2021-041

Considering the following network and its partition in green and pink nodes:

The modularity \( M \) of a partition can be computed using:

$$ M = \sum_{c=1}^{n_c} \left[ \frac{L_c}{L} - \left( \frac{k_c}{2L} \right)^2 \right], $$

where \( n_c \) is the number of sets in the partition and, for each set \( S \), we denote by \( L_c \) the number of links connecting two nodes in the set \( S \), and by \( k_c \) the sum of all degrees of nodes in \( S \). What is the value of \( M \) for the partition depicted above (round to 2 decimal places)?

1. \( M = 0.45 \)
2. \( M = 0.12 \)
3. \( M = 0 \)
4. \( M = 0.5 \)
5. None of the above

Original idea by: Tiago Almeida

2021-040

In the SIR disease spreading model, three different states are defined, namely, Susceptible (Healthy), Infected (Sick) and Removed (Immune/Dead). The fractions of individual per state, over time, present the following evolution curves and differential equations:

Choose the alternative that correctly associates each curve with its state and rate equation.

1. 3 - c) (Susceptible),  2 - b) (Infected),  1 - a) (Removed)
2. 1 - a) (Susceptible),  2 - b) (Infected),  3 - c) (Removed)
3. 3 - a) (Susceptible),  2 - b) (Infected),  1 - c) (Removed)
4. 1 - c) (Susceptible),  2 - b) (Infected),  3 - a) (Removed)
5. None of the above.

Original idea by: Diego Moreira.

2021-039

Consider the two partitions below on the same network:

What are the modularities \( M_a \) and \( M_b \) of the partitions in pictures \( a \) and \( b \), respectively? Round the result to two decimal places, if necessary.

1. \( M_a = 0.20 \) and \( M_b = 0.30 \)
2. \( M_a = 0.20 \) and \( M_b = 0.34 \)
3. \( M_a = 0.24 \) and \( M_b = 0.30 \)
4. \( M_a = 0.24 \) and \( M_b = 0.34 \)
5. None of the above.

Original idea by: Bruno Almêda

2021-038

The Hierarchical clustering algorithm helps to uncover the community structure of large real networks. Look at the Dendrogram below and choose the correct community organization, based on the red line cut:

1. 11 communities found: A | B | C | D | E | F | G | H | I | J | K
2. 8 communities found: A, B | C | D | E, F | G | H | I, J | K
3. 5 communities found: A, B, C | D | E, F, G | H | I, J, K
4. 3 communities found: A, B, C | D, E, F, G | H, I, J, K
5. None of the above

 

Original idea by: Mauricio Schiezaro

2021-037

The values of \( \langle k \rangle \) for a network to have a critical threshold \( f_c > 0 \) are:

1. \( \langle k \rangle > 2 \) for any random network and \( \langle k \rangle > 1 \) for any other
2. \( \langle k \rangle > 1 \) for any network
3. \( \langle k \rangle > 2 \) for any network
4. \( \langle k \rangle > 1 \) for any random network and \( \langle k \rangle > 2 \) for any other
5. None of the above

Original idea by Thales Nazatto

2021-036

Consider a Random Network such that \( N = 1000 \) and \( p = 0.004 \). We need to break apart its giant component, and the only way to do that is by randomly removing nodes. Under these conditions, how many nodes should we remove at minimum to have a good chance of achieving our goal?

1. 750
2. 720
3. 840
4. 810
5. None of the above.

Original idea by: José Nascimento

2021-035

Consider an Erdos–Rényi Network with 2500 links and 1000 nodes. What is its critical threshold, that is, the fraction of nodes that need to be removed in order to achieve the breakdown of the giant component?

1. 50%
2. 60%
3. 80%
4. 99%
5. None of the above

Original idea by: Vitor Estima

2021-034

Given a scale-free network with \( \gamma = 3.1 \) and \( k_{min} = 2 \), find the percentage of nodes that need to be removed in order to fragment the network.

Tip: calculate the critical threshold \( f_c \) to find this percentage.

The percentage found is (rounded to nearest multiple of 1%):

1. 100%
2. 70%
3. 90%
4. 95%
5. None of the above

Original idea by Angelica Oliveira