wins that each team has
had
𝐿𝑜𝑠𝑒𝑟𝑠
𝐴 𝐵 𝐶 𝐿 𝑁 𝑆
𝑊 𝐴
𝑖 𝐵
𝑛 𝐶
𝑛 𝐿
𝑒 𝑁
𝑟 𝑆
0 1 1 0 1 0
0 0 0 0 1 1
0 1 0 1 0 1
1 1 0 0 1 0
0 0 1 0 0 0
1 0 0 1 1 0

29. Breaking the tie
What happens when multiple vertices have the same out-
degree?
 We then look at the next level of dominance, which is the
number of two-stage pathways from one vertex to another,
 This is the matrix 𝐴2
 Called the two-step dominance matrix
 The overall dominance is found using the matrix 𝐴 + 𝐴2
 Add up all the rows of 𝐴 + 𝐴2: The highest score is the best
ranking

30. Example
One-stage dominance
Matrix
Two-stage dominance matrix
𝐴 =
0 1 1 0 1 0
0 0 0 0 1 1
0 1 0 1 0 1
1 1 0 0 1 0
0 0 1 0 0 0
1 0 0 1 1 0
𝐴2
=
0 1 1 1 1 2
1 0 1 1 1 0
2 1 0 1 3 1
0 1 2 0 2 1
0 1 0 1 0 1
1 2 2 0 2 0

31. Total Dominance Matrix
𝐴 + 𝐴2
=
0 2 2 1 2 2
1 0 1 1 2 1
2 2 0 2 3 2
1 2 2 0 3 1
0 1 1 1 0 1
2 2 2 1 3 0

32. Total Dominance Matrix
𝐴 + 𝐴2
=
0 2 2 1 2 2
1 0 1 1 2 1
2 2 0 2 3 2
1 2 2 0 3 1
0 1 1 1 0 1
2 2 2 1 3 0
Sum of the rows gives us
our ranking
𝐴
𝐵
𝐶
𝐿
𝑁
𝑆
9
6
13
9
4
10
3𝑟𝑑
5𝑡ℎ
1𝑠𝑡
3𝑟𝑑
6𝑡ℎ
2𝑛𝑑
The final ranking of the teams was C, S, (A and L), B, N