4
Graph Theory
(Networks)
The Mathematics of
Relationships
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 1
4.3 Directed Graphs
• Understand how directed graphs
model relationships that go in
only one direction
• Use directed graphs to model
influence
• Model the spread of a disease by
a directed graph
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 2
Directed Graphs
• When an edge has a direction it is called a
directed edge.
• A graph in which all edges are directed is
called a directed graph.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 3
Directed Graphs
• Example:
We want to model how rumors are spread among four people: Lauren,
Heidi, Whitney, and Audrina. Assume that we have gathered the following
information:
1. If Lauren hears a rumor, she will communicate it to Heidi, but Heidi will
not tell rumors she hears to Lauren.
2. Heidi and Whitney will tell any rumors that they hear to each other.
3. Heidi will tell rumors she hears to Audrina, but Audrina does not relay
rumors she hears to Heidi.
Model this situation with a directed graph.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 4
Directed Graphs
• Solution:
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 5
Directed Graphs
• Example:
Consider the directed graph
shown in the figure.
a) What is the length of the
directed path ACDE?
b) Is ABCE a directed path?
• Solution:
a) ACDE is a directed path of length 3 from A to E.
b) ABCE is not a directed path from A to E because
CE has the wrong direction.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 6
Modeling Influence
• Example:
A zoning board has eight members. The table below
shows how each member influences the others. Which
board member has the most influence?
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 7
Modeling Influence
• Solution: Use a directed graph to model
the influence relations.
Because Alvarez and Davis each exert direct influence over
three people, we might say that both are equally influential.
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 8
Modeling Influence
However, let us consider what we
will call two-stage influence.
We observe that Alvarez influences
Cohen, who in turn influences Ellis.
Therefore, we will say that Alvarez
has two-stage influence over Ellis.
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 9
Modeling Influence
We build a table that shows the amount
of one- and two-stage influence for
each pair of board members.
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 10
Modeling Influence
Alvarez is the most influential council
member, followed by Davis and Ferraro.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 11
Modeling Influence
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 12
Modeling Disease
• Example:
A virologist has quarantined eight people who have contracted a deadly
virus and hopes that no others have contracted this virus. He believes that
one person introduced the virus and communicated it to the others. Use
the information in the table to determine if the virologist’s hypothesis is
correct.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 13
Modeling Disease
• Solution: Use a directed graph to model
the contact relationships.
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 14
Modeling Disease
Represent each patient with a vertex and draw a
directed edge from vertex X to vertex Y, if X could
have transmitted the virus to Y.
(continued on next slide)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 15
Modeling Disease
It is impossible for the virus to have started
with Amanda and spread within the group to
Brian. There is no directed path from A to B.
In fact, checking each of the eight people,
we see that it was impossible for the virus
to start with any one of them and then
spread to all the others.
If our information and assumptions are
correct, there is at least one other person in
the city who has the virus but who has not
yet been identified.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section 4.3, Slide 16