CS4980 Homework 3 Solution

Reminders:
1. Out of 100 points. 3 Questions. Contains 4 pages.
2. If you use Late days, mark how many you are using (out of maximum 4 available) at the top ofyour answer PDF.
3. There could be more than one correct answer. We shall accept them all.
4. Whenever you are making an assumption, please state it clearly.
5. You will submit a solution pdf LASTNAME.pdf containing your answers and the plots as well as a zip file LASTNAME.zip that contains your code and any output files.
7. Additionally, you will submit one zip file LASTNAME.zip that contains your code and any results files. Code and results for each question should be contained in a separate sub-directory (Eg: Q1) and there should be a README.txt file for each sub-directory explaining any packages to install, command to run the code files and location of the expected output. Please follow the naming convention strictly.
8. If a question asks you to submit code please enter the file path (Eg: Q1/Q-1.3.1.py) in the solution pdf.
9. If needed, you can download all the datasets needed for this homework from ICON. Informationabout the datasets will be given in the README.txt file.
1. (30 points) Submodularity
Q 1.1 (7 points) Prove that the coverage function is a monotone submodular function. Recall that the coverage function takes a collection of sets and outputs the size of their union |S1 ∪ S2 ∪ ··· ∪ Sn|.
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Q 1.2 (8 points) Let f be a monotone submodular function. Show that for any two sets S and T: f(T) − f(S) ≤ Pe∈T (f(S + e) − f(S)).
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Q 1.3 (8 points) Let f be a monotone submodular function. Show that the following function is also submodular: h(A) = min(f(A),f(V/A)) where V is the universal set of all elements.
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Q 1.4 (7 points) Let f be a monotone submodular function and let g be a concave function. Show that the following function is also submodular: h(A) = g(f(A)).
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2. (40 points) Sensors for detection in Network Model
We are going to empirically look at various strategies of selecting social network sensors to detect epidemic outbreaks. We will use the graph in facebook.txt which contains a small subset of friendship network of a Facebook group. We will use SI model with β = 0.005 to simulate the infection.
Q 2.1 (6 points) As a warmup, simulate the SI model for upto T = 100 time-steps for 100 runs and plot the average fraction of S,I vs t curve. Select 4 nodes at random to be infected at t = 0. Also plot the average number of daily infected people for t = 0,...,200. Report the average time t∗ at which the maximum daily infections is maximum.
Hint: Use the implementation of SI model in sis model.py file. If you use another language you can convert the logic of the code in the file.
Note:
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Q 2.2 (15 points) Since the graphs are large, it is not always feasible to keep track of the states of all nodes in practice. Therefore, we will select a smaller subset of nodes to track during the epidemic which we call as sensors. We will implement three strategies for sensor selection:
• RANDOM: We choose k nodes uniformly at from the graph
• FRIENDS: We choose k nodes uniformly at random and for each of them we select a random friend. We will use these friends as the sensors.
• CENTRAL: We select the top k nodes with largest eigenvector centrality 2. You can use functions like nx.eigenvector centrality numpy.
Set k = 100 for all strategies. The file rand nodes.npy contains a random list of nodes. Select the first k nodes from the list to implement the RANDOM strategy. For FRIENDS strategy, use the k selected nodes from RANDOM strategy to randomly select their friends. Simulate the SI model for T = 100 steps and note the fraction I˜(t) of the sensors that are infected at each time-step for each strategy. Note that sensors can also be infected at t = 0.
Plot average I˜(t) vs t for all three strategies averaged over 20 runs. Also plot the average number of daily infections I˜d(t) = I˜(t) − I˜(t − 1) over time.
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Q 2.3 (6 points) Report the peak time t˜∗ and peak daily infection for all 3 strategies (the time t where I˜d(t) is maximum is peak time).
The value t∗ − t˜∗ is the lead time i.e., the difference in time between detection of epidemic peak among sensors and time when it peaks in entire population. Report the lead-time for all 3 strategies.
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Q 2.4 (5 points) Compare the lead-time of various strategies and explain difference in leadtime of various strategies.
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Q 2.5 (8 points) Now repeat Q 2.2 for k = 50 and k = 500. For each strategy submit a I˜(t) vs t plot comparing different values of k. How does the lead time change with value of k for each strategy?
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3. (30 points) Flu Surveillance using Google Symptoms Data
Typically, epidemiologists focus on specific weeks of a year where the flu is prevalent called a flu season. A flu season starts at week 40 of a given year and ends at week 20 of the next year. We enumerate the weeks of a Flu season as Epiweeks.
Q 3.1 (2 points) We will start by considering the state of Iowa. Extract the % Unweighted
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Q 3.2 (10 points) CDC lists various symptoms for Flu . We will consider the following symptoms: Fever, Low-grade fever, Cough, Sore throat, Headache, Fatigue, Muscle weakness.
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Q 3.3 (3 points) We will use Pearson Correlation Coefficient (PCC) to measure how correlated each of symptom’s trend is to ILI. Evaluate PCC for each of the symptoms with
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Q 3.5 (5 points) There maybe some lead-time between reported ILI and the symptoms trend. Here, we define the lead-time ts of a symptom s to be the value of t′ that maximizes the PCC between ILI time-series from week t′ + 1 to T and time-series of symptom s from 1 to T − t′ where T = 52 week. Intuitively, we measure the delay between symptoms signal and ILI signal.
For the most correlated symptoms you calculated in Q3.3 for each of 7 states, find the lead-time for the respective symptom. Submit the code.
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