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