Description

Reminders:
1. Out of 100 points. 4 Questions. Contains 6 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. (25 points) Metapopulation Model
We will model the SIR metapopulation model briefly introduced in Lecture 3.
Let there be M regions with total populations . We let Si(t),Ii(t),Ri(t) be the total population in respective component for region i at time t.
Assume that σij is the population flow from region i to j. Also, we assume the parameters β,γ are same across all regions.
Q 1.1 (6 points) Similar to Lecture 3 Slide 4, write down the equations to derive effective population sizes Sieff(t),Iieff(t),Rieff(t) at time t from and flow parameters σij.
Solution:
Since SIR parameters β,γ are same across all regions we can write the change in S,I,R compartment sizes in 1 time-step as:

Q 1.2 (6 points) Instead of 1 time-step, derive equations for change in ∆t time; i.e., derive the formulas for Si(t + ∆t) − Si(t),Ii(t + ∆t) − Ii(t),Ri(t + ∆t) − Ri(t). Hint: This is similar in nature to Q1.2 in HW 1.
Solution:
Q 1.3 (3 points) Set ∆t → 0 and derive the ODE equations for the metapopulation model.
Solution:
Q 1.4 (10 points) Implement the ODE of metapopulation model. The code should take the parameters β,γ, the flow parameters {σi,j}i,j∈[1,M]×[1,M], total population and the initial population of each compartment for all regions and max time and output the population sizes of till max time.
We have provided a starter code in python script metapop.py that implements the model. You have to fill in the missing parts of the method def ode(self, times, init, parms): that involves calculating the effective population.
Once you have completed the method, run the script to generate the SIR plots for each population. Submit the completed code and the plots.
Note: You don’t need to change other functions in the script, only fill in the code in the space indicated.
Solution:
2. (26 points) Viral propagation You are managing a network of devices inside your company connected to each other via a local intranet connection. Occasionally, some of the computers in the network get infected with a virus which can spread across the network as users share information with each other. You job is to simulate the spread of the virus under different assumptions on the diffusion mechanism.
Network dataset: Your dataset contains multiple networks G = {G1,G2,…,G9} between routers (this dataset is sourced from actual internet router dataset . Each of the networks are snapshots of the connections made across routers on some day collected over 3 months. The network files are named as network1.txt, network2.txt,…, network9.txt.
Aggregating all networks: We aggregate all the 9 snapshots of the networks into a single network G(ρ). The aggregation process is simple: For each pair of nodes (u,v) if at least ρ networks in G have an edge between them, we add an edge (u,v) to G(ρ). Here ρ is a parameter of our model.
We also assume an SIS network model with usual parameters β,γ on the graph G(ρ).
Q 2.1 (7 points) Decreasing ρ will increase the number of edges in the graph and make it more denser and connected. In class we saw that λ, the highest eigenvalue of the adjacency matrix is a good proxy for the connectedness of the graph. Let us observe this empirically for this dataset.
For each value of ρ in {1,2,3,…,9}, write code to construct G(ρ). Find the largest eigenvalues λ of the adjacency matrix for each graph G(ρ). Submit the plot between ρ and γ. How does λ change with increase in ρ?
Hint: Since the graphs have over 10,000 nodes, please consider using sparse representations of adjacency matrix and efficient built in methods to compute eigenvalues on the sparse graph. For example scipy.sparse.linalg module has efficient implementations to compute eigenvalues that enable you to perform the computations within fractions of a second.
Solution:
Q 2.2 (7 points) Set ρ = 1. Use the EoN package to set up an SIS model on G(ρ). Specifically use the EoN.fast SIS function . Set γ = 0.08,β = 0.001 and max time T = 10. Set node 1 as the only infected node at t = 0.
Now plot a curve with I(t) on y-axis vs t on x-axis for 0 ≤ t ≤ T after simulating the model 50 times (similar to HW1 Q1.3). Now set β = 0.0001 and plot the I(t) vs t curve. Do you notice any difference? Explain.
Solution:
Q 2.3 (7 points) Now set β = 0.01,γ = 0.16 and ρ = 1. We will try to reduce the effective strength of the graph by another method of iteratively removing nodes using the following rule:
Find the node with the largest degree among all nodes (in case of a tie, choose the node with the highest index). Then remove the node.
Using this rule find the number of node removals required to reduce the effective strength of the graph to less than 1. Also produce a plot of number of nodes removed in x-axis and effective strength on y-axis till effective strength reduces to less than 1.
Solution:
Q 2.4 (5 points) We will denote as K the number of nodes you had to remove in Q2.3 before effective strength decreased below 1. Consider the two graphs Ga and Gb where Ga is the initial graph you constructed by removing all K nodes and Gb the graph you constructed by removing K − 1 nodes (hence, effective strength is > 1). Simulate the SIS model on Ga and Gb with the same parameters by randomly choosing a single node to be infected at t = 0. Perform 100 simulations for both Ga and Gb for T = 10 steps.
Plot I(t) vs t for both the graphs.
Solution:
3. (30 points) Finding Culprits
Let’s use the k-regular Cayley tree we used in the previous HW.
Q 3.1 (10 points) Let’s consider the k = 3 tree with 22 nodes (see Figure 1). You can use edge list from cayley.txt to construct the graph.
Implement a simple SI model using EoN.fast SI or equivalent package in your language of choice. Set β = 0.2 in the SI model and run it on this graph. Let only the central node 0 be infected at t = 0. Run the SI model for 10 steps and record the set of nodes I which are infected at the end of 10 steps.

