Graph Theory (Networks)

1. 4 Graph Theory (Networks) The Mathematics of Relationships

2. Directed Graphs 4.3 • 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

3. 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.

4. 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.

5. Directed Graphs • Solution:

6. Directed Graphs • Example: • Solution: Consider the directed graph shown in the figure. a) What is the length of the directed path ACDE? b) Is ABCE a directed path? 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.

7. 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?

8. 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)

9. 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)

10. 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)

11. Modeling Influence Alvarez is the most influential council member, followed by Davis and Ferraro.

12. Modeling Influence

13. 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.

14. Modeling Disease • Solution: Use a directed graph to model the contact relationships. (continued on next slide)

15. 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)

16. 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.