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COMS 4995: Intro Networks & Crowds

英国CS作业代写 “tortuous”: this question either requires an advanced notion, a proof that is long or inventive, or it is still open.

Homework #4 – Influence & Representing Graphs  英国CS作业代写

Why  there  is  three  parts  in this assignment: Each part fulfills one of the objectives of the class:

• Manipulateconcepts: Getting Familiar with the technical concepts used in class, by reproducing similar  Being proficient by manipulating the object to answer some small-size problem.

You are expected to answer this question rigorously, the answer can be quite short as long as it contains all the required argument to justify your answer.  英国CS作业代写

• Experience the concepts in real case: Being able to reproduce these concepts on real or synthetic data. Study  their properties in real examples.
• Connectthe concepts to real-life: Interpret a problem you find in light of the notions you have  Develop some critical eye to determine how the concepts introduced are useful in practice.

How to read this assignment : Exercise levels are indicated as follows

(↣) “elementary”: the answer is not strictly speaking obvious, but it fits in a single sentence, and it is an immediate application of results covered in the lectures.

Use them as a checkpoint: it is strongly advised to go back to your notes if the answer to one of these questions does not come to you in a few minutes.

(↷) “intermediary”: The answer to this question is not an immediate translation of results covered in class, it can be deduced from them with a reasonable effort.  英国CS作业代写

Use them as a practice: how far are you from the answer? Do you still feel uncomfortable with some of the notions? which part could you complete quickly?

(↬) “tortuous”: this question either requires an advanced notion, a proof that is long or inventive, or it is still open.

Use them as an inspiration: can you answer any of them? does it bring you to another problem that you can answer or study further? It is recommended to work on this question only AFTER you are done with the rest!

Part A — Manipulating the Concepts  英国CS作业代写

Exercise 1: Game coordination and threshold model (you could skip or complete it for 4 pt of extra homework credit)

Motivation: This exercice allows you to bridge simple coordination games, also referred to as cooper- ative game, with threshold model seen in the course. A coordination game is any type of game where players chooses from the same set of strategies and turn out to have higher payoff when they make the same choice.

Assume that each player can choose either strategy A or B. Strategy A may for example denote the adoption of an innovation. We suppose that players are connected to each other according to a graph  tt = (V, E) where V denotes the set of all players and an edge (u, v) is in E if and only if two players are connected.

Every edge (u, v) corresponds to a game where players u and v are involved. In particular they all receive a pay-off for this game which only depends on which strategy each of this player has chosen according to the following table:

where we assume that a and b are two real numbers such that a > b.

The total payoff receives by  a user is the result of her payoff on all games she is involved in (i.e., ,   she is involved by one game for each incident edge).  英国CS作业代写

1. (↣) Given that all players except u has decided their strategy (which could either be A or B),and that u knows their choices, under which conditions will u rationally choose to play strategy A or B?

We consider an infinite sequence of games indexed by time t = 0, 1, 2, . . . . We assume that at time t a user decides her strategy to maximize her payoff when all other players plays the same strategy as at time t − 1 (which is her last observation).

2. (↣) Let us assume that a set of players always choose to play strategy A independently of others, and that all other players initially play strategy B. How would you describe the evolution of thissystem with time t? Can you compare it to one seen in the course?

Exercise 2: Adoption with neighbor effect and renewed decision (you could skip it or com- plete it for 10pt of homework extra credit)  英国CS作业代写

Motivation Adoption of an innovation (like an online service) could be promoted by encouraging users to start the service for free. One can distinguish a permanent promotion (where users could access the service for free forever) and temporary promotion where they have a free period. Clearly the first form of promotion (which is the one we studied in class) can only do better. On the other hand, and especially for service paid by subscription, the second option seems cheaper overall to organize. The exercise answers the following question “Could this form of promotion be significantly less efficient?”

In this exercise, we propose to show that, according to macroscopic metric (i.e., , the ability for a finite set of players to create an infinite cascade of adoption), the two are equivalent.

As in the previous exercice, we consider a set of users V that are connected together along edges of a graph tt = (V, E). We consider an infinite amount of time slots t = 0, 1, 2, . . . . We assume that all users have an adoption threshold θ which characterizes their behavior as follows: during a time slot t, a user observes how many of her friends used the service and renew her subscription for time slot t + 1 if only if at least a fraction θ of her friend have used the service during time slot t.

In the temporary promotion model we  consider an extreme version where initially a set of users S₀  are proposed to use the service for free for a single time slot.

At  the end of this time slot, they may or  may not renew the service depending on what they have  observed and the threshold rule defined above,  just like any other nodes in the network. The only effect of the promotion is to increase the set of users during the first time slot.  英国CS作业代写

In the targeted permanent promotion model  we  assume that an initial set of users S₀ are proposed  the service for free for an unlimited amount of time, while all other users who may use the service or not decides to do so according to the threshold rule defined above.In the general permanent promotion model, we assume that any user who decides to use the service once will receive a free subscription in all subsequent time slot. All users who have never used the service may decide to adopt it or not according to the threshold rule defined above.

