Problems 2.1-2.8
Mathematical Foundations

Questions are given in black, responses are given in blue (and are preceded by a bullet point).

1) For the types of relations listed in Problem 3 in Chapter 1, are the ties implied by these relations directed or undirected?

  • Trade: typically directed

  • Financial transactions: typically directed

  • Children’s preferences: directed

  • Co-attendance: undirected

  • Trust: directed

  • Advice-seeking: directed

  • Confidant: directed, although we expect high reciprocity

  • Lending money: directed

  • Conflict among groups: undirected

  • Who one enjoys: directed

  • Who one wants to work with: directed

  • Sexual relationships: undirected

  • Lab proximity: undirected

  • Observed interactions: as stated, undirected

  • Voting the same: undirected

2) Re-express the simple graph in Figure 2.5 as an adjacency matrix.

Figure 2.5

Adjacency matrix for Figure 2.5

3) Re-express the graph in Figure 2.5 as a geodesic distance matrix. What do those distances mean?

The values in the cells give the length of the shortest path from the row node to the column node. For instance, the 4 in cell (a,g) indicates that the shortest path from node a to node g is 4 links long.

4) For the graph in Figure 2.5, provide an example of each of the following:

a. A path: f-e-c-d

b. A trail that is not a path: a-b-c-e-b

c. A walk that is not a trail: a-b-c-e-b-c-d

5. Re-express the directed valued graph in Figure 2.4 as a valued proximity matrix (values from a node to another node are placed closest to the sending node, so that Ying nominated Juan with a value 3, while Juan nominated Ying with a value 1.

Note: we have assumed that if there is no tie, then there is no value. Often, however, the lack of tie would correspond to a zero value (as in, say, the number of times u called v).

6. For the Hawthorne bank wiring room games graph in Figure 2.2, if the edge between nodes W7 and W5 were removed, how many components would the graph now have? If we were to calculate geodesic distances for the new graph, what would be the distance between W9 and W3? Explain.

Figure 2.2

If the edge between W7 and W5 were removed, there would be 4 components: two large ones and two isolates. The distance between W9 and W3 would be undefined, as there is no path from one to the other. Sometimes, people like to regard that distance as infinite.

7. For each of the network examples in Chapter 1, Problem 3, are the associated matrices one-mode or two-mode?

a) Trade: one-mode

b) Financial transactions: one-mode

c) Children’s preferences: one-mode

d) Attendance at events: two-mode. Co-attendance: one-mode

e) Trust: one-mode

f) Advice-seeking: one-mode

g) Confidant: one-mode

h) Lending money: one-mode

i) Conflict among groups: one-mode

j) Who one enjoys: one-mode

k) Who one wants to work with: one-mode

l) Sexual relationships: one-mode

m) Lab location: two-mode; lab proximity: 1-mode

n) Observed interactions: one-mode

o) Votes on policies: two-mode; voting the same way: 1-mode

8. Given the friendship relation and ‘is the boss of’ relation, use matrix multiplication to hand-calculate the ‘is friends with the boss of’ relation.

Recall that if cell (a, c) has a value greater than 0 in the ‘is friends with the boss of’ matrix, it means that a is friends with c’s boss. From a power perspective, how would you view the row sums of the ‘friend of boss of’ matrix?

Assuming friendship is an undirected relation, if a person is friends with many people’s bosses, they might have some power over those people.