S18 SOCI 111: Social Networks Homework 4 DUE: 6/6/19 81 points total Instructions: As before, these problems give you an opportunity to use the concepts and techniques we’ve learned in class to analyze interesting social systems. This time I’ve leaned more toward interesting conceptual puzzles rather than brute calculations. I encourage you to work in groups (of no more than 5), but each student should prepare her or his own solutions, along with a note at the top of your assignment crediting other members of the working group. Please make sure you truly understand the solution to a problem, and please: DO NOT COPY SOLUTIONS. This includes copying solutions from earlier years. We will figure it out. Do not go to the testbank at SAC, do not go to Course Hero, and do not get solutions from friends who have taken the course in the past. This will impede your learning, it will not prepare you for the final exam, and I will treat it as academic misconduct. In general, you should show your work rather than just writing down a number. This makes it easier for us to see that you know what you are doing, and easier for you to see when you’ve made a trivial mistake. On the other hand, don’t just throw everything vaguely relevant at the wall and hope that something sticks. You need to be judicious! Please turn in a hard copy of your solutions to the TA at the beginning of class. Your submission should be neat and legible: you can type it up or write it out by hand, but we aren’t going to engage in decipherment here. Please follow the breakdown of questions into parts (a, b, c, etc.) to make these easier for your TA to grade. Q1. Why is it sometimes rational to imitate the behavior of others? Briefly describe two reasons, and give an example of each. (6 points) Q2. You are in charge of marketing a new smartphone app, SeeFood, that allows users to share pictures of food with each other. Uptake of the app will obey the following dynamical rule: If the company spends $50,000 on marketing, it can get a fraction z’ of the population to Adopt the app at time 0. This fraction z’ is between B and C—just above B, in fact. If the company spends $300,000 on marketing, it can get a fraction z’’ of the population to Adopt at time 0 (mostly by giving the App away). This fraction z’’ is close to D. You think that the company should spend the $300,000, as it will get you close to total adoption. Your intern argues that you should spend $50,000 on marketing instead. If adoption settles at fraction A, your company will make $0 in revenue; at fraction B, your company will make $100,000 in revenue; at fraction C, your company will make $200,000 in revenue; and at fraction D, your company will make $500,000 in revenue. a) Explain why you are wrong, by showing that you will actually lose money if you spend the $300,000 to get a fraction z’’ of the population to Adopt. Remember, in.

1. S18 SOCI 111: Social Networks
Homework 4
DUE: 6/6/19
81 points total
Instructions: As before, theseproblems give you an
opportunity to use the concepts and
techniques we’ve learned in class to analyze
interesting social systems. This time I’ve
leaned more toward interesting conceptual puzzles
rather than brutecalculations. I
encourage you to work in groups (of no more than 5),
but each student should prepare
her or his own solutions, along with a note at
the top of your assignment crediting
othermembers of the working group. Please make
sure you truly understand the
solution to a problem, and please: DO NOT COPY
SOLUTIONS. This includes copying
solutions from earlier years. We will figure it
out. Do not go to the testbank at SAC, do
not go to Course Hero, and do not get
solutions from friends who have taken the course
in the past. This will impede your learning, it will
not prepare you for the final exam, and
I will treat it as academic misconduct.
In general, you should showyour work rather than
just writing down a number. This
makes it easier for us to see that you know

2. what you are doing, and easier for you to
see when you’ve made a trivial mistake.
On the otherhand, don’t just throw
everything
vaguely relevant at the wall and hope that something
sticks. You need to be judicious!
Please turn in a hard copy of your solutions to
the TA at the beginning of class. Your
submission should be neat and legible: you can
type it up or writeit out by hand, but
we
aren’t going to engage in decipherment here.
Please follow the breakdown of questions
into parts (a, b, c, etc.) to make theseeasier
for your TA to grade.
Q1. Why is it sometimes rational to imitate
the behavior of others? Briefly describe
two reasons, and give an example of each. (6
points)

3. Q2. You are in charge of marketing a new
smartphone app, SeeFood, that allows
usersto share pictures of food with each other.
Uptake of the app will obey the
following dynamicalrule:
If the company spends $50,000 on marketing, it
can get a fraction z’ of the population to
Adopt the app at time 0. This fraction z’ is
between B and C—just above B, in fact. If
the
company spends $300,000 on marketing, it can
get a fraction z’’ of the population to
Adopt at time 0 (mostly by giving the App
away). This fraction z’’ is closeto D. You
think
that the company should spend the $300,000, as it
will get you closeto total adoption.
Yourintern argues that you should spend
$50,000 on marketing instead. If adoption
settles at fraction A, your company will make $0
in revenue; at fraction B, your company
will make $100,000 in revenue; at fraction C,
your company will make $200,000 in
revenue; and at fraction D, your company will make
$500,000 in revenue.

