### Word Count : 3000 words

For this entire coursework, There are three Tasks

**Task 1**

(18 Marks) Use the dataset given in the link

https://networkrepository.com/aves-sparrow-social.php to answer this exercise.

a) (10 marks) Show the plot of the modularity against the number of communities on your network for both the Girvan-Newman and the Greedy algorithms in the same image. Compare the best partition according to modularity for both methods. Relate your results to the nature of your network.

b) (8 marks) Use the Louvain algorithm to find a partition of this network in a number of communities equal to the last digit of your student number unless it is 0 or 1, in which case find 2 communities. Provide explicitly the obtained partition and compare it to the results obtained on item a both qualitatively and in terms of modularity. Relate this result to the nature of the network.

**Task 2**

(48 marks) For this exercise, choose an undirected network from any online repository with a number of nodes between 500 and 1000. Some things to pay attention:

• Provide the appropriate reference for the dataset.

• There are some repositories linked on the module’s Blackboard page, but you are free to choose networks from other repositories.

• When choosing your network, it is a good idea to first read the exercises and seek for a network that will be appropriate for the required analysis

a) (11 marks) By simulating random failures on this network for different values of failure rate, obtain an approximate value for the critical failure rate and show the appropriate plot of the results.

If you compare with the value obtained according to the Molloy-Reed Criterion, what can you conclude?

b) (12 marks) Simulate simultaneous targeted attacks for two different centralities, node betweenness and closeness, and plot the appropriate results of the simulation together in the same graph. Explain which one is more efficient in disrupting the network justifying your answer appropriately. Does it make sense taking into consideration the nature of your network?

c) (5 marks) Using the results from item a and b, analyse the robustness of the network against random failures and targeted attacks, comparing both and relating them to the nature of your network.

d) (9 marks) Consider the failure propagation model on the giant component of your network. Find the approximate value of 𝜑 that characterises the transition between small and whole network avalanches to test the robustness against cascading failures by simulation. Is the network robust?

What would that mean for your network taking into consideration its nature?

e) (11 marks) Consider the SIR model with parameters 𝛼 = 0.9 and 𝛽 = 0.2 for your network.

Assuming that a vaccinated node becomes removed, explicitly show that selective vaccination works better on your network than random vaccination. (You are not allowed to use the EoN library or any other pre-coded disease spread library).

**Task 3**

(14 Marks) Consider the traffic network below, where drivers start their journey on node 𝑆 and end it on node 𝑇. Times to cross each link are given in the drawing. The shortcuts AB and CD take a negligible amount of time to cross (you can consider it effectively as zero) no matter how much traffic they carry. A data scientist suggested to the City Council that it might be possible to decrease the time drivers spend on traffic during rush hours simply by closing one or both of the shortcuts during that time.

The City Council has commissioned a research to decide on what to do. Their engineers were able to establish that 𝑎𝑛 < 𝛽, where 𝑛 is the maximum amount of cars crossing the network at any time.

Some questions remain to consider.

a) (5 marks) Consider all possible scenarios – closing none, one or both shortcuts – and find the relation between the parameters 𝛼 and 𝛽 of the model that will support each case.

b) (9 marks) Consider the case where both shortcuts are open compared to that that both are blocked and the total number of drivers in the network is 𝑛 = 1000. Choose parameters values such that Braess’ Paradox is observed and consider an agent based simulation with the following rules:

(i) Each discrete time step corresponds to one day.

(ii) In the first day, all drivers choose one of the possible routes at random and take it. All drivers

cross the network at the same time.

(iii) At the end of each day, the time to cross each route is recorded and made public for all drivers.

(iv) In the following day, every driver takes the route with less time.

**Reference Style : BU Harvard**