# Social Network Homework

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 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 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 other node has a threshold of q = 1/2 for switching to behavior A.

a) If other nodes 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 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 these students needs to decide whether to join, by registering, creating a player account, and taking a few other steps necessary 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 story concrete, let’s suppose that each game goes through the following “life cycle” within this group of students:

(a) The game has some initial players in the group, who have discovered it and are already involved in it.

(b) Each other student 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 dorm and 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 other floor. 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 other students 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 three possible answers to this question: yes, no, or there is not information in the set-up of the question to be able to tell.

a) Say which answer you think is correct: yes, no, or there is 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 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 other hand, 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 other words, you can’t be Unequal in some interactions 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)

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 think that makes it easier or harder for the system to switch over to sharing Equal value? You might think about how this relates to Marx and Engels’ famous statement that “[the] proletarians have nothing to 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 think the adoption curve will look like in country A? Explain your answer. (You can draw the curve if you like). (3 points)

d) What factors will make it easy or hard for rubella vaccination to spread in country B? Explain. (2 points)

e) What do you think the 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 AND an 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, there is 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, there is 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 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 these people 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 steps t. 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. You are totally confident that the disease couldn’t have been introduced into this group from other sources, 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)