Complex Networks Coupled centrality measures

Complex Networks Homework 1 Coupled centrality measuresMatteo Mazzamurro

1 Instructions

• Deadline: Monday 7th April, 9am.
• Marks: This homework is worth 50% of your coursework mark, that is 10of your final mark. Please note that you need to pass both you courseworkand your final exam to pass the course.Submission: Please submit your solutions as a PDF on QM+. Handwritten or typed solutions (e.g., LATEX) are both acceptable.

• Please read carefully the introduction as it contains essential informationto solve the problems.

2 Introduction Consider a citation network: a directed network where nodes represent journal

articles. A link from article i to article j indicates that i cites j. Articles thatcontain valuable information are helpful, but so are good review articles, which

allow to effectively identify valuable 代 写Complex Networks  Coupled centrality measuresarticles. Hence, in a citation network, twotypes of articles can be considered important:

• Authorities: articles that contain valuable information, and are thus cited

in helpful review articles.

• Hubs: articles that cite many authorities, helping scientists to find them.

We define a pair of coupled centrality measures to identify important articles:

• The authority centrality xi of an article i is proportional to the sum ofthe hub centralities of the articles citing i:

• The hub centrality yi of an article i is proportional to the sum of th13 Questions

1. Let G be a citation network with n articles. Let x = (x1, x2, . . . , xn) Tbe the (column) vector of authority centralities of articles in G, and lety = (y1, y2, . . . , yn) T be the (column) vector of hub centralities of articlesin G. By using the definition of adjacency matrix A of G, rewrite equations(1) and (2) in matrix form. [6 Marks]

for a certain value of λ. Specify the value of λ. [6 Marks]This allows to conclude that the authorities and hub centralities are given,respectively, by the eigenvectors of AAT and AT A with the same eigenvalue.

Justify why the matrices AAT and AT A have the same set of eigenvalues[8 Marks]We want all entries in x and y to be positive. This is possible by the PerronFrobenius theorem, which ensures that this is the case if we take λ to be the

argest (i.e., most positive) eigenvalue.Let us look at a specific example. Consider the following network G, with 5

4. Write down the adjacency matrix A of G. [3 Marks]Compute the authority and hub centrality for each node/article. (Hint:

Compute either the hub centrality or the authority centrality by finding the

characteristic polynomial of an appropriate matrix, and then the eigenvec

tor associated with the largest eigenvalue. Deduce the other centrality using

your answers for question 1. Note that the centralities are defined only up to a constant). [12 Marks]

6. Interpret the centrality values you found in Question 5. Which nodes aremore important according to each measure? [6 Marks]

7. Which nodes are likely to be review articles? Which review article wasprobably written first? Why? [3 Marks]

8. Explain why the eigenvector centrality would not be meaningful for thisnetwork. [6 Marks]