The information from the digraph can be converted into matrix form (a dominance matrix) as below:

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1 SME Week Investigation Application Dominance Networks Read the information provided and study the examples. Use the information and worked examples provided to answer the questions that follow. A dominance matrix, which can be calculated to represent a dominance network is one in which for every pair of vertices, one either dominates or is dominated by the other. That is: (a) (b) (c) a dominance network cannot be a zero matrix. there can be no loops in a directed network. a vertex cannot dominate itself. Our assumption appears to be reasonable that is: The square of a matrix represents the two stage paths from one vertex to another vertex. Therefore, the cube of a matrix represents the stage paths and the th power of a matrix the stage paths and so on. We can obtain the matrix representing both one stage and two stage paths by adding together the matrices M and M². EXAMPLE : Consider players Alan, Brian, Carlo and Denis (A, B, C, D), who on past performances have shown that A defeats D and B, D defeats B, C defeats A and D, and B defeats C. This situation can be represented on a digraph a network diagram that has arrows on the edges, where AB indicates that A defeats B. The information from the digraph can be converted into matrix form (a dominance matrix) as below: where s are used to indicate defeats and s to indicate otherwise. Obviously A can t defeat A so a is used along the leading diagonal.

2 When the elements of each of the rows are added they yield a dominance vector, showing how many players each has defeated. According the above matrix, there is a tie between A & C for the first position. There is a tie between B & D too. No clear leader can be picked at this stage, so we proceed to the next stage and compute M+M : Step. Calculate the next higher power of M. M² Step. Add the new power of M to the sum of all the lower powers of M. M M² M + M² M + M² + Step. Add the elements in each row. M + M² Here, the dominance vector can be represented by V The position is now clearer: C is the most dominant, followed by A then B and then D.

3 EXAMPLE : This network represents the dominance relationships in a group of people e.g. A is dominant over B. A B D C From the network we can determine the matrix which represents this network. M The larger the sum of the rows, the greater the dominance within the group. Clearly person A is the most dominant but the dominance order between the other three persons is not able to be determined. From the matrix M we can conclude that A is the most dominant person. However, the position of the other three is not clear, nor is the relationship between A and C. ( A dominates most people, but C dominates A). Step. Calculate the next higher power of M. M M M² M² Step. Add the new power of M to the sum of all the lower powers of M. M M² M + M² M + M² + Step. Add the elements in each row. M + M² The dominance order is person A, person C but the dominance order of person B and person D is not able to be determined.

4 The sum of the first two stages, M + M² makes the situation clearer. A is obviously the leader (dominates each of the others in the group in one or two stages). C is obviously pretty important too. Step. Calculate the next higher power of M. M² M M M Step. Add the new power of M to the sum of all the lower powers of M. M + M² M M + M² + M M + M² + M + Step 6. Add the elements in each row. M + M² + M 6 Once the sum of each of the rows is different then there is an order of dominance established within the group. Here, the dominance vector can be represented by 6 V The position is now clearer: A is the most dominant, followed by C then B and then D. In general, to show dominance, we continue to add to M increasing powers of M until the sum of the rows in each case is different.

5 Your task is to: Use dominance matrices to rank the following teams from their performance in last season s A-League (Australian Football) competition after only 8 rounds. The participating teams were: Adelaide United, Brisbane Roar, Central Coast Mariners, Gold Coast United, Melbourne Heart, Melbourne Victory, Newcastle Jets, Perth Glory, Sydney FC, and Wellington Phoenix. Below are the first 8 rounds of the Hyundai A-League fixtures for -. Match day October 8, : pm Ausgrid Stadium Newcastle Jets Melbourne Heart - October 8, 6: pm Etihad Stadium Melbourne Victory Sydney FC - October 8, 7: pm Suncorp Stadium Brisbane Roar Central Coast Mariners - October 9, : pm Skilled Park Gold Coast United Wellington Phoenix - October 9, : pm NIB Stadium Perth Glory Adelaide United - Match day October, 7: pm Hindmarsh Stadium Adelaide United Melbourne Victory - October, : pm Bluetongue Stadium Central Coast Mariners Gold Coast United - October, 7: pm Sydney Football Stadium Sydney FC Brisbane Roar - October 6, : pm Westpac Stadium Wellington Phoenix Newcastle Jets - October 6, : pm AAMI Park Melbourne Heart Perth Glory - Match Day October, 7: pm Suncorp Stadium Brisbane Roar Gold Coast United - October, : pm Hindmarsh Stadium Adelaide United Sydney FC - October, 7: pm Etihad Stadium Melbourne Victory Melbourne Heart - October, : pm NIB Stadium Perth Glory Wellington Phoenix - October, : pm Ausgrid Stadium Newcastle Jets Central Coast Mariners -

