Recommended
More Related Content
What's hot
What's hot (20)
Similar to Lesson 7 dominance matrices
Similar to Lesson 7 dominance matrices (20)
More from Jonathan Templin
More from Jonathan Templin (13)
Recently uploaded
Recently uploaded (20)
Lesson 7 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