2021-033

Consider networks A and B shown in the figures below. In both networks, the three most connected hubs are shown with red nodes.

The networks possess the following characteristics:

• Number of nodes: 32
• Average degree: 2.375
• The average degrees of the neighborhood of each red node in both graphs are:
  • Network A: { 1.5 , 1.4 , 1.5 }
  • Network B: { 3.5 , 2.8 , 3.5 }
• The degree correlation coefficient (Pearson correlation coefficient) for both networks is a negative value.

Which alternative is compatible with the information given above?

1. Network A is assortative and Network B is neutral
2. Network A is disassortative and its degree correlation coefficient is larger than network B's
3. Network B has a larger degree correlation coefficient than network A.
4. Both networks are assortative.
5. None of the above

Original idea by:‌ Soroor Salavati

2021-032

Carlos decided to construct two networks of relationships, one with the managers in his company (figure a) and another with the families in his neighborhood (figure b). He obtained the following results:

Which of the alternatives below represent a credible assessment of the network type and correlation coefficient r for networks (a) and (b), respectively, with respect to degree correlations?

1. Assortative with r = -0.276 and Disassortative with r = 0.247
2. Assortative with r = 0.276 and Disassortative with r = -0.247
3. Disassortative with r = 0 and Assortative with r = -0.247
4. Disassortative with r = -0.276 and Assortative with r = 0.247
5. None of the above.

Original idea by: Diego Moreira.

2021-031

Choose the incorrect alternative below:

1. The Initial Attractiveness model addresses an issue with the Barabási-Albert model, namely, that an isolated node cannot acquire links, by adding a constant to the preferential attachment function.  Instead of just \( k \), we now have \( k + A \).
2. The Internal Links model addresses an issue with the Barabási-Albert model, namely, to add new links among preexisting nodes, and not only among the new node and former nodes.
3. The Accelerated Growth model adds new nodes and links.  The number of links to add is not a constant, but grows over time.
4. In real networks, nodes and links can disappear. The Node Deletion model only removes nodes, and does not add new nodes. Nodes are removed based on a rate \( r \). Depending on \( r \) being positive or negative, more or less nodes are removed.
5. E. None of the above

Original idea by: Mauricio Schiezaro

2021-030

According to the Barabási-Albert model, the ability of a node to acquire more links grows with the node's age. But in reality newer nodes can acquire a large number of links and become hubs, despite their age. To try and explain this behavior, the Bianconi-Barabási model was created. Choose the incorrect alternative regarding this model.

1. A new property of a node is introduced, namely, its fitness, that represents its ability to turn a random link into a permanent link.
2. The new model is composed of two main steps. In the first step, a new node is added in the network with a random fitness.
3. The new model is composed of two main steps. In the second step, links are created using preferential attachment.
4. Preferential attachment in this model is driven by the product of the node's fitness by its degree.
5. None of the above. 

 

Original idea by: Adolfo Aires Schneider

2021-029

In a network that follows the Barabási-Albert model, at each time step t one new node is added to the network and connects to m other nodes that were added to the network before. Suppose that a new node n is added at t = 4 in a network which m = 3. What is the expected degree of n when t = 10000?

1. 50
2. 120
3. 150
4. 300
5. None of the above

Original idea by: José Nascimento

2021-028

 When a Bianconi-Barabási model reduces to a Barabási-Albert model?

1. When the degree distribution is a normal distribution
2. When the fitness distribution is uniformly distributed in the interval [0, 1]
3. When the model generates a power law degree distribution
   
4. When all fitnesses are equal
5. None of the above

Original idea by: Thales Nazatto

2021-027

Consider the figure below. It shows the snapshot of a network at time "t". A new node "U"‌ will enter the network at time "t+1", linking to one of the existing nodes in the network under the Barabasi-Albert (BA)‌ model. Determine which node in the network is most likely to get the link with the new node "U", and also calculate the corresponding probability.

1. Node B with probability (1/3)
2. Node B with probability (1/4)
3. Node C with probability (1/3)
4. Node E with probability (1/4)
5. None of the above

Original idea by:‌ Soroor Salavati

2021-026

Considering scale-free networks whose degree distribution follows a power law of the form $$ p_k \sim k^{-\gamma}, $$which of the alternatives below is correct:

Notation:

\( k_{max} = \) maximum degree in the network

\( N = \)  number of nodes in the network

1. For \( \gamma = 2 \): The second moment of the distribution is finite, thus in many ways the network behaves as a random network.
2. For \( \gamma = 3 \): \( k_{max} \) increases faster than \( N \)
3. For \( \gamma < 2 \): The size of the biggest hub is \( O(N) \)
4. For \( 2 < \gamma < 3 \): The average path length increases slower than logarithmically
5. None of the above.

Original idea by: Diego Moreira.