4. a) Explain why you are wrong, by showing that
you will actually lose money
if you spend the $300,000 to get a fraction
z’’ of the population to Adopt.
Remember, in all cases, you need to subtract
the cost of your campaign
from the revenue. (5 points)
b) Show why it is better to follow the intern’s
suggestion and spend $50,000
to get a fraction z’ of the population to
Adopt. You should demonstrate
that this will result in a net profit for your
company. (5 points)
c) Explain why fraction C is a stable
equilibrium. (2 points)
(QUESTION 3) Consider our model of the
diffusion of a new behavior through a social
network. Recall that to define this model, we
have a network, a behavior B that
everyone starts with,and a threshold q for
switching to a new behavior A. Any node will
switch to A if at least a fraction q of its
neighbors have adopted A. If exactly q of its

5. neighbors have adopted A, we assume it adopts A,
too.
Consider the network depicted below; each node starts
with the behavior B.
Q3. Now suppose that the two shaded nodes (D
and I) both adopt the new behavior
A, and each othernode has a threshold of q =
1/2 for switching to behavior A.
a) If othernodes follow the threshold rule for
choosing behaviors, which nodes will
eventually switch to the new behavior A? You do
not have to explain your
answer. (2 points)
b) Describe and explain a possible history of
the system (which nodes adopt at the
first time step, which nodes adopt at the second
time step, etc.).(3 points)
c) Name a cluster of density greater than 1 –
q = 1/2 that blocks behavior A from
spreading to all nodes, starting from D and I, at
threshold q. (2 points)
d) Explain why the density of this cluster is

6. greater than 1 – q = 1/2. (3 points)
Q4. A group of 20 students living on the
third and fourth floors of a college dorm
like
to play online games. When a new game
appears on campus, each of thesestudents
needs to decide whether to join, by registering,
creating a player account, and taking a
few otherstepsnecessary in order to start playing.
When a student evaluates whether to join a
new online game, she bases her decision on
how many of her friends in this group are
involved in the game as well. (Not all pairs of
people in this 20-person group are friends, and it
is more important whether your
friends are playing than whether many people in
the group overall are playing.)
To make the storyconcrete, let’s suppose that each
game goes through the following
“life cycle” within this group of students:

7. (a) The game has someinitial players in the
group, who have discovered it and
are already involved in it.
(b) Each otherstudent outside this set of initial
players is willing to join the game
if at least half of her friends in the group
are playing it.
(c) Rule (b) is applied repeatedly over time,as in
our model from Chapter 19 for
the diffusion of a new behavior through a social
network.
Suppose that in this group of 20 students, 10
live on the third floor of the dormand 10
live on the fourth floor. Suppose that each student
in this group has two friends on their
own floor, and one friend on the otherfloor.
Now, a new game appears, and five
students all living on the fourth floor each begin
playing it.
HINT: Draw a picture of a social
network that satisfies the above description –
this
might help you work through the possible answers.
The question is: if the otherstudents use the rule
above to evaluate whether to join the
game, will this new game eventually be adopted
by all 20 students in the group? There
are threepossible answers to this question: yes, no, or
thereis not information in the
set-up of the question to be able to tell.

8. a) Say which answer you thinkis correct: yes,
no, or thereis not information in
the set-up of the question to be able to tell. (2
points)
b) The spread of the game can be blocked by
a cluster with a certain density p.
Explain what value of p will block the spread
of the game and why. (4 points)
c) Can you find a cluster with that density p in
the network, outside the group
of initial adopters? If so, explain why. If not,
explain why not. (4 points)
Q5. Recall that the threshold q in our diffusion
models can be derived from a
coordination game. In the original version we
studied, the (A,A) equilibrium gives payoff
a to both players and the (B,B) equilibrium
gives payoff b. Now let’s modify this basic
model in two ways. First, suppose that we
have two different types of nodes: Workers
and Bosses. Workers only play with Bosses; Workers
never play with Workers, and

9. Bosses never play with Bosses. Now imagine that
currently, both Workers and Bosses
play strategy Unequal, in which they divide the
total value produced by employment
unequally, so that the Worker gets payoff uW
and the Boss gets payoff uB, which is
greater than uW. Now imagine another strategy Equal
is introduced, in which Workers
and Bosses divide the total value equally, so
that the Worker and the Boss both get the
same payoff e. Now e is greater than uW so in
any specific interaction the Workers would
prefer strategy Equal; on the otherhand, e is
less than uB so the Bosses would prefer
strategy Unequal. Workers and Bosses have to play
the same strategy in every
interaction (in otherwords, you can’tbe Unequal in
someinteractions and Equal in
others); as in the earlier model, if a Worker
and a Boss play a different strategy (e.g.,
the
Worker plays Equal and the Boss players
Unequal) the payoff to each is 0. Note:
this
question seems tricky, but you are doing some
serious, cutting-edge stuff here. And
you can solve it by following the threshold
derivation in class closely. You may also
find it helpful to put in concrete numbers for
the different payoffs, though you should
answer using the general payoffs.
a) First, draw the payoff matrix for the game
played between Workers and
Bosses. (2 points)