Figure 1: Cayley Tree
Repeat this for 100 times. Record these 100 sets of nodes as X = {I1,I2,…,I100} in a file.
The file should contain 100 lines and each line i should contain the list of nodes infected Ii which are separated by a space.
Solution:
Find the accuracy of this algorithm i.e., find the fraction of time the algorithm infers node 0 as the center of subgraph Ii over X = {I1,I2,…,I100}.
Solution:
Q 3.3 (6 points) Repeat the same steps as Q3.2 and Q3.3 except make node 1 (a point at depth 1) as the starting point. Compute the accuracy of the algorithm.
Solution:
Q 3.4 (4 points) Does the accuracy of the estimator change (comparing the answer you get in Q3.2 and Q3.3)? Show your result and give the answer. If so, what does it tell you about the estimator on this type of graph (strengths and weaknesses; this is slightly an open-ended question). Do you think the rumor centrality metric we studied in class will probably perform better?
Solution:
4. (19 points) Steiner trees for Missing Infections
If G = (V,E) is our undirected social network with edge weights, and a set of terminals I ⊆ V , then a Steiner tree is a tree in G that spans I (i.e., contains at least all nodes in I; it might contain extra nodes from V not in I too, but at least it has to span all of I).
The minimum weighted Steiner tree, is the Steiner tree for a given I that has the least sum of the weights of all the edges included in it. There are several algorithms for finding such a Steiner tree given the set of terminals I and graph G. See this link for more details on the Steiner tree problem https://en.wikipedia.org/wiki/Steiner_tree_problem.
For the following question, assume that all the edges have equal weights.
Q 4.1 (8 points) Let’s try to compute a couple of minimum Steiner trees by hand in unweighted graphs. Feel free to use whatever method you want (except of course just using a software package). These can be done with intuition as well. Just draw and show us the final answer.
Q 4.1.1 (4 points) Terminal set = {7,9,3,6} for graph below.
Solution:

Q 4.1.2 (4 points) Terminal set = {12,14,9,10} for graph below.
Solution:

Q 4.2 (9 points) In the missing infections problem you are given a set of observed infections (the known set) K. For using Steiner trees to compute missing infections, the idea is very simple. Set K as the terminal nodes set I in the minimum weighted Steiner tree problem over the graph G. The resulting minimum weighted Steiner tree is a good rough approximation of how the disease might have spread. In particular the extra nodes you find in your min. Steiner tree are your missing nodes.
First generage the graph G as using the following function :
G = networkx.random graphs.extended barabasi albert graph(50, 1, 0.2, 0.1, seed=10).
Obtain an approximate minimum Steiner tree using this existing Networkx function .
Submit both a visualization and the adjacency list of the minimum Steiner tree for each of the following initial known infected nodes.
1. K = {10,14,3}
2. K = {10,14,3,4,2}
3. K = {10,14,3,4,2,31,49}
4. K = {10,14,3,4,2,31,49,21,25,36,43}
Plot the visualization in the pdf and submit the adjacency file as a .npy 2-D numpy array file.
Solution:
Q 4.3 (2 points) Briefly in 1-2 lines, what maybe the pros and cons in using a min. Steiner tree to identify missing infections? Suggest improvements for the same. This is an open-ended question so feel free to give intuitive answers.
Solution: