# [Solved] BLG549E Graph Theory main-

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You need to have Python and Pip installed in your computer to install Jupyter Notebook.

Windows

On command prompt (cmd.exe with admin mode):

C:**path**> python −m pip install jupyter

After changing your current folder to the folder which you want to work on (see ’cd’ command):

C:**your_working_folder**> jupyter notebook

Then it will be launched on your default browser

## Ubuntu/Linux/Unix/Mac

On Terminal: \$ pip install notebook

Then launch with:

\$ jupyter notebook

# Introduction

You will carry out all the tasks below using Ipython Notebook. Simply add all your work to the provided template file HW1_template.ipynb using jupyter notebook. In this homework, you will code up several experiments in Oldham et al. paper (You can click anywhere on this sentence instead of a small, hard-to-click word “here” to find the paper).To this aim, first simulate 40 networks: 20 unconstrained weighted using Erdos-Renyi generative model  and 20 constrained unweighted using MaslovSneppen algorithm . For more details, check the attached supplementary material. For this task, you can use ready-made pieces of code. But all needs to be commented out. The number of nodes in each network category (e.g., ER) should equal to 200.

## Part A Simulate random weighted and unweighted networks

1. Briefly explain how Erdos-Renyi generative model works.
2. What are the key properties of Erdos-Renyi graphs?
3. Briefly explain how Maslov-Sneppen algorithm works.
4. What are the key properties of Maslov-Sneppen graphs?
5. Visualize two random graphs you simulated (ER and MS).

Figure 1: Example visualization of two graphs.

## Part B: Analyzing Erdos-Renyi and Maslov-Sneppen graphs using centrality measures

1. Code up the necessary steps to reproduce plots A and B in Figure 2 using the Centrality Measure Correlation(CMC)s of the 40 networks you simulated: 20 weighted Erdos-Renyi and 20 unweighted Maslov-Sneppen networks.
2. What conclusions can you derive from the plots?

Figure 2: Distributions of Centrality Measure Correlations (CMCs) for example unweighted and weighted networks. Distributions of CMCs for every pair of centrality measures for five example unweighted (panel A); and weighted networks (panel B). Networks have been ordered from highest (left) to lowest (right) median CMC. Both the figure and the caption is from 

1. Code up the necessary steps to reproduce Figure 3 (A to D) using the between-network CMCs of the 40 networks you simulated: 20 weighted ErdosRenyi and 20 unweighted Maslov-Sneppen networks.
2. What conclusions can you derive from the plots?

Figure 3: Mean and standard deviation of between-network CMCs. Panels A and B show the between-network CMC mean and standard deviation for unweighted measures, respectively. Panels C and D show the between-network CMCs mean and standard deviation for weighted measures, respectively. Both the figure and the caption is from 

1. Plot the CMC distributions using bar plots as in the Figure 4 for your 20 unweighted and 20 weighted networks. What do you notice?

Figure 4: Example bar plot of CMC distributions.

Part C: Association between mean within-network Centrality Measure Correlation(CMC) and network properties

1. Choose two different network properties (e.g., modularity and density) and regenerate similar plots to Figure 5 by plotting the mean within-network CMC against the selected network property. Erdos-Renyi networks should have the same color (e.g., transparent blue) and Maslov-Sneppen networks can be assigned a different color (e.g., transparent red). You can use existing codes to compute the network properties.
2. Interpret your plots. What conclusions can you derive?

Figure 5: Association between mean within-network CMC and network properties in unweighted networks. The association between the mean within-network CMC (the average of all CMCs within a single network) and each of the global topological properties. Networks are coloured by their natural category (blue = social, grey = technological, brown = biological, orange = informational, purple = transportation; green = economic). Both the figure and the caption is from 

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