2021-066

Evaluate each of the following statements:

1 - Under random node removal, a random network will break into smaller components in a gradual, linear way.

2 - Scale-free networks, regardless of the degree distribution exponent, are robust against random failures, since their biggest component will only vanish when the fraction of removed nodes gets close to 1.

3 - The breakdown of a scale-free network is the same for random node removal and hub removal.

1. 1-False, 2-False, 3-False
2. 1-True, 2-True, 3-False
3. 1-False, 2-True, 3-False
4. 1-True, 2-True, 3-True
5. None of the above

Original idea by: Thales Rogério

2021-065

Consider a scale-free telecommunication network with \( N = 10^4 \) nodes, with degrees ranging from 2 to 400.  This network is under a failure process, closely following the branching model. Which of the alternatives below display power-law exponents compatible with these data?

1. Degree distribution exponent \( \gamma = 2.43 \) and avalanche exponent \( \alpha = 1.7 \)
2. Degree distribution exponent \( \gamma = 2.738 \) and avalanche exponent \( \alpha = 1.575 \)
3. Degree distribution exponent \( \gamma = 3 \) and avalanche exponent \( \alpha = 1.5 \)
4. Degree distribution exponent \( \gamma = 3.1 \) and avalanche exponent \( \alpha = 1.5 \)
5. None of the above

Original idea by: Matheus Cerqueira

2021-064

According to the Molloy-Reed criterion, what is the threshold for \( \kappa = \langle k^2 \rangle / \langle k \rangle \) regarding the existence of a giant component in a network with a given degree distribution?

1. 0
2. 1
3. 2
4. 3
5. None of the above

Original idea by: Leandro Stival

2021-063

It is incorrect to say that:

1. The degree correlation function helps us capture the presence or absence of correlations in real networks.
2. In assortative networks, hubs tends to connect to other hubs, hence the higher is the degree of a node, the higher is the average degree of its nearest neighbors.
3. In a perfectly assortative network, each node links only to nodes with the same degree.
4. In disassortative network, hubs prefer to link to high-degree nodes.
5. None of the above

Original idea by: Adson N Alves

2021-062

Given two networks G1 and G2 generated by the Barabási-Albert model with \( N = 1000 \) and \( N = 100 \) nodes, respectively, find out which network likely has the smallest diameter. Also, give their expected diameters, rounded to two decimal places.

1. G1 has the smallest diameter, 6.29, while G2 has diameter 6.64
2. G2 has the smallest diameter, 3.02, while G1 has diameter 3.57
3. G1 has the smallest diameter, 3.22, while G2 has diameter 5.44
4. G2 has the smallest diameter, 5.31, while G1 has diameter 5.44
5. None of the above

Original idea by: Victor Antonio Menuzzo

2021-061

Given a network with 200 nodes and minimum degree 1, what is the expected maximum degree of a node in the network? For this problem, consider \( \gamma = 3 \) and round your answer to the nearest integer.

1. 14
2. 17
3. 10
4. 22
5. None of the above

Original idea by: Thales Rogério

2021-060

Scale-free networks are networks whose degree distribution follows a power law, that is, the probability \( p_k \) of a node having degree \( k \) is proportional to \( k^{-\gamma} \) for a certain constant \( \gamma \).

About their degree distributions and properties, which of the following statements is not correct:

1. The scale-free property comes from a finite first moment and an infinite second moment, and for \( \gamma \lt 3 \) this remains unchanged;
2. Unlike random networks, scale-free networks have a 'tail' in their degree distribution, which supports the presence of hubs;
3. Most real scale-free networks are found in the regime \( 2 \lt \gamma \lt 3 \);
4. Networks with \( \gamma \lt 2 \) are not graphical. In this regime, the largest hub will tend to have degree greater than the number of nodes \( N \), when \( N \) grows;
5. None of the above.

Original idea by: Matheus Cerqueira