1. 15. Directed graphs and networks
15A Reachability and dominance in directed networks

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

3. Reachability
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.

4. Matrix representation
Example, Ex 15A, Q.2
You can represent the number of one-stage pathways with a matrix. This matrix is labelled A, and is called the adjacency matrix, or the one-stage matrix.
A matrix of two-stage pathways is labelled A2. 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.

5. Dominance
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.

6. 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

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