10. b) Imagine a Worker who currently plays
Unequal and who interacts with d
Bosses. A fraction 1-p of those Bosses play
Unequal, and a fraction p play
Equal. What is the payoff to the Worker
for playing Unequal? (2 points)
c) Imagine that same worker in part b. What is
the payoff to the Worker for
playing Equal instead? (2 points)
d) What fraction of the Bosses would have to
play Equal before the Worker
would switch from Unequal to Equal? Write
down this threshold as a
function of e and uW. Explain how you figured
out the correct threshold.
(2 points)
e) Imagine that the Worker’s payoff when he
plays Unequal becomes very
small; what does that do to the threshold? (2
points)
f) Do you thinkthat makes it easier or harder
for the system to switch over
to sharing Equal value? You might thinkabout
how this relates to Marx
and Engels’ famous statement that “[the] proletarians
have nothing to

11. lose but their chains.” (1 points)
Q6. In your own words, name and briefly explain
four properties of an innovation
that can affect how easily it diffuses. Give an
example of each property. (12 points)
Q7. You have been put in charge of a child
immunization program for rubella,
covering two different countries. Neither country has
rubella vaccination at the
moment. In country A, vaccination for other
diseases is a well-established practice. In
country B, vaccination for any disease is
completely unknown.
a) In which country is the practice of rubella
vaccination likely to spread
more easily? (2 points)
b) What factors will make it easy or hard for
rubella vaccination to spread in
country A? Explain. (2 points)
c) What do you thinkthe adoption curve will look
like in country A? Explain
your answer. (You can draw the curve if you like).
(3 points)

12. d) What factors will make it easy or hard for
rubella vaccination to spread in
country B? Explain. (2 points)
e) What do you thinkthe adoption curve will look
like in country B? Explain
your answer. (You can draw the curve if you like).
(3 points)
Q8. The world is in danger, menaced by an
epidemic of werewolves ANDan epidemic
of vampires. Suppose that each werewolf comes in
contact with 100 humans before it is
killed (with a silver bullet, naturally).
For each person contacted by a werewolf, thereis
a 3% chance that the victim will become a
werewolf as well. Suppose that each vampire
comes in contact with 200 humans before it is
killed (with a stake, of course). For
each
person contacted by a vampire, thereis a 2%
chance that the victim will become a
vampire.
a) Is it likely that the epidemic of werewolves
will continue to spread?
Explain why or why not. (4 points)
b) Is it likely that the epidemic of vampires will
continue to spread? Explain

13. why or why not. (4 points)
c) Which is the greater danger to the world?
Werewolves or vampires?
Explain your answer. (2 points)
Suppose you are studying the spread of a rare
disease among the set of people pictured
below:
The contacts among thesepeople are as depicted in
the network in the figure, with a
time interval on each edge showing when the period
of contact occurred. Assume that
the period of observation runs from time 0 to
time 20. For the sake of concreteness,
assume each time step is a week. Assume also
that an infected individual is no longer
infections (i.e., capable of spreading the disease) after an
unknown number of time
stepst. This period is the same for all individuals.
Note that the disease is not necessarily
transmitted when two individuals are in contact,
even if one is infectious.
Q9. Suppose that v is the only individual who had
the disease at time 0. Suppose also
that you know that v got the disease at time 0
exactly. At the end of the observation
period, every individual except x has the disease.

14. You are totally confident that the
disease couldn’t have been introduced into this group
from othersources, so you know
it must have spread through the contacts represented
in this network.
a) Is the observed pattern of spreading consistent with a
recovery time
t = 5? Explain why or why not. (3 points)
b) Is the observed pattern of spreading consistent with a
recovery time
t = 6? Explain why or why not. (3 points)
c) Is the observed pattern of spreading consistent with a
recovery time
t = 7? Explain why or why not. (3 points)
d) What is the minimal recovery time consistent with
the observed pattern
of spreading? (1 point)
686 CHAPTER 21. EPIDEMICS
v
u x
y
z

15. [1,3]
[5,9]
[14,18]
[12,16]
w
s
[7,12]
[4,8]
[10,16]
Figure 21.22: Contacts among a set of people, with time
intervals showing when the contacts
occurred.
(b) Suppose that you find, in fact, that all nodes have the
disease at time 20. You’re
fairly certain that the disease couldn’t have been introduced into
this group from
other sources, and so you suspect instead that a value you’re
using as the start
or end of one of the time intervals is incorrect. Can you find a
single number,
designating the start or end of one of the time intervals, that

16. you could change
so that in the resulting network, it’s possible for the disease to
have flowed from
s to every other node?
a b c d e
Figure 21.23: A contact graph on five people.
2. Imagine that you know a contact graph on a set of people, but
you don’t know exactly
the times during which contacts happened. Suppose you have a
hypothesis that a
particular disease passed between certain pairs of people, but
not between certain other
pairs. (Let’s call the first set of pairs positive, and the second
set of pairs negative.)
It’s natural to ask whether it’s possible to find a set of time
intervals for the edges that
support this hypothesis in a strong sense: they make it possible
for the disease to flow
between the positive pairs, but not between the negative pairs.
Let’s try this genre of question out on the simple contact graph
among five people
shown in Figure 21.23.