6 Match Day October 8, 7: pm Suncorp Stadium Brisbane Roar Adelaide United 7 - October 9, : pm Bluetongue Stadium Central Coast Mariners Perth Glory - October 9, 7: pm AAMI Park Melbourne Heart Sydney FC - October, : pm Skilled Park Gold Coast United Newcastle Jets - October, : pm Westpac Stadium Wellington Phoenix Melbourne Victory - Match Day November, 7: pm Hindmarsh Stadium Adelaide United Melbourne Heart - November, 7: pm Westpac Stadium Wellington Phoenix Central Coast Mariners - November, : pm Ausgrid Stadium Newcastle Jets Perth Glory - November, 7: pm Etihad Stadium Melbourne Victory Brisbane Roar - November 6, : pm Sydney Football Stadium Sydney FC Gold Coast United - Match Day 6: November, 7: pm Hindmarsh Stadium Adelaide United Gold Coast United - November, : pm AAMI Park Melbourne Heart Newcastle Jets - November, 7: pm NIB Stadium Perth Glory Sydney FC - November, 7: pm Bluetongue Stadium Central Coast Mariners Melbourne Victory - November, : pm Suncorp Stadium Brisbane Roar Wellington Phoenix - Match Day 7: November 8, 7: pm Skilled Park Gold Coast United Melbourne Heart - November 9, : pm Eden Park Wellington Phoenix Adelaide United - November 9, : pm Ausgrid Stadium Newcastle Jets Brisbane Roar - November 9, 7: pm Sydney Football Stadium Sydney FC Central Coast Mariners - November, : pm Etihad Stadium Melbourne Victory Perth Glory -

7 Match Day 8: November, 7: pm Hindmarsh Stadium Adelaide United Newcastle Jets - November 6, : pm Bluetongue Stadium Central Coast Mariners Melbourne Heart - November 6, 6: pm Suncorp Stadium Brisbane Roar Perth Glory - November 7, : pm Westpac Stadium Wellington Phoenix Sydney FC - November 7, : pm AAMI Park Melbourne Victory Gold Coast United - Summarise the above results into a dominance matrix, as shown in the above examples, to determine rank of the teams from the best to the worst. Award a to both teams when there is a draw. For games yet to be played also insert a. You may use the Graphics- Calculator/Excel/Online-Calculator for computation purposes. ALL steps/calculations, each with a brief description, must be shown for maximum marks. Graphics Calculators are recommended to assist you with any calculations and an online Calculator is available here: Flinear%Fmatrix.en&+cmdresume&+presetA Final Standings Table after /6 games: Pos Team Pld Pts Central Coast Mariners Brisbane Roar 6 Melbourne Heart Wellington Phoenix 6 Melbourne Victory Sydney FC 9 7 Adelaide United Perth Glory 8 9 Newcastle Jets 7 Gold Coast United Above is the final standings table. Comment on how your predicted rankings compare to the actual results. Does the use of dominance matrices for the first eight rounds provide a good indicator of the final rankings of the teams? Suggest another application where dominance matrices could be used, or indeed are being used, in the sporting field or elsewhere. Football results obtained from: