Chapter 4 - Scale-Free Property

Consider the Scale-Free Network Model with minimum degree k_min​ = 1.

Define f(N,γ) as the function that computes the second moment of the degree distribution, ⟨k²⟩ , for a finite network with N nodes and degree exponent γ. Use the Continuum Formalism.

Compute the following ratios:

1. f(10⁴, 7/3) / f(10⁴, 5)

2. f(10⁸, 7/3) / f(10⁸, 5)

3. f(10¹², 7/3) / f(10¹², 5)

The values for (1), (2), and (3) are approximately, respectively:

A. 1, 10_000,  1_000_000_000
B. 1, 100_000_000, 1_000_000
C. 100, 10_000, 1_000_000
D. 100, 100_000_000, 1_000_000_000
E. None of the above

Note: underscores are used only for readability of large numbers.

Original idea by: Rodolfo Bitu

Chapter 3 - Random Networks

Consider networks following the Random Network Model with N=10,000 nodes and probability p of a pair of nodes to be connected.

Choose the WRONG alternative.
Hint: ln⁡(10,000) ≅ 10

A. When p = 1/20,000​, the network is in the subcritical regime, which means it does not have a giant component.
B. When p = 1/10,000​, the network is in the critical regime, which means it has a giant component.
C. When p = 1/4,000​, the network is in the connected regime, which means the giant component absorbs all nodes and components.
D. When p = 1/5,000​, the network is in the supercritical regime, which means the giant component contains a considerable fraction of the nodes.
E. None of the above.

Original idea by: Rodolfo Bitu

Chapter 2 - Graph Theory

 

G is a bipartite graph with vertex partition:

U = {1,2,3,4,5,6}

V = {7,8,9}

Consider the projections of G onto the sets U and V and the following graphs. 

Select the alternative that contains only true statements:

A. The projection of G onto U is graphs II and IV.

B. By removing one edge from graph IV, it becomes the projection of G onto V.

C. The projection of G onto U is graph III.

D. The projection of G onto V is graph II.

E. None of the above.

Original idea by: Rodolfo Bitu