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15. Directed graphs and networksPowerPoint Presentation

15. Directed graphs and networks

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15. Directed graphs and networks

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15. Directed graphs and networks

15A Reachability and dominance in directed networks

- A directed graph is a graph or network where every edge has a direction.

- Reachability is the concept of how it is possible to go from one vertex in a directed network to another.
- Example, Ex 15A, Q.1
- A one-stage pathway is one that includes one edge only.
- A two-stage pathway is one that includes two edges only.
- The indegree is the number of edges moving into a vertex and the outdegree is the number of edges moving away from a vertex.

- Example, Ex 15A, Q.2
- You can represent the number of one-stage pathways with a matrix. This matrix is labelledA, and is called the adjacency matrix, or the one-stage matrix.
- A matrix of two-stage pathways is labelledA2. In Further Maths, we only use matrices up to two-stage pathways.
- The sum of each row is equal to the outdegree of each vertex. The sum of each column is equal to the indegree of each vertex.

- If there are more ways to go from A to B than there are to go from B to A, we say that A is dominant over B. Similarly, if B has edges moving to C and D, then B is dominant over C and D. The vertices in order of dominance would be A then B then C and then D.

- Sometimes, it is not obvious which is the dominant vertex.
- If we add the matrix A (the one-stage matrix) to the matrix A2 (the two-stage matrix) we can work out the order of dominance by summing the rows of the resultant matrix. The row with the highest sum is the dominant matrix.
- Example, Ex 15A, Q. 6

- Ex 15A, Q. 3, 4, 5, 7, 8, 10