2022-178

Consider a network described by the Bianconi–Barabási model, with degree dynamics following a power law.  Consider also a node A joining this network at time t=10 and another node B joining the network at time t=100.  If node B has twice the fitness of node A, at which time point t will the degree of node B start to be greater than or equal to the degree of node A?

1. t = 100
2. t = 1000
3. t = 10000
4. t = 100000
5. None of the above

Original idea by: Meer Muhammad Khan

2022-177

Aging can influence the topology of a network.  Denote by \( \Pi(k_i,t-t_i) \) the probability of a new node arriving at time \( t \) connecting to node \( i \) of degree \( k_i \), where \( t_i \) is the time node \( i \) was added to the network.  Aging can be modeled by choosing \( \Pi(k_i,t-t_i) \sim k_i(t-t_i)^{-\nu} \), where \( \nu \) is a tunable parameter.  In this case, what is not correct to state regarding the tunable parameter \( \nu \):

1. If \( \nu < 0 \), new nodes will link to older nodes.
2. If \( \nu \rightarrow -\infty \) each new node connects to the oldest node.
3. If \( \nu > 0 \) new nodes are encouraged to attach to younger nodes.
4. If \( \nu \rightarrow \infty \) each node will connect to its immediate predecessor.
5. None of the above.

Original idea by: Marcelo Silva

2022-176

You are analyzing a network that follows the Bianconi-Barabási Model. You know that a node A has degree 5 and fitness 0.2, and a node B has degree 16 and fitness 0.5. How much more likely is a new node to connect to B than to A?

1. 2.5 times more likely
2. 3.2 times more likely
3. 8 times more likely
4. 12 times more likely
5. None of the above

Original idea by: Luan Silveira

2022-175

The sequence of images below shows some steps for generating the network based on the Barabási-Albert model. Consider that starting with \( m_0 \) nodes, the empty circles mark the newly added node to the network in each timestep \( t \), and the new node's number of links is \( m = 2 \).

Analyze the following statements about the network:

1. Choosing which node will be connected uses the preferential attachment criteria.

2. After \( t \) timesteps, this model generates a network with \( t + m_0 \) nodes.

3. The degree distribution of this network follows the Poisson distribution.

4. This network has the scale-free property.

5. The diameter of this network with N nodes can be estimated by the equation \( D \sim \frac{\ln N}{\ln \ln N} \)

Select the correct alternative:

1. All statements are correct. 
2. Only statements 1, 2, and 3 are correct.  
3. Only statements 3, 4, and 5 are correct.
4. Only statements 1, 2, 4, and 5 are correct. 
5. None of the above.

Original idea by: Rubens de Castro Pereira

2022-174

In the Non-linear preferential attachment, the preferential attachment probability becomes \( \Pi(k) \sim k^{\alpha} \). For a starting network with 1 node, approximately how many nodes we have to add on the network for it to achieve \( k_{max} = 100 \), considering the cases where \( \alpha \) is 0.5, 1, and 1.5, respectively?

1. 22026 nodes; 10000 nodes; 100 nodes, respectively
2. 1024 nodes; 1000 nodes; 100 nodes respectively
3. 22026 nodes; 10000 nodes; 200 nodes, respectively
4. 1024 nodes; 1000 nodes; 200 nodes, respectively
5. None of the above.

Original idea by: Pedro Henrique Di Francia Rosso

2022-173

Consider a scale-free network that has evolved according to the BA model and choose the best possible option given below. Let P and Q be two nodes such that the rate at which node P acquires new links is twice the rate at which node Q acquires new links. If node P joined the network at time 100 units, find the time at which node Q joined the network.

1. 800
2. 600
3. 400
4. 200
5. None of the above

Original idea by: Muhammad Idrees

2022-172

By considering the Barabási–Albert model for a network, the degree of a node is k=50, and it joined the network at time t=10, what will be the degree of another node which joined the network at time t=100?

1. 10.53
2. 15.81
3. 21.09
4. 26.37
5. None of the above

Original idea by: Meer Muhammad Khan