Networked Life-Homework 4 Information Cascades and Computing Centrality and Betweenness Solved

Exercise 1. Information cascades   
Consider the case of information cascades, when private signals are correct with probability

p. Calculate the probability that no cascade forms after 2 steps and then a bad cascade forms after the next 2 steps. Here you can assume that the correct number of 𝜃 is 1.

 

 

 

Exercise 2.  

 

Consider the case of picking the good restaurant as in the slides, but there is web site that makes a suggestion that is correct with probability 𝑞 0.5,but 𝑞 < 𝑝, where 𝑝 is the probability of the private signals of the agents to be correct (as in the slides). Calculate the probability of good and of bad cascades after two steps. Can we have the case after a number of arrivals that no cascade forms, if so, what is the probability of this case? Explain. Is the case of 𝑞 𝑝 interesting? (Assume that the first restaurant is the good one, i.e., 𝜃 = 1.)

 

 

 

Exercise 3. A citation network and matrix multiplication (Exercise 7.3 in book)
 

  

 

Consider a set of eight papers with their citation relationships represented by the graph in Figure 7.16. Each paper is a node, and a directed edge from node 𝑖 to node 𝑗 means paper 𝑖 cites paper 𝑗.

 

a)      Write down the adjacent matrix 𝐀 (which we will talk much more about in the next chapter), where the (𝑖, 𝑗) entry is 1 if node 𝑖 points to node 𝑗, and 0 otherwise.

 

b)      Compute the matrix 𝐂 defined as:

𝐂 = 𝐀𝐓𝐀,

 

and compare the values 𝐶78 and 𝐶75. In general, what is the physical interpretation of the entries 𝐶𝑖𝑗?

 

c)      Now compute  

 

𝐀𝟐 = 𝐀𝐀, 

𝐀𝟑 = 𝐀𝟐𝐀. 

Is there anything special about 𝐀𝟑? In general, what do the entries in 𝐀m (where 𝑚 = 1,2,…) represent?

 

 

 

 

 

Exercise 4. Computing centrality and betweenness (Exercise 8.1 in book)   
 

 

  

a)      Compute the degree, closeness, and eigenvector centrality of each node in the graph in Figure 8.19.

b)      Compute the node betweenness centrality of nodes 2 and 3.

c)      Compute the link betweenness centrality of the links (3,4) and (2,5).

 

 

 

 

 

 

 

 

Exercise 5. Contagion (Exercise 8.2 in book)  
 

 

 

 

Consider the contagion model in the graph in Figure 8.20 with 𝑝 = 0.3.

a)      Run the contagion model with node 1 initialized at state-1 and the other nodes initialized at state-0.

b)      Run the contagion model with node 3 initialized at state-1 and the other nodes initialized at state-0.

c)      Contrast the results from (a) and (b) and explain in terms of the cluster densities of the sets of initially state-0 nodes.

 

 

 

 

Exercise 6. SIRS infection model (Exercise 8.3 in book)   
 

We consider an extension to the SIR model that allows nodes in the state 𝑅 to go to state 𝑆. This model, known as the SIRS model, accounts for the possibility that a person loses the acquired immunity over time.

 

 

 

Consider the state diagram in Figure 8.21. We can write out the set of differential equations as:

 

𝑑𝑆(𝑡)

= −𝛽𝑆(𝑡)𝐼(𝑡) + 𝑣𝑅(𝑡) 

𝑑𝑡

𝑑𝐼(𝑡)

= 𝛽𝑆(𝑡)𝐼(𝑡) − 𝛾𝑅(𝑡) 

𝑑𝑡

 𝑑𝑅(𝑡)

= 𝛾𝐼(𝑡) − 𝑣𝑅(𝑡) 

𝑑𝑡

 

Modify the Matlab code “exercise4.m” for the numerical solution of the SIR model. Solve for 𝑡 = 1,2,… ,200 (set the tspan vector in code accordingly) with the following parameters and initial conditions: 𝛽 = 1, 𝛾 = 1/3, 𝑣 = 1/50, 𝐼(0) = 0.1, 𝑆(0) = 0.9 and 𝑅(0) = 0. Describe and explain your observations. (Please paste your MATLAB code into the solution or insert as a picture, no need to submit a separate .m file)