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

2021-059

Considering a scale-free network with \( N = 10^3 \) and \( \gamma = 2.1 \), which formula better describes the average distance between two nodes?

1. constant
2. \( \ln N \)
3. \( \ln N / \ln \ln N \)
4. \( \ln \ln N \)
5. None of the above.

Original idea by: Leandro Stival

2021-058

According to the characteristics of the Barabási–Albert (BA) model, analyze the following statements and mark the correct alternative.

1. The Barabási-Albert model captures the fact that, in real networks, nodes arrive one after the other, offering a dynamic description of the evolution of a network. This generates a link competition in which older nodes prevails over younger ones, eventually turning into hubs.
2. The degree distribution is stationary (that is, time-invariant), explaining why networks with a different history, size, and age develop a similar degree distribution.
3. The Barabási-Albert model is a recipe for randomly generating scale-free networks: the network grows by adding new nodes over time, and these new nodes connect to existing nodes in the network with a probability proportional to their degree.
4. In the Barabási-Albert model, each new node has more nodes to link than previous nodes. Consequently, over time, existing nodes compete for links with a growing pool of other nodes.
5. The random network model plays a particularly important role in network science but differs from real networks in two important characteristics: growth and preferential attachment.

1. Only statements I, II and III are true.
2. Statement IV is true.
3. Statement V is true.
4. All statements are true.
5. None of the above.

Original idea by: Andressa Cristina dos Santos