Module 5 – Networks and Decision Mathematics Chapter 24 – Directed Graphs

24.1 Introduction, reachability and dominance 0 A graph where direction is indicated for every edge is called a directed graph (or digraph). 0 Alternative terms commonly used are network for digraph, node for vertex and arc for edge.

0 E.g. The results of a competition can be represented by the figure below: 0 D defeats A 0 D defeats C 0 A defeats B 0 A defeats C 0 B defeats C 0 This information can be represented with an adjacency matrix.

Reachability 0 As the name suggests, reachability is the concept of how it is possible to go from one vertex in a directed network to another. 0 One point in a network is said to be reachable from another point in a directed graph if a path exists between the two points. 0 For example, in the directed graph shown previously: B is reachable from A in one step along the path AB. C is reachable from A in one step along the path AC. C is also reachable from A in two steps along the path A-B-C. D is not reachable from A.

0 Finally, we can combine this information into a single reachability matrix R 0 We can then calculate the total reachability of each of the vertices by adding each of the columns and recording the sums in table. 3

There is only one way you can get to vertex A. There are two different ways that you can get to B. There are six different ways you can get to C from other vertices. D with a total reachability of 0 is not reachable from any other vertex in the matrix. 6

Dominance 0 If an edge in a directed network moves from A to B, then it can be said that A is dominant, or has a greater influence, over B. 0 We often wish to find the dominant vertex in a network; that is, the vertex that holds the most influence over all the other vertices. 0 This may be clearly seen by inspection, by examining the pathways between the vertices. It is generally the vertex that has the most edges moving away from it. 0 However, the dominant vertex in a directed network isn’t always easily determined by inspection.

0 For example: A group of five tennis players A, B, C, D and E, play each other in a round-robin competition to see who is the best player. 0 The results are as follows: 0 A defeated C and D 0 B defeated A, C and E 0 C defeated D 0 D defeated B 0 E defeated A, C and D 1. Who is the best player? B or E? 2. Note: B and E each had three wins (i.e. there are three arrows leaving each). Who is the best player?

0 Matrix D1 records the number of one-step dominances between the players. 0 Matrix D2 records the number of two-steps dominances between the players. 0 Matrix D3 is not necessary as the top players (B or E) have just three wins. 0 B is the best player (highest dominance)