Sheet 2 MATH 168 Winter 2020

Due: upload to Gradescope by Tuesday 21 January 2020 at 11am.

Reading:

• Chapter 7.1 of Newman’s book

• Axioms for , P. Boldi, S. Vigna (2013) (link on the course website)

Submit your asnwers to the following problems:

Problem 1 Read the paper Axioms for Centrality, P. Boldi, S. Vigna (2013) (link on the course website), and write a couple of sentences for each of the following points:

• What are the take-aways of the paper?

• What are some interesting follow-up questions?

Problem 2 Consider the following networks: the Internet, the World Wide Web, a friend- ship network, a neural network and a power grid. Explain briefly in your own words what type of information is measured by centrality, betweenness centrality, closeness cen- trality and Katz centrality in these networks. Can you give some examples for which the centrality measures would give different rankings for the same network? (A couple of ex- amples are enough, you don’t need to give examples for each network.)

Problem 3 Compute the degree centrality, betweenness centrality and closeness centrality for the nodes in the following network:

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Indicate which nodes have the highest centrality value, and discuss the results obtained.

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Note: For this problem it is best if you use some software, either to compute shortest paths, or to directly compute the centrality values. Widely used software packages for networks are NetworkX (https://networkx.github.io) and the Brain Connectivity Toolbox (https: //sites.google.com/site/bctnet/), though there are numerous other software packages. In your solutions please indicate how you computed the values (which software package, which implementation from that package, etc).

Problem 4 (This is problem 7.5 in Newman’s book) Suppose that a directed network takes the form of a tree with all edges pointing inward towards a central node:

What is the PageRank centrality of the central node in terms of the single parameter α appearing in the definition of PageRank and the distances di from each node i to the central node?

Problem 5 (This is Problem 7.1 in Newman’s book) Consider a connected k-regular undi- rected network (i.e., a network in which every node hasScanned degree by CamScannerk and there is only one ).

a) Show that the uniform vector 1 = (1, 1, 1,... ) is an eigenvector of the with eigenvalue k. In a connected network there is only one eigenvector with all elements positive and hence the eigenvector 1 gives, by definition, the of the k-regular network, and the are the same for every .

b) Find the Katz centralities of all nodes in the network as a function of k.

c) You should find that, like the eigenvector centralities, the Katz centralities of all nodes are the same. Name a centrality measure that could give different centrality values for different nodes in a regular network.

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Problem 6 (This is Problem 7.3 in Newman’s book) We saw in the lecture that the Katz centrality in vector form satisfies the equation x = αAx + 1.

a) Show that the Katz centrality can also be written in series form as x = 1 + αA1 + α2A21 + ...

b) Hence, argue that in the limit where α is small but non-zero, the Katz centrality is essentially equivalent to degree centrality.

c) Conversely, in the limit α → 1/κ1, where κ1 is the largest (most positive) eigenvalue of the adjacency matrix, argue that x becomes proportional to the leading eigenvector, which is simply the eigenvector centrality.

Thus, the Katz centrality can be thought of as a one-parameter family of centralities, parametrized by α ∈ [0, 1/κ1], which includes the degree centrality and the eigenvector centrality at the two limits of the range and interpolates between them everywhere in between.

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