Let us denote by S_t for t = 0, 1, . . . the set of users that decide to subscribe to the service during time slot t in the temporary promotion model.  Let us define for each subset A ⊆ V  the function f_θ as

f_θ(A) = { v ∈ V   |   at least a fraction q ≥ θ of neighbors of v are in A } .  英国CS作业代写

1. (↣)Showthat for any t ≥ 0, S_t =f ^(t)_θ (S₀) where f ^(t)_θ  denotes the function f_θ applied t times.Let us denote respectively by St′ and S^′′_t for t = 0, 1, . . . the set of users that decide to subscribe to

the service during time slot t in the targeted permanent promotion model and the general permanent promotion model.

2. (↷)Show that S^′ =S^′′_t= g^(t)(S₀) where for any subset A ⊆ V , g_θ(A) = f_θ(A) ∪ A.  What can you deduce w.r.t. the efficiency of the general permanent promotion strategy?

We consider an infinite graph tt = (V, E) where all nodes have finite degree. For a given threshold θ, we say that a set S₀ is an infectious set for the temporary promotion model if, starting from S₀ the sequence S_t of nodes that subscribe the service eventually reach all nodes. Formally, S₀ is infectious if

∀v ∈ V, ∃k ≥ 0  such that  ∀t ≥ k, v ∈ S_t .

Note that this definition is shown here for the temporary promotion model (i.e., using the sequence (S_t)_t≥0) and that the same definition could be used with S_t^′ and S^′′_t to define infectious set in the two other models.

3. (↷)Give an example of a graph tt = (V, E) and a set S₀ that is infectious for the general permanent promotion model but not for the temporary promotion

Let S₀ be a finite subset that is infectious for the general permanent promotion model. We  define S⁺  as the subset that contains all nodes in S₀ as well as all neighbors of nodes in  S₀.  Since S₀ is infectious  for the general permanent promotion model, there exists t₀ such that S⁺ ⊆ St′_t0 , where S_t^′ is the sequence of subscribing nodes starting from S₀.

4. (↷)Show that the subset T  = St′0   is infectious for the temporary promotion model (e.,   that the sequence (S_t)_t≥0 starting from T eventually contains the whole set).

Part B — Experiencing the Concepts  英国CS作业代写

Exercise 3: Using network features to predict community membership (20 pt) In this exercise we will predict community membership using node2vec features and structured features extracted from an email communication network. You can download the network here: https://snap.stanford.edu/ data/email-Eu-core.html or find it on coursework. The provided network wasgenerated using email communications from a large European research institution. There is an edge (u, v) in the network if person u sent person v at least one email. Note that we represent the graph as undirected (i.e. person u sending an email to v is equivalent to person v sending an email to u). The code for constructing the network is provided on coursework. Each individual belongs to one of 42 departments (i.e. community membership) at the research institute. The task is to predict which department a certain individual belongs to given the network.

1.(↣)Use some or all of the centrality measures we have discussed in class to predict which de-  partment each person belongs. The code for prediction and evaluation is provided on coursework as ‘1.py’. Report the cross-validation accuracy you  Submit your code as ‘1 sol.py’. Please do not modify the coding framework.

2.(↷) We have discussed biased random walk used in training node2vec in class. In this question youwill compute the transition probabilities for node v given that the walk just moves to v from t. The coding framework is provided on coursework as ‘2.py’. Please fill in the missing

• • d graph[v][t] should contain a list or numpy array containing the normalized probabilities of transitioning to neighbors of v given last-visited node t. The list or numpy array should be ordered byneighbors(v).
  • We normalize the probabilities such that the transition probabilities sum to 1 given v (current node) and t (last-visitednode).  英国CS作业代写

Submit your code as ‘2 sol.py’. Please do not modify the coding framework.

3.(↷) Install node2vec libracy from https://github.com/eliorc/node2vec using pip install node2vec. Generate the embedding using the followingparameters:

• • Node2Vec: dimensions=50, walk length=40, num walks=10, q=4, p=1
  • fit: window=10, min count=1

Use the trained node2vec embedding to predict the community membership and compare the per- formance with accuracy from part 1. Same as part 1, please use the provided prediction and  evaluation code ‘3.py’. Report the cross-validation accuracy. What do you observe? Why do you think structured/node2vec features perform better? Submit your code as ‘3 sol.py’. Please do not modify the coding framework.

4. (↷) Tune the parameters to improve the performance of the node2vec embedding. What do you observe?  Why do you think changing this/these parameter increases the accuracy?  Submit your  code as ‘4py’.

No part C this time.

Part C — Concepts at large

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