Chapter 4 Networks in Their Surrounding Contexts - Cornell University

116 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS
As a final point, we note that while the model is mathematically precise and selfcontained,
the discussion has been carried out in terms of simulations and qualitative observations.
This is because rigorous mathematical analysis of the Schelling model appears to
be quite difficult, and is largely an open research question. For partial progress on analyzing
properties of the Schelling model, see the work of Young [420], who compares properties of
different arrangements in which all agents are satisfied; Möbius and Rosenblat [302], who
perform a probabilistic analysis; and Vinković and Kirman [401], who develop analogies to
models for the mixing of two liquids and other physical phenomena.
4.6 Exercises
b
a
c
e
Figure 4.20: A social network where triadic closure may occur.
1. Consider the social network represented in Figure 4.20. Suppose that this social network
was obtained by observing a group of people at a particular point in time and
recording all their friendship relations. Now suppose that we come back at some point
in the future and observe it again. According to the theories based on empirical studies
of triadic closure in networks, which new edge is most likely to be present? (I.e. which
pair of nodes, who do not currently have an edge connecting them, are most likely to
be linked by an edge when we return to take the second observation?) Also, give a
brief explanation for your answer.
2. Given a bipartite affiliation graph, showing the membership of people in different social
foci, researchers sometimes create a projected graph on just the people, in which we
join two people when they have a focus in common.
(a) Draw what such a projected graph would look like for the example of memberships
on corporate boards of directors from Figure 4.4. Here the nodes would